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10.1: Plane - Mathematics


Area (A) is measured in square units, perimeter (P) is measured in units, and circumference (C) is measured in units.

Square

[P=4 s ]

[A=s^{2}]

Rectangle

[P=2 l+2 w ]

[A=l w]

Parallelogram

[P=2 a+2 b]

[A=b h]

Trapezoid

[P=a+b+c+d]

[A=frac{1}{2} h(a+b)]

Triangle

[P=a+b+c]

[A=frac{1}{2} b h]

Circle

[C=2 pi ]

[r=pi r^{2}]


Identify the sequence graphed below and the average rate of change from n = 0 to n = 2.

We must find the equation of this function first. This is a geometric sequence as each term is a constant ratio of the previous term.

5/10=1/2, And this sequence can be expressed as:

a(n)=10(1/2)^(n-1) So the term when n=0 would be 20

The average rate of change is just the change in y divided by the change in x, in this case:

and the average rate of change from n=0 to n=2 is:

We are given the points as:

Clearly after looking the point we see that these point follow a geometric sequence.

Since with the increase in x-value by 1 unit there is a decrease in the y-value by a factor of 1/2

Let the points be denoted by:

Hence, we have the sequence as:

Similarly we can check the other points as well

Now, the average rate of change from n=0 to n=2 is calculated as:


Contents

Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles. [1]

Decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, [2] sometimes displaying geometric patterns. [3] [4]

In 1619 Johannes Kepler made an early documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi he was possibly the first to explore and to explain the hexagonal structures of honeycomb and snowflakes. [5] [6] [7]

Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the plane features one of seventeen different groups of isometries. [8] [9] Fyodorov's work marked the unofficial beginning of the mathematical study of tessellations. Other prominent contributors include Aleksei Shubnikov and Nikolai Belov (1964), [10] and Heinrich Heesch and Otto Kienzle (1963). [11]

Etymology Edit

In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. [12] The word "tessella" means "small square" (from tessera, square, which in turn is from the Greek word τέσσερα for four). It corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay.

Tessellation in two dimensions, also called planar tiling, is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules. These rules can be varied. Common ones are that there must be no gaps between tiles, and that no corner of one tile can lie along the edge of another. [13] The tessellations created by bonded brickwork do not obey this rule. Among those that do, a regular tessellation has both identical [a] regular tiles and identical regular corners or vertices, having the same angle between adjacent edges for every tile. [14] There are only four shapes that can form such regular tessellations: the equilateral triangle, square, regular hexagon and the concave hexagon, or chevron. Any one of these four shapes can be duplicated infinitely to fill a plane with no gaps. [6]

Many other types of tessellation are possible under different constraints. For example, there are eight types of semi-regular tessellation, made with more than one kind of regular polygon but still having the same arrangement of polygons at every corner. [15] Irregular tessellations can also be made from other shapes such as pentagons, polyominoes and in fact almost any kind of geometric shape. The artist M. C. Escher is famous for making tessellations with irregular interlocking tiles, shaped like animals and other natural objects. [16] If suitable contrasting colours are chosen for the tiles of differing shape, striking patterns are formed, and these can be used to decorate physical surfaces such as church floors. [17]

More formally, a tessellation or tiling is a cover of the Euclidean plane by a countable number of closed sets, called tiles, such that the tiles intersect only on their boundaries. These tiles may be polygons or any other shapes. [b] Many tessellations are formed from a finite number of prototiles in which all tiles in the tessellation are congruent to the given prototiles. If a geometric shape can be used as a prototile to create a tessellation, the shape is said to tessellate or to tile the plane. The Conway criterion is a sufficient but not necessary set of rules for deciding if a given shape tiles the plane periodically without reflections: some tiles fail the criterion but still tile the plane. [19] No general rule has been found for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations. [18]

Mathematically, tessellations can be extended to spaces other than the Euclidean plane. [6] The Swiss geometer Ludwig Schläfli pioneered this by defining polyschemes, which mathematicians nowadays call polytopes. These are the analogues to polygons and polyhedra in spaces with more dimensions. He further defined the Schläfli symbol notation to make it easy to describe polytopes. For example, the Schläfli symbol for an equilateral triangle is <3>, while that for a square is <4>. [20] The Schläfli notation makes it possible to describe tilings compactly. For example, a tiling of regular hexagons has three six-sided polygons at each vertex, so its Schläfli symbol is <6,3>. [21]

Other methods also exist for describing polygonal tilings. When the tessellation is made of regular polygons, the most common notation is the vertex configuration, which is simply a list of the number of sides of the polygons around a vertex. The square tiling has a vertex configuration of 4.4.4.4, or 4 4 . The tiling of regular hexagons is noted 6.6.6, or 6 3 . [18]

Introduction to tessellations Edit

Mathematicians use some technical terms when discussing tilings. An edge is the intersection between two bordering tiles it is often a straight line. A vertex is the point of intersection of three or more bordering tiles. Using these terms, an isogonal or vertex-transitive tiling is a tiling where every vertex point is identical that is, the arrangement of polygons about each vertex is the same. [18] The fundamental region is a shape such as a rectangle that is repeated to form the tessellation. [22] For example, a regular tessellation of the plane with squares has a meeting of four squares at every vertex. [18]

The sides of the polygons are not necessarily identical to the edges of the tiles. An edge-to-edge tiling is any polygonal tessellation where adjacent tiles only share one full side, i.e., no tile shares a partial side or more than one side with any other tile. In an edge-to-edge tiling, the sides of the polygons and the edges of the tiles are the same. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks. [18]

A normal tiling is a tessellation for which every tile is topologically equivalent to a disk, the intersection of any two tiles is a connected set or the empty set, and all tiles are uniformly bounded. This means that a single circumscribing radius and a single inscribing radius can be used for all the tiles in the whole tiling the condition disallows tiles that are pathologically long or thin. [23]

A monohedral tiling is a tessellation in which all tiles are congruent it has only one prototile. A particularly interesting type of monohedral tessellation is the spiral monohedral tiling. The first spiral monohedral tiling was discovered by Heinz Voderberg in 1936 the Voderberg tiling has a unit tile that is a nonconvex enneagon. [1] The Hirschhorn tiling, published by Michael D. Hirschhorn and D. C. Hunt in 1985, is a pentagon tiling using irregular pentagons: regular pentagons cannot tile the Euclidean plane as the internal angle of a regular pentagon, 3 π / 5 , is not a divisor of 2 π . [24] [25] [26]

An isohedral tiling is a special variation of a monohedral tiling in which all tiles belong to the same transitivity class, that is, all tiles are transforms of the same prototile under the symmetry group of the tiling. [23] If a prototile admits a tiling, but no such tiling is isohedral, then the prototile is called anisohedral and forms anisohedral tilings.

A regular tessellation is a highly symmetric, edge-to-edge tiling made up of regular polygons, all of the same shape. There are only three regular tessellations: those made up of equilateral triangles, squares, or regular hexagons. All three of these tilings are isogonal and monohedral. [27]

A semi-regular (or Archimedean) tessellation uses more than one type of regular polygon in an isogonal arrangement. There are eight semi-regular tilings (or nine if the mirror-image pair of tilings counts as two). [28] These can be described by their vertex configuration for example, a semi-regular tiling using squares and regular octagons has the vertex configuration 4.8 2 (each vertex has one square and two octagons). [29] Many non-edge-to-edge tilings of the Euclidean plane are possible, including the family of Pythagorean tilings, tessellations that use two (parameterised) sizes of square, each square touching four squares of the other size. [30] An edge tessellation is one in which each tile can be reflected over an edge to take up the position of a neighbouring tile, such as in an array of equilateral or isosceles triangles. [31]

Wallpaper groups Edit

Tilings with translational symmetry in two independent directions can be categorized by wallpaper groups, of which 17 exist. [32] It has been claimed that all seventeen of these groups are represented in the Alhambra palace in Granada, Spain. Though this is disputed, [33] the variety and sophistication of the Alhambra tilings have surprised modern researchers. [34] Of the three regular tilings two are in the p6m wallpaper group and one is in p4m. Tilings in 2D with translational symmetry in just one direction can be categorized by the seven frieze groups describing the possible frieze patterns. [35] Orbifold notation can be used to describe wallpaper groups of the Euclidean plane. [36]

Aperiodic tilings Edit

Penrose tilings, which use two different quadrilateral prototiles, are the best known example of tiles that forcibly create non-periodic patterns. They belong to a general class of aperiodic tilings, which use tiles that cannot tessellate periodically. The recursive process of substitution tiling is a method of generating aperiodic tilings. One class that can be generated in this way is the rep-tiles these tilings have surprising self-replicating properties. [37] Pinwheel tilings are non-periodic, using a rep-tile construction the tiles appear in infinitely many orientations. [38] It might be thought that a non-periodic pattern would be entirely without symmetry, but this is not so. Aperiodic tilings, while lacking in translational symmetry, do have symmetries of other types, by infinite repetition of any bounded patch of the tiling and in certain finite groups of rotations or reflections of those patches. [39] A substitution rule, such as can be used to generate some Penrose patterns using assemblies of tiles called rhombs, illustrates scaling symmetry. [40] A Fibonacci word can be used to build an aperiodic tiling, and to study quasicrystals, which are structures with aperiodic order. [41]

Wang tiles are squares coloured on each edge, and placed so that abutting edges of adjacent tiles have the same colour hence they are sometimes called Wang dominoes. A suitable set of Wang dominoes can tile the plane, but only aperiodically. This is known because any Turing machine can be represented as a set of Wang dominoes that tile the plane if and only if the Turing machine does not halt. Since the halting problem is undecidable, the problem of deciding whether a Wang domino set can tile the plane is also undecidable. [42] [43] [44] [45] [46]

Truchet tiles are square tiles decorated with patterns so they do not have rotational symmetry in 1704, Sébastien Truchet used a square tile split into two triangles of contrasting colours. These can tile the plane either periodically or randomly. [47] [48]

Tessellations and colour Edit

Sometimes the colour of a tile is understood as part of the tiling at other times arbitrary colours may be applied later. When discussing a tiling that is displayed in colours, to avoid ambiguity one needs to specify whether the colours are part of the tiling or just part of its illustration. This affects whether tiles with the same shape but different colours are considered identical, which in turn affects questions of symmetry. The four colour theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at a curve of positive length. The colouring guaranteed by the four colour theorem does not generally respect the symmetries of the tessellation. To produce a colouring which does, it is necessary to treat the colours as part of the tessellation. Here, as many as seven colours may be needed, as in the picture at right. [49]

Tessellations with polygons Edit

Next to the various tilings by regular polygons, tilings by other polygons have also been studied.

Any triangle or quadrilateral (even non-convex) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary quadrilateral can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting. [50]

If only one shape of tile is allowed, tilings exists with convex N-gons for N equal to 3, 4, 5 and 6. For N = 5 , see Pentagonal tiling, for N = 6 , see Hexagonal tiling,for N = 7 , see Heptagonal tiling and for N = 8 , see octagonal tiling.

For results on tiling the plane with polyominoes, see Polyomino § Uses of polyominoes.

Voronoi tilings Edit

Voronoi or Dirichlet tilings are tessellations where each tile is defined as the set of points closest to one of the points in a discrete set of defining points. (Think of geographical regions where each region is defined as all the points closest to a given city or post office.) [51] [52] The Voronoi cell for each defining point is a convex polygon. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Delaunay triangulations are useful in numerical simulation, in part because among all possible triangulations of the defining points, Delaunay triangulations maximize the minimum of the angles formed by the edges. [53] Voronoi tilings with randomly placed points can be used to construct random tilings of the plane. [54]

Tessellations in higher dimensions Edit

Tessellation can be extended to three dimensions. Certain polyhedra can be stacked in a regular crystal pattern to fill (or tile) three-dimensional space, including the cube (the only Platonic polyhedron to do so), the rhombic dodecahedron, the truncated octahedron, and triangular, quadrilateral, and hexagonal prisms, among others. [55] Any polyhedron that fits this criterion is known as a plesiohedron, and may possess between 4 and 38 faces. [56] Naturally occurring rhombic dodecahedra are found as crystals of andradite (a kind of garnet) and fluorite. [57] [58]

Tessellations in three or more dimensions are called honeycombs. In three dimensions there is just one regular honeycomb, which has eight cubes at each polyhedron vertex. Similarly, in three dimensions there is just one quasiregular [c] honeycomb, which has eight tetrahedra and six octahedra at each polyhedron vertex. However, there are many possible semiregular honeycombs in three dimensions. [59] Uniform polyhedra can be constructed using the Wythoff construction. [60]

The Schmitt-Conway biprism is a convex polyhedron with the property of tiling space only aperiodically. [61]

Tessellations in non-Euclidean geometries Edit

It is possible to tessellate in non-Euclidean geometries such as hyperbolic geometry. A uniform tiling in the hyperbolic plane (which may be regular, quasiregular or semiregular) is an edge-to-edge filling of the hyperbolic plane, with regular polygons as faces these are vertex-transitive (transitive on its vertices), and isogonal (there is an isometry mapping any vertex onto any other). [63] [64]

A uniform honeycomb in hyperbolic space is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family. [65]

In architecture, tessellations have been used to create decorative motifs since ancient times. Mosaic tilings often had geometric patterns. [4] Later civilisations also used larger tiles, either plain or individually decorated. Some of the most decorative were the Moorish wall tilings of Islamic architecture, using Girih and Zellige tiles in buildings such as the Alhambra [66] and La Mezquita. [67]

Tessellations frequently appeared in the graphic art of M. C. Escher he was inspired by the Moorish use of symmetry in places such as the Alhambra when he visited Spain in 1936. [68] Escher made four "Circle Limit" drawings of tilings that use hyperbolic geometry. [69] [70] For his woodcut "Circle Limit IV" (1960), Escher prepared a pencil and ink study showing the required geometry. [71] Escher explained that "No single component of all the series, which from infinitely far away rise like rockets perpendicularly from the limit and are at last lost in it, ever reaches the boundary line." [72]

Tessellated designs often appear on textiles, whether woven, stitched in or printed. Tessellation patterns have been used to design interlocking motifs of patch shapes in quilts. [73] [74]

Tessellations are also a main genre in origami (paper folding), where pleats are used to connect molecules such as twist folds together in a repeating fashion. [75]

Tessellation is used in manufacturing industry to reduce the wastage of material (yield losses) such as sheet metal when cutting out shapes for objects like car doors or drinks cans. [76]

Tessellation is apparent in the mudcrack-like cracking of thin films [77] [78] – with a degree of self-organisation being observed using micro and nanotechnologies. [79]

The honeycomb is a well-known example of tessellation in nature with its hexagonal cells. [80]

In botany, the term "tessellate" describes a checkered pattern, for example on a flower petal, tree bark, or fruit. Flowers including the fritillary [81] and some species of Colchicum are characteristically tessellate. [82]

Many patterns in nature are formed by cracks in sheets of materials. These patterns can be described by Gilbert tessellations, [83] also known as random crack networks. [84] The Gilbert tessellation is a mathematical model for the formation of mudcracks, needle-like crystals, and similar structures. The model, named after Edgar Gilbert, allows cracks to form starting from randomly scattered over the plane each crack propagates in two opposite directions along a line through the initiation point, its slope chosen at random, creating a tessellation of irregular convex polygons. [85] Basaltic lava flows often display columnar jointing as a result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Northern Ireland. [86] Tessellated pavement, a characteristic example of which is found at Eaglehawk Neck on the Tasman Peninsula of Tasmania, is a rare sedimentary rock formation where the rock has fractured into rectangular blocks. [87]

Other natural patterns occur in foams these are packed according to Plateau's laws, which require minimal surfaces. Such foams present a problem in how to pack cells as tightly as possible: in 1887, Lord Kelvin proposed a packing using only one solid, the bitruncated cubic honeycomb with very slightly curved faces. In 1993, Denis Weaire and Robert Phelan proposed the Weaire–Phelan structure, which uses less surface area to separate cells of equal volume than Kelvin's foam. [88]

Tessellations have given rise to many types of tiling puzzle, from traditional jigsaw puzzles (with irregular pieces of wood or cardboard) [89] and the tangram [90] to more modern puzzles which often have a mathematical basis. For example, polyiamonds and polyominoes are figures of regular triangles and squares, often used in tiling puzzles. [91] [92] Authors such as Henry Dudeney and Martin Gardner have made many uses of tessellation in recreational mathematics. For example, Dudeney invented the hinged dissection, [93] while Gardner wrote about the rep-tile, a shape that can be dissected into smaller copies of the same shape. [94] [95] Inspired by Gardner's articles in Scientific American, the amateur mathematician Marjorie Rice found four new tessellations with pentagons. [96] [97] Squaring the square is the problem of tiling an integral square (one whose sides have integer length) using only other integral squares. [98] [99] An extension is squaring the plane, tiling it by squares whose sizes are all natural numbers without repetitions James and Frederick Henle proved that this was possible. [100]

Floret pentagonal tiling, dual to a semiregular tiling and one of 15 monohedral pentagon tilings.


10.1 Parametric Equations

Why would we want to parameterize a curve? Imagine that you are located in a Cartesian plane at (1, 0). You start walking and your position after time t is given by . The path you will trace out will be the unit circle. But more importantly, you can give your location at any time because your position is given by the pair of parametric equations. So while the Cartesian equation of the unit circle describes the route you walk, the parametric equations describe your location at any time.

A curve in the plane is said to be parameterized if the coordinates of the points on the curve, ( x , y ), are represented as functions of a variable t . Namely, x = f( t ), y = g( t )  t D. where D is a set of real numbers. The variable t is called a parameter and the relations between x , y and t are called parametric equations. The set D is called the domain of f and g and it is the set of values t takes.

Conversely, given a pair of parametric equations with parameter t , the set of points (f( t ), g( t )) form a curve in the plane.

As an example, the graph of any function can be parameterized. For, if y = f( x ), then let t = x so that x = t , y = f( t ) This pair of parametric equations, with parameter t , has graph identical to that of the function y = f( x ). The domain of the parametric equations is the same as the domain of f.

Example: The parametric equations x = t , y = t 2 , t any real number are an example of how to parameterize the graph of the function y = x 2 . We arrived at this pair of parametric equations as described above. The picture below shows the graph of the parametric equations. We have labeled three points on the graph that are obtained from different values of the parameter.

If the parameter t represented time measured in seconds, and the x and y-coordinates are the position of a moving object, then where would the object be after 3 seconds? If then and The object would be at .

In the example no restrictions were placed on the parameter. The curve defined parametrically by the equations was identical to the graph of the function. By restricting the values of the parameter, we can parameterize part of the graph of the function.

Example: The parametric equations x = t , y = t 2 t [&minus1,2] give an example of how to parameterize part of the graph of the function y = x 2 . The part of the graph we get is from x = &minus1 to x = 2. The diagram shows the graph of the parametric equations. We have marked three points on the curve corresponding to three values of the parameter.

There are several techniques we use to sketch a curve generated by a pair of parametric equations. The most basic is to evaluate f( t ) and g( t ) for several values of t . We then plot the points (f( t ), g( t )) in the plane and through them draw a smooth curve (assuming this is valid!). Sketching the parametric curve by plotting points is identical to the elementary graphing techniques of graphing functions and is illustrated in the following two diagrams.

In this example the parametric equations are x = 2 t and and we have evaluated t at &minus2, &minus1.5, &minus1, &minus0.5, &minus0.25, 0, 0.25, 0.5, 1.5 and 2. We have determined the corresponding values of x and y and plotted these points. The diagram shows the result of plotting these points.
Through these plotted points we have drawn a smooth curve and the result is shown in the diagram to the right. We'll see shortly how to deduce that the Cartesian equation of this curve is

With this method, it is not clear how many values of t to choose. We chose enough values until we felt we had a clear idea of what the curve would look like. We also chose negative values of the parameter as well as fractional values. Usually, it is not a good idea to rely only on positive or only on integer values of the parameter. This makes the point-plotting method seem laborious and tedious, and indeed it is. This task is better suited to a computer or graphing calculator. Because of this, we employ other methods to help draw the graph and use point-plotting mainly to place certain special or interesting points on the graph. One of these methods, the method of "eliminating the parameter", is discussed at the end of this section.

It is time to practice this technique. In this exercise, you are given a pair of parametric equations. Select 6 valid values of t and substitute each into both equations to get the coordinates of a point in the plane. Plot all six points in the plane. The "Delete" button deletes the points in reverse order. The "Delete all" button will remove all the points you have plotted. Once you are satisfied, press the "Check" button. If you have at least two points correct you can either correct the points or try sketching the graph. To sketch the graph, hold down the mouse button and drag the mouse. Finally, press "Graph" and you will be shown the correct graph.
You are also asked to provide the orientation of the curve. Make the best selection of the four choices. You should only choose clockwise or counterclockwise for the parameterizations using sine and cosine. All other curves have orientation which can be described as left to right or right to left.

Plotting a Parametric Curve

A second technique to identifying the curve of the parametric equations is to try to eliminate the parameter from the equations. This will result in an equation involving only x and y which we may recognize. For example, let's look again at the previous example. Since x = 2 t then solving for t gives t = x /2. Substitute for t into the equation for y to get y = ( x /2) 2 = x 2 /4. This we recognize as the graph of a parabola and we can sketch its graph using function graphing techniques.
Be careful, however, to take into account any restrictions on the value of the parameter. If, in this example, we add the condition t > 0, then the curve defined by the parametric equations would be the graph of y = x 2 /4 on the positive x axis.

In this set of exercises you are given two parametric equations. You are to eliminate the parameter to find an expression between y and x . All answers in this set can be written in the form y = f( x ). Click "New" for a new problem. Type your function in the typing area. Once you have entered the expression, press "Check" to see if your answers are correct. The "Help" button will provide a hint. The "Solve" button will reveal the solution if your checked answer is incorrect.
Eliminating the Parameter

A parametric representation of a curve is not unique. That is, a curve C may be represented by two (or more) different pairs of parametric equations.

Example: We saw earlier that the parametric equations x = t , y = t 2 t [&minus1,2] parameterize part of the graph of the function y = x 2 . from x = &minus1 to x = 2.

However, the equations x = 1 &minus 2 u , y = (1 &minus 2 u ) 2 u [&minus1/3,1] also parameterize the same part of the graph of the function y = x 2 . from x = &minus1 to x = 2.

Example: Consider x = cos( t ), y = sin( t ) t [0, 2]. To see what curve these equations define, let's square both x and y , and add both terms. x 2 + y 2 = cos 2 ( t ) + sin 2 ( t ) = 1. This is the equation of the unit circle and so the two parametric equations are a parameterization of the unit circle.

Now, consider x = sin( t ), y = cos( t ) t [0, 2]. We apply the same procedure to eliminate the parameter, namely square x and y , and add the terms. x 2 + y 2 = sin 2 ( t ) + cos 2 ( t ) = 1.
This parametric curve is also the unit circle and we have found two different parameterizations of the unit circle.

Notice how we obtained an equation in terms of x and y in the previous example. We eliminated the parameter but not in a direct way by solving one of the equations. The technique is useful in many parametric equations involving sine and cosine. That is, solve the equations for sin( t ) and cos( t ) then square and add the equations.


10.1: Plane - Mathematics

Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems.

A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle. In polar coordinates a point in the plane is identified by a pair of numbers $(r, heta)$. The number $ heta$ measures the angle between the positive $x$-axis and a ray that goes through the point, as shown in figure 10.1.1 the number $r$ measures the distance from the origin to the point. Figure 10.1.1 shows the point with rectangular coordinates $ds (1,sqrt3)$ and polar coordinates $(2,pi/3)$, 2 units from the origin and $pi/3$ radians from the positive $x$-axis.

Just as we describe curves in the plane using equations involving $x$ and $y$, so can we describe curves using equations involving $r$ and $ heta$. Most common are equations of the form $r=f( heta)$.

Example 10.1.1 Graph the curve given by $r=2$. All points with $r=2$ are at distance 2 from the origin, so $r=2$ describes the circle of radius 2 with center at the origin.

Example 10.1.2 Graph the curve given by $r=1+cos heta$. We first consider $y=1+cos x$, as in figure 10.1.2. As $ heta$ goes through the values in $[0,2pi]$, the value of $r$ tracks the value of $y$, forming the "cardioid'' shape of figure 10.1.2. For example, when $ heta=pi/2$, $r=1+cos(pi/2)=1$, so we graph the point at distance 1 from the origin along the positive $y$-axis, which is at an angle of $pi/2$ from the positive $x$-axis. When $ heta=7pi/4$, $ds r=1+cos(7pi/4)=1+sqrt2/2approx 1.71$, and the corresponding point appears in the fourth quadrant. This illustrates one of the potential benefits of using polar coordinates: the equation for this curve in rectangular coordinates would be quite complicated.

Each point in the plane is associated with exactly one pair of numbers in the rectangular coordinate system each point is associated with an infinite number of pairs in polar coordinates. In the cardioid example, we considered only the range le hetale2pi$, and already there was a duplicate: $(2,0)$ and $(2,2pi)$ are the same point. Indeed, every value of $ heta$ outside the interval $[0,2pi)$ duplicates a point on the curve $r=1+cos heta$ when le heta Figure 10.1.3. The point $(-2,pi/4)=(2,5pi/4)=(2,-3pi/4)$ in polar coordinates.

The relationship between rectangular and polar coordinates is quite easy to understand. The point with polar coordinates $(r, heta)$ has rectangular coordinates $x=rcos heta$ and $y=rsin heta$ this follows immediately from the definition of the sine and cosine functions. Using figure 10.1.3 as an example, the point shown has rectangular coordinates $ds x=(-2)cos(pi/4)=-sqrt2approx 1.4142$ and $ds y=(-2)sin(pi/4)=-sqrt2$. This makes it very easy to convert equations from rectangular to polar coordinates.

Example 10.1.3 Find the equation of the line $y=3x+2$ in polar coordinates. We merely substitute: $rsin heta=3rcos heta+2$, or $ds r= <2over sin heta-3cos heta>$.

Example 10.1.4 Find the equation of the circle $ds (x-1/2)^2+y^2=1/4$ in polar coordinates. Again substituting: $ds (rcos heta-1/2)^2+r^2sin^2 heta=1/4$. A bit of algebra turns this into $r=cos(t)$. You should try plotting a few $(r, heta)$ values to convince yourself that this makes sense.

Example 10.1.5 Graph the polar equation $r= heta$. Here the distance from the origin exactly matches the angle, so a bit of thought makes it clear that when $ hetage0$ we get the spiral of Archimedes in figure 10.1.4. When $ heta Figure 10.1.4. The spiral of Archimedes and the full graph of $r= heta$.

Converting polar equations to rectangular equations can be somewhat trickier, and graphing polar equations directly is also not always easy.

Example 10.1.6 Graph $r=2sin heta$. Because the sine is periodic, we know that we will get the entire curve for values of $ heta$ in $[0,2pi)$. As $ heta$ runs from 0 to $pi/2$, $r$ increases from 0 to 2. Then as $ heta$ continues to $pi$, $r$ decreases again to 0. When $ heta$ runs from $pi$ to $2pi$, $r$ is negative, and it is not hard to see that the first part of the curve is simply traced out again, so in fact we get the whole curve for values of $ heta$ in $[0,pi)$. Thus, the curve looks something like figure 10.1.5. Now, this suggests that the curve could possibly be a circle, and if it is, it would have to be the circle $ds x^2+(y-1)^2=1$. Having made this guess, we can easily check it. First we substitute for $x$ and $y$ to get $ds (rcos heta)^2+(rsin heta-1)^2=1$ expanding and simplifying does indeed turn this into $r=2sin heta$.


Contents

Relation to Euclidean geometry Edit

Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, Euclidean and hyperbolic. All theorems of absolute geometry, including the first 28 propositions of book one of Euclid's Elements, are valid in Euclidean and hyperbolic geometry. Propositions 27 and 28 of Book One of Euclid's Elements prove the existence of parallel/non-intersecting lines.

This difference also has many consequences: concepts that are equivalent in Euclidean geometry are not equivalent in hyperbolic geometry new concepts need to be introduced. Further, because of the angle of parallelism, hyperbolic geometry has an absolute scale, a relation between distance and angle measurements.

Lines Edit

Single lines in hyperbolic geometry have exactly the same properties as single straight lines in Euclidean geometry. For example, two points uniquely define a line, and line segments can be infinitely extended.

Two intersecting lines have the same properties as two intersecting lines in Euclidean geometry. For example, two distinct lines can intersect in no more than one point, intersecting lines form equal opposite angles, and adjacent angles of intersecting lines are supplementary.

When a third line is introduced, then there can be properties of intersecting lines that differ from intersecting lines in Euclidean geometry. For example, given two intersecting lines there are infinitely many lines that do not intersect either of the given lines.

These properties are all independent of the model used, even if the lines may look radically different.

Non-intersecting / parallel lines Edit

Non-intersecting lines in hyperbolic geometry also have properties that differ from non-intersecting lines in Euclidean geometry:

For any line R and any point P which does not lie on R, in the plane containing line R and point P there are at least two distinct lines through P that do not intersect R.

This implies that there are through P an infinite number of coplanar lines that do not intersect R.

These non-intersecting lines are divided into two classes:

  • Two of the lines (x and y in the diagram) are limiting parallels (sometimes called critically parallel, horoparallel or just parallel): there is one in the direction of each of the ideal points at the "ends" of R, asymptotically approaching R, always getting closer to R, but never meeting it.
  • All other non-intersecting lines have a point of minimum distance and diverge from both sides of that point, and are called ultraparallel, diverging parallel or sometimes non-intersecting.

Some geometers simply use the phrase "parallel lines" to mean "limiting parallel lines", with ultraparallel lines meaning just non-intersecting.

These limiting parallels make an angle θ with PB this angle depends only on the Gaussian curvature of the plane and the distance PB and is called the angle of parallelism.

For ultraparallel lines, the ultraparallel theorem states that there is a unique line in the hyperbolic plane that is perpendicular to each pair of ultraparallel lines.

Circles and disks Edit

In hyperbolic geometry, the circumference of a circle of radius r is greater than 2 π r .

Then the circumference of a circle of radius r is equal to:

And the area of the enclosed disk is:

Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than 2 π , though it can be made arbitrarily close by selecting a small enough circle.

If the Gaussian curvature of the plane is −1 then the geodesic curvature of a circle of radius r is: 1 tanh ⁡ ( r ) < anh(r)>>> [1]

Hypercycles and horocycles Edit

In hyperbolic geometry, there is no line all of whose points are equidistant from another line. Instead, the points that all have the same orthogonal distance from a given line lie on a curve called a hypercycle.

Another special curve is the horocycle, a curve whose normal radii (perpendicular lines) are all limiting parallel to each other (all converge asymptotically in one direction to the same ideal point, the centre of the horocycle).

Through every pair of points there are two horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the line-segment between them.

Given any three distinct points, they all lie on either a line, hypercycle, horocycle, or circle.

The length of the line-segment is the shortest length between two points. The arc-length of a hypercycle connecting two points is longer than that of the line segment and shorter than that of a horocycle, connecting the same two points. The arclength of both horocycles connecting two points are equal. The arc-length of a circle between two points is larger than the arc-length of a horocycle connecting two points.

If the Gaussian curvature of the plane is −1 then the geodesic curvature of a horocycle is 1 and of a hypercycle is between 0 and 1. [1]

Triangles Edit

Unlike Euclidean triangles, where the angles always add up to π radians (180°, a straight angle), in hyperbolic geometry the sum of the angles of a hyperbolic triangle is always strictly less than π radians (180°, a straight angle). The difference is referred to as the defect.

The area of a hyperbolic triangle is given by its defect in radians multiplied by R 2 . As a consequence, all hyperbolic triangles have an area that is less than or equal to R 2 π. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.

As in Euclidean geometry, each hyperbolic triangle has an incircle. In hyperbolic geometry, if all three of its vertices lie on a horocycle or hypercycle, then the triangle has no circumscribed circle.

As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent.

Regular apeirogon Edit

A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides.

In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line.

However, in hyperbolic geometry, a regular apeirogon has sides of any length (i.e., it remains a polygon).

The side and angle bisectors will, depending on the side length and the angle between the sides, be limiting or diverging parallel (see lines above). If the bisectors are limiting parallel the apeirogon can be inscribed and circumscribed by concentric horocycles.

If the bisectors are diverging parallel then a pseudogon (distinctly different from an apeirogon) can be inscribed in hypercycles (all vertices are the same distance of a line, the axis, also the midpoint of the side segments are all equidistant to the same axis.)

Tessellations Edit

Like the Euclidean plane it is also possible to tessellate the hyperbolic plane with regular polygons as faces.

There are an infinite number of uniform tilings based on the Schwarz triangles (p q r) where 1/p + 1/q + 1/r < 1, where p, q, r are each orders of reflection symmetry at three points of the fundamental domain triangle, the symmetry group is a hyperbolic triangle group. There are also infinitely many uniform tilings that cannot be generated from Schwarz triangles, some for example requiring quadrilaterals as fundamental domains. [2]

Though hyperbolic geometry applies for any surface with a constant negative Gaussian curvature, it is usual to assume a scale in which the curvature K is −1.

This results in some formulas becoming simpler. Some examples are:

  • The area of a triangle is equal to its angle defect in radians.
  • The area of a horocyclic sector is equal to the length of its horocyclic arc.
  • An arc of a horocycle so that a line that is tangent at one endpoint is limiting parallel to the radius through the other endpoint has a length of 1. [3]
  • The ratio of the arc lengths between two radii of two concentric horocycles where the horocycles are a distance 1 apart is e : 1. [3]

Cartesian-like coordinate systems Edit

In hyperbolic geometry, the sum of the angles of a quadrilateral is always less than 360 degrees, and hyperbolic rectangles differ greatly from Euclidean rectangles since there are no equidistant lines, so a proper Euclidean rectangle would need to be enclosed by two lines and two hypercycles. These all complicate coordinate systems.

There are however different coordinate systems for hyperbolic plane geometry. All are based around choosing a point (the origin) on a chosen directed line (the x-axis) and after that many choices exist.

The Lobachevski coordinates x and y are found by dropping a perpendicular onto the x-axis. x will be the label of the foot of the perpendicular. y will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other).

Another coordinate system measures the distance from the point to the horocycle through the origin centered around ( 0 , + ∞ ) and the length along this horocycle. [4]

Other coordinate systems use the Klein model or the Poincare disk model described below, and take the Euclidean coordinates as hyperbolic.

Distance Edit

Construct a Cartesian-like coordinate system as follows. Choose a line (the x-axis) in the hyperbolic plane (with a standardized curvature of −1) and label the points on it by their distance from an origin (x=0) point on the x-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates x and y by dropping a perpendicular onto the x-axis. x will be the label of the foot of the perpendicular. y will be the distance along the perpendicular of the given point from its foot (positive on one side and negative on the other). Then the distance between two such points will be [ citation needed ]

This formula can be derived from the formulas about hyperbolic triangles.

In this coordinate system, straight lines are either perpendicular to the x-axis (with equation x = a constant) or described by equations of the form

where A and B are real parameters which characterize the straight line.

Since the publication of Euclid's Elements circa 300 BCE, many geometers made attempts to prove the parallel postulate. Some tried to prove it by assuming its negation and trying to derive a contradiction. Foremost among these were Proclus, Ibn al-Haytham (Alhacen), Omar Khayyám, [5] Nasīr al-Dīn al-Tūsī, Witelo, Gersonides, Alfonso, and later Giovanni Gerolamo Saccheri, John Wallis, Johann Heinrich Lambert, and Legendre. [6] Their attempts were doomed to failure (as we now know, the parallel postulate is not provable from the other postulates), but their efforts led to the discovery of hyperbolic geometry.

The theorems of Alhacen, Khayyam and al-Tūsī on quadrilaterals, including the Ibn al-Haytham–Lambert quadrilateral and Khayyam–Saccheri quadrilateral, were the first theorems on hyperbolic geometry. Their works on hyperbolic geometry had a considerable influence on its development among later European geometers, including Witelo, Gersonides, Alfonso, John Wallis and Saccheri. [7]

In the 18th century, Johann Heinrich Lambert introduced the hyperbolic functions [8] and computed the area of a hyperbolic triangle. [9]

19th-century developments Edit

In the 19th century, hyperbolic geometry was explored extensively by Nikolai Ivanovich Lobachevsky, János Bolyai, Carl Friedrich Gauss and Franz Taurinus. Unlike their predecessors, who just wanted to eliminate the parallel postulate from the axioms of Euclidean geometry, these authors realized they had discovered a new geometry. [10] [11] Gauss wrote in an 1824 letter to Franz Taurinus that he had constructed it, but Gauss did not publish his work. Gauss called it "non-Euclidean geometry" [12] causing several modern authors to continue to consider "non-Euclidean geometry" and "hyperbolic geometry" to be synonyms. Taurinus published results on hyperbolic trigonometry in 1826, argued that hyperbolic geometry is self consistent, but still believed in the special role of Euclidean geometry. The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832.

In 1868, Eugenio Beltrami provided models (see below) of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was.

The term "hyperbolic geometry" was introduced by Felix Klein in 1871. [13] Klein followed an initiative of Arthur Cayley to use the transformations of projective geometry to produce isometries. The idea used a conic section or quadric to define a region, and used cross ratio to define a metric. The projective transformations that leave the conic section or quadric stable are the isometries. "Klein showed that if the Cayley absolute is a real curve then the part of the projective plane in its interior is isometric to the hyperbolic plane. " [14]

For more history, see article on non-Euclidean geometry, and the references Coxeter [15] and Milnor. [16]

Philosophical consequences Edit

The discovery of hyperbolic geometry had important philosophical consequences. Before its discovery many philosophers (for example Hobbes and Spinoza) viewed philosophical rigour in terms of the "geometrical method", referring to the method of reasoning used in Euclid's Elements.

Kant in the Critique of Pure Reason came to the conclusion that space (in Euclidean geometry) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences. [17]

It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the Boeotians", which would ruin his status as princeps mathematicorum (Latin, "the Prince of Mathematicians"). [18] The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in mathematical rigour, analytical philosophy and logic. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.

Geometry of the universe (spatial dimensions only) Edit

Because Euclidean, hyperbolic and elliptic geometry are all consistent, the question arises: which is the real geometry of space, and if it is hyperbolic or elliptic, what is its curvature?

Lobachevsky had already tried to measure the curvature of the universe by measuring the parallax of Sirius and treating Sirius as the ideal point of an angle of parallelism. He realised that his measurements were not precise enough to give a definite answer, but he did reach the conclusion that if the geometry of the universe is hyperbolic, then the absolute length is at least one million times the diameter of the earth's orbit ( 2 000 000 AU , 10 parsec). [19] Some argue that his measurements were methodologically flawed. [20]

Henri Poincaré, with his sphere-world thought experiment, came to the conclusion that everyday experience does not necessarily rule out other geometries.

The geometrization conjecture gives a complete list of eight possibilities for the fundamental geometry of our space. The problem in determining which one applies is that, to reach a definitive answer, we need to be able to look at extremely large shapes – much larger than anything on Earth or perhaps even in our galaxy. [21]

Geometry of the universe (special relativity) Edit

Special relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. [22] [23] Minkowski geometry replaces Galilean geometry (which is the three-dimensional Euclidean space with time of Galilean relativity). [24]

In relativity, rather than considering Euclidean, elliptic and hyperbolic geometries, the appropriate geometries to consider are Minkowski space, de Sitter space and anti-de Sitter space, [25] [26] corresponding to zero, positive and negative curvature respectively.

Hyperbolic geometry enters special relativity through rapidity, which stands in for velocity, and is expressed by a hyperbolic angle. The study of this velocity geometry has been called kinematic geometry. The space of relativistic velocities has a three-dimensional hyperbolic geometry, where the distance function is determined from the relative velocities of "nearby" points (velocities). [27]

The hyperbolic plane is a plane where every point is a saddle point. There exist various pseudospheres in Euclidean space that have a finite area of constant negative Gaussian curvature.

By Hilbert's theorem, it is not possible to isometrically immerse a complete hyperbolic plane (a complete regular surface of constant negative Gaussian curvature) in a three-dimensional Euclidean space.

Other useful models of hyperbolic geometry exist in Euclidean space, in which the metric is not preserved. A particularly well-known paper model based on the pseudosphere is due to William Thurston.

The art of crochet has been used (see Mathematics and fiber arts § Knitting and crochet) to demonstrate hyperbolic planes with the first being made by Daina Taimiņa. [28]

In 2000, Keith Henderson demonstrated a quick-to-make paper model dubbed the "hyperbolic soccerball" (more precisely, a truncated order-7 triangular tiling). [29] [30]

Instructions on how to make a hyperbolic quilt, designed by Helaman Ferguson, [31] have been made available by Jeff Weeks. [32]

There are different pseudospherical surfaces that have for a large area a constant negative Gaussian curvature, the pseudosphere being the best well known of them.

But it is easier to do hyperbolic geometry on other models.

There are four models commonly used for hyperbolic geometry: the Klein model, the Poincaré disk model, the Poincaré half-plane model, and the Lorentz or hyperboloid model. These models define a hyperbolic plane which satisfies the axioms of a hyperbolic geometry. Despite their names, the first three mentioned above were introduced as models of hyperbolic space by Beltrami, not by Poincaré or Klein. All these models are extendable to more dimensions.

The Beltrami–Klein model Edit

The Beltrami–Klein model, also known as the projective disk model, Klein disk model and Klein model, is named after Eugenio Beltrami and Felix Klein.

For the two dimensions this model uses the interior of the unit circle for the complete hyperbolic plane, and the chords of this circle are the hyperbolic lines.

For higher dimensions this model uses the interior of the unit ball, and the chords of this n-ball are the hyperbolic lines.

  • This model has the advantage that lines are straight, but the disadvantage that angles are distorted (the mapping is not conformal), and also circles are not represented as circles.
  • The distance in this model is half the logarithm of the cross-ratio, which was introduced by Arthur Cayley in projective geometry.

The Poincaré disk model Edit

The Poincaré disk model, also known as the conformal disk model, also employs the interior of the unit circle, but lines are represented by arcs of circles that are orthogonal to the boundary circle, plus diameters of the boundary circle.

  • This model preserves angles, and is thereby conformal. All isometries within this model are therefore Möbius transformations.
  • Circles entirely within the disk remain circles although the Euclidean center of the circle is closer to the center of the disk than is the hyperbolic center of the circle. are circles within the disk which are tangent to the boundary circle, minus the point of contact. are open-ended chords and circular arcs within the disc that terminate on the boundary circle at non-orthogonal angles.

The Poincaré half-plane model Edit

The Poincaré half-plane model takes one-half of the Euclidean plane, bounded by a line B of the plane, to be a model of the hyperbolic plane. The line B is not included in the model.

The Euclidean plane may be taken to be a plane with the Cartesian coordinate system and the x-axis is taken as line B and the half plane is the upper half (y > 0 ) of this plane.

  • Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B.
  • The length of an interval on a ray is given by logarithmic measure so it is invariant under a homothetic transformation ( x , y ) ↦ ( x , λ y ) , λ > 0.
  • Like the Poincaré disk model, this model preserves angles, and is thus conformal. All isometries within this model are therefore Möbius transformations of the plane.
  • The half-plane model is the limit of the Poincaré disk model whose boundary is tangent to B at the same point while the radius of the disk model goes to infinity.

The hyperboloid model Edit

The hyperboloid model or Lorentz model employs a 2-dimensional hyperboloid of revolution (of two sheets, but using one) embedded in 3-dimensional Minkowski space. This model is generally credited to Poincaré, but Reynolds [33] says that Wilhelm Killing used this model in 1885

  • This model has direct application to special relativity, as Minkowski 3-space is a model for spacetime, suppressing one spatial dimension. One can take the hyperboloid to represent the events that various moving observers, radiating outward in a spatial plane from a single point, will reach in a fixed proper time.
  • The hyperbolic distance between two points on the hyperboloid can then be identified with the relative rapidity between the two corresponding observers.
  • The model generalizes directly to an additional dimension, where three-dimensional hyperbolic geometry relates to Minkowski 4-space.

The hemisphere model Edit

The hemisphere model is not often used as model by itself, but it functions as a useful tool for visualising transformations between the other models.

The hemisphere model uses the upper half of the unit sphere: x 2 + y 2 + z 2 = 1 , z > 0. +y^<2>+z^<2>=1,z>0.>

The hyperbolic lines are half-circles orthogonal to the boundary of the hemisphere.

The hemisphere model is part of a Riemann sphere, and different projections give different models of the hyperbolic plane:

    from ( 0 , 0 , − 1 ) onto the plane z = 0 projects corresponding points on the Poincaré disk model from ( 0 , 0 , − 1 ) onto the surface x 2 + y 2 − z 2 = − 1 , z > 0 +y^<2>-z^<2>=-1,z>0> projects corresponding points on the hyperboloid model from ( − 1 , 0 , 0 ) onto the plane x = 1 projects corresponding points on the Poincaré half-plane model onto a plane z = C projects corresponding points on the Beltrami–Klein model. from the centre of the sphere onto the plane z = 1 projects corresponding points on the Gans Model

The Gans model Edit

In 1966 David Gans proposed a flattened hyperboloid model in the journal American Mathematical Monthly. [34] It is an orthographic projection of the hyperboloid model onto the xy-plane. This model is not as widely used as other models but nevertheless is quite useful in the understanding of hyperbolic geometry.

  • Unlike the Klein or the Poincaré models, this model utilizes the entire Euclidean plane.
  • The lines in this model are represented as branches of a hyperbola. [35]

The band model Edit

The band model employs a portion of the Euclidean plane between two parallel lines. [36] Distance is preserved along one line through the middle of the band. Assuming the band is given by < z ∈ C : | Im ⁡ z | < π / 2 > :|operatorname z|<pi /2>> , the metric is given by | d z | sec ⁡ ( Im ⁡ z ) z)> .

Connection between the models Edit

All models essentially describe the same structure. The difference between them is that they represent different coordinate charts laid down on the same metric space, namely the hyperbolic plane. The characteristic feature of the hyperbolic plane itself is that it has a constant negative Gaussian curvature, which is indifferent to the coordinate chart used. The geodesics are similarly invariant: that is, geodesics map to geodesics under coordinate transformation. Hyperbolic geometry generally is introduced in terms of the geodesics and their intersections on the hyperbolic plane. [37]

Once we choose a coordinate chart (one of the "models"), we can always embed it in a Euclidean space of same dimension, but the embedding is clearly not isometric (since the curvature of Euclidean space is 0). The hyperbolic space can be represented by infinitely many different charts but the embeddings in Euclidean space due to these four specific charts show some interesting characteristics.

Since the four models describe the same metric space, each can be transformed into the other.

Every isometry (transformation or motion) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In n-dimensional hyperbolic space, up to n+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)

All the isometries of the hyperbolic plane can be classified into these classes:

  • Orientation preserving
    • the identity isometry — nothing moves zero reflections zero degrees of freedom. — two reflections through mutually perpendicular lines passing through the given point, i.e. a rotation of 180 degrees around the point two degrees of freedom. around a normal point — two reflections through lines passing through the given point (includes inversion as a special case) points move on circles around the center three degrees of freedom.
    • "rotation" around an ideal point (horolation) — two reflections through lines leading to the ideal point points move along horocycles centered on the ideal point two degrees of freedom.
    • translation along a straight line — two reflections through lines perpendicular to the given line points off the given line move along hypercycles three degrees of freedom.
    • reflection through a line — one reflection two degrees of freedom.
    • combined reflection through a line and translation along the same line — the reflection and translation commute three reflections required three degrees of freedom. [citation needed]

    M. C. Escher's famous prints Circle Limit III and Circle Limit IV illustrate the conformal disc model (Poincaré disk model) quite well. The white lines in III are not quite geodesics (they are hypercycles), but are close to them. It is also possible to see quite plainly the negative curvature of the hyperbolic plane, through its effect on the sum of angles in triangles and squares.

    For example, in Circle Limit III every vertex belongs to three triangles and three squares. In the Euclidean plane, their angles would sum to 450° i.e., a circle and a quarter. From this, we see that the sum of angles of a triangle in the hyperbolic plane must be smaller than 180°. Another visible property is exponential growth. In Circle Limit III, for example, one can see that the number of fishes within a distance of n from the center rises exponentially. The fishes have an equal hyperbolic area, so the area of a ball of radius n must rise exponentially in n.

    The art of crochet has been used to demonstrate hyperbolic planes (pictured above) with the first being made by Daina Taimiņa, [28] whose book Crocheting Adventures with Hyperbolic Planes won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year. [38]

    HyperRogue is a roguelike game set on various tilings of the hyperbolic plane.

    Hyperbolic geometry is not limited to 2 dimensions a hyperbolic geometry exists for every higher number of dimensions.

    Hyperbolic space of dimension n is a special case of a Riemannian symmetric space of noncompact type, as it is isomorphic to the quotient

    The orthogonal group O(1, n) acts by norm-preserving transformations on Minkowski space R 1,n , and it acts transitively on the two-sheet hyperboloid of norm 1 vectors. Timelike lines (i.e., those with positive-norm tangents) through the origin pass through antipodal points in the hyperboloid, so the space of such lines yields a model of hyperbolic n-space. The stabilizer of any particular line is isomorphic to the product of the orthogonal groups O(n) and O(1), where O(n) acts on the tangent space of a point in the hyperboloid, and O(1) reflects the line through the origin. Many of the elementary concepts in hyperbolic geometry can be described in linear algebraic terms: geodesic paths are described by intersections with planes through the origin, dihedral angles between hyperplanes can be described by inner products of normal vectors, and hyperbolic reflection groups can be given explicit matrix realizations.

    In small dimensions, there are exceptional isomorphisms of Lie groups that yield additional ways to consider symmetries of hyperbolic spaces. For example, in dimension 2, the isomorphisms SO + (1, 2) ≅ PSL(2, R) ≅ PSU(1, 1) allow one to interpret the upper half plane model as the quotient SL(2, R)/SO(2) and the Poincaré disc model as the quotient SU(1, 1)/U(1) . In both cases, the symmetry groups act by fractional linear transformations, since both groups are the orientation-preserving stabilizers in PGL(2, C) of the respective subspaces of the Riemann sphere. The Cayley transformation not only takes one model of the hyperbolic plane to the other, but realizes the isomorphism of symmetry groups as conjugation in a larger group. In dimension 3, the fractional linear action of PGL(2, C) on the Riemann sphere is identified with the action on the conformal boundary of hyperbolic 3-space induced by the isomorphism O + (1, 3) ≅ PGL(2, C) . This allows one to study isometries of hyperbolic 3-space by considering spectral properties of representative complex matrices. For example, parabolic transformations are conjugate to rigid translations in the upper half-space model, and they are exactly those transformations that can be represented by unipotent upper triangular matrices.

    1. ^ ab"Curvature of curves on the hyperbolic plane". math stackexchange . Retrieved 24 September 2017 .
    2. ^
    3. Hyde, S.T. Ramsden, S. (2003). "Some novel three-dimensional Euclidean crystalline networks derived from two-dimensional hyperbolic tilings". The European Physical Journal B. 31 (2): 273–284. CiteSeerX10.1.1.720.5527 . doi:10.1140/epjb/e2003-00032-8.
    4. ^ ab
    5. Sommerville, D.M.Y. (2005). The elements of non-Euclidean geometry (Unabr. and unaltered republ. ed.). Mineola, N.Y.: Dover Publications. p. 58. ISBN0-486-44222-5 .
    6. ^
    7. Ramsay, Arlan Richtmyer, Robert D. (1995). Introduction to hyperbolic geometry . New York: Springer-Verlag. pp. 97–103. ISBN0387943390 .
    8. ^ See for instance,
    9. "Omar Khayyam 1048–1131" . Retrieved 2008-01-05 .
    10. ^
    11. "Non-Euclidean Geometry Seminar". Math.columbia.edu . Retrieved 21 January 2018 .
    12. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447–494 [470], Routledge, London and New York:

    "Three scientists, Ibn al-Haytham, Khayyam and al-Tūsī, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between this postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."


    Parallel Postulates

    Let the following be postulated: that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

    — Euclid, Elements , Postulate 5 [ A : Euclid]

    Parallel Lines on the Plane Are Special

    Up to this point we have not had to assume anything about parallel lines. No version of a parallel postulate has been necessary either on the plane or on a sphere or on a hyperbolic plane. We defined the concrete notion of parallel transport and proved in Problem 8.2 that on the plane parallel transported lines do not intersect. Now in this chapter we will look at two important properties on the plane:

    If two lines on the plane are parallel transports of each other along some transversal, then they are parallel transports along any transversal. (Problem 10.1 )

    On the plane the sum of the interior angles of a triangle is always 180 °. (Problem 10.4 )

    Neither of these properties is true on a sphere or on a hyperbolic plane and thus both need an additional assumption on the plane for their proofs. The various assumptions that permit proofs of these two statements are collectively termed the Parallel Postulates . Only the two statements above are needed from this chapter for the rest of the book. Therefore, it is possible to omit this chapter and assume one of the above two statements and then prove the other. However, parallel postulates have a historical importance and have a central position in many geometry textbooks and in many expositions about non-Euclidean geometries. The problems in this chapter are an attempt to help people unravel and enhance their understanding of parallel postulates. Comparing the situations on the plane with a sphere and a hyperbolic plane is a powerful tool for unearthing our hidden assumptions and misconceptions about the notion of parallel on the plane.

    Since we have so many (often unconscious) connotations and assumptions attached to the word "parallel," we find it best to avoid using the term parallel as much as possible in this discussion. Instead we will use terms like "parallel transport," "non-intersecting," and "equidistant."

    Problem 10.1. Parallel Transport on the Plane

    Show that if l 1 and l 2 are lines on the plane such that they are parallel transports along a transversal l , then they are parallel transports along any transversal. Prove this using any assumptions you find necessary. Make as few assumptions as you can, and make them as simple as possible. Be sure to state your assumptions clearly.

    What part of your proof does not work on a sphere or on a hyperbolic plane ?

    This problem is by no means as trivial as it, at first, may appear. In order to prove this theorem, you will have to assume something — there are many possible assumptions, so use your imagination. But at the same time, try not to assume any more than is necessary. If you're having trouble deciding what to assume, try to solve the problem in a way that seems natural to you and see what develops.

    On a sphere, try the same construction and proof you used for the plane. What happens? You should find that your proof does not work on a sphere or on a hyperbolic plane. So, what is it about your proof (or a sphere and hyperbolic plane) that creates difficulties?

    Again, you may be tempted to use "the sum of the angles of a triangle is 180 ° " as part of your proof. As in many other cases before, there is nothing wrong with doing the problems out of order — you can use Problem 10.4 to prove Problem 10.1 as long as you don't also use Problem 10.1 to prove Problem 10.4 . Most people find it much easier to prove Problem 10.1 , first, and then use it to prove Problem 10.4 .

    Problem 10.1 emphasizes the differences between parallelism on the plane and parallelism on a sphere. On the plane, non-intersecting lines exist, and one can "parallel transport" everywhere. Yet, as was seen in Problems 8.2 and 8.3 , on a sphere two lines are cut at congruent angles if and only if the transversal line goes through the center of the lune formed by them. That is, on a sphere two lines are locally parallel only when they can be parallel transported through the center of the lune formed by them. Be sure to draw a picture of the lune locating the center and the transversal. On a sphere (and on a hyperbolic plane) it is impossible to slide the transversal along two parallel transported lines keeping both angles constant — which is something you can do on the plane. In Figure 10.1a and Figure 10.1b, the line t' is a parallel transport of line t along line l , but it is not a parallel transport of t along l ¢ .

    Figure 10.1 a. Parallel transport on a sphere along l , but not along l'.

    Figure 10.1b. Parallel transport on a hyperbolic plane along l but not along l'.

    Pause, explore, and write out your ideas before reading further.

    Parallel Circles on a Sphere

    Figure 10.2. Special equidistant circles.

    The latitude circles on the earth are sometimes called "Parallels of Latitude." They are parallel in the sense that they are everywhere equidistant as are concentric circles on the plane. In general, transversals do not cut equidistant circles at congruent angles. However, there is one important case where transversals do cut the circles at congruent angles. Let l and l ¢ be latitude circles which are the same distance from the equator on opposite sides of it. See Figure 10.2. Then, every point on the equator is a center of half-turn symmetry for these pair of latitudes. Thus, as in Problems 8.3 and 10.1 , every transversal cuts these latitude circles in congruent angles.

    Parallel Postulates

    One of Euclid's assumptions constitutes Euclid's Fifth ( or Parallel ) Postulate ( EFP ) , which says:

    If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles .

    For a picture of EFP, see Figure 10.3.

    Figure 10.3 . Euclid's Parallel Postulate.

    You probably did not assume EFP in your proof of Problem 10.1 . You are in good company — many mathematicians, including Euclid, have tried to avoid using it as much as possible. However, we will explore EFP because, historically, it is important, and because it has some very interesting properties as you will see in Problem 10.3 . On a sphere, all straight lines intersect twice which means that EFP is trivially true on a sphere. But in Problem 10.3 , you will show that EFP is also true in a stronger sense on spheres. You will also be able to prove that EFP is false on a hyperbolic plane.

    Thus, EFP does not have to be assumed on a sphere — it can be proved! However, in most geometry text books, EFP is substituted by another postulate which, it is claimed, is equivalent to EFP. This postulate is Playfair's Parallel Postulate ( PPP ) , and it can be expressed in the following way:

    For every line l and every point P not on l , there is a unique line l ¢ which passes through P and is parallel to l .

    Figure 10.4 . Playfair's Parallel Postulate.

    Note that, on a sphere, since any two great circles intersect, there are no lines l ¢ which are parallel to l in the "not intersecting" sense. Therefore, Playfair's Postulate is not true on spheres. On the other hand, if we change "parallel" to "parallel transport" then every great circle through P is a parallel transport of l along some transversal. What happens on a hyperbolic plane? In Problem 10.2 , you will explore the relationships among EFP, Playfair's Postulate, and the assumptions you used in Problem 10.1 .

    Problem 10.2. Parallel Postulates on the Plane

    On the plane, are EFP, Playfair's Postulate, and your postulate from Problem 10.1 equivalent ? Why ? Or, why not ?

    To show that EFP and Playfair's Postulate are equivalent on the plane, you need to show that you can prove EFP if you assume Playfair's Postulate and vice versa. Do the same for your postulate from Problem 10.1 . If the three postulates are equivalent, then you can prove the equivalence by showing that

    EFP Þ PPP Þ Your Postulate Þ EFP

    or in any other order. It will probably help you to draw lots of pictures of what is going on. Note that PPP is not true on a sphere but EFP is true, so therefore your proof that EFP implies PPP on the plane must use some property of the plane that does not hold on a sphere. Look for it.

    Problem 10.3. The P P on Sphere & Hyperbolic Plane

    a. Show that EFP is true on a sphere in a strong sense that is, if lines l and l ¢ are cut by a transversal t such that the sum of the interior angles a + b on one side is less than two right angles, then, not only do l and l ¢ intersect, but they also intersect "closest" to t on the side of a and b . You will have to determine an appropriate meaning for "closest . "

    To help visualize the postulates, draw these "parallels" o n an actual sphere. There are really two parts to this proof — first, you must come up with a definition of "closest" and, then, prove that EFP is true for this definition. The two parts may come about simultaneously as you come up with a proof. This problem is closely related to Euclid's Exterior Angle Theorem, but can also be proved without using EEAT. One case that you should look at specifically is pictured in Figure 10.5. It is not necessarily obvious how to define the "closest" intersection.

    Figure 10.5 . Is EFP true on a sphere?

    b. On a hyperbolic plane let l be a geodesic and let P be a point not on l , then show that there is an angle q with the property that any line l ¢ passing through P is parallel to (not intersecting) l if the line l ¢ does not form an angle less than q with the line from P which is perpendicular to l . (See Figure 10.6.)

    Figure 10.6. Multiple parallels on a hyperbolic plane.

    c. Using the notion of parallel transport, change Playfair's Postulate so that the changed postulate is true on both spheres and hyperbolic planes. Make as few alterations as possible and keep some form of uniqueness.

    d. Either prove your postulate from Problem 10.1 on a sphere and on a hyperbolic plane or change it, with as few alterations as possible, so that it is true on these surfaces. You may need to make different changes for the two surfaces.

    Suggestions for b. and c.

    Above we noted that Playfair's Postulate is not true on a sphere or a hyperbolic space, and in Problem 10.1 you should have decided whether or not your postulate is true on spheres or on hyperbolic spaces. The next step is to come up with a modified versions of the postulate, that are true on a sphere or on a hyperbolic plane. Try to limit the modifications you make so that the new postulate preserves the spirit of the old one. You can draw ideas from any of the previous problems to obtain suitable modifications. Then, prove that your modified versions of the postulate are true.

    Parallelism in Spherical and Hyperbolic Geometry

    Playfair's Postulate assume both the existence and uniqueness of parallel lines. In Problem 8.2 , it was proven that if one line is a parallel transport of another, then the lines do not intersect on the plane or on a hyperbolic plane that is, they are parallel. Thus, it is not necessary to assume the existence of parallel lines. On a sphere any two lines intersect. However, in Problem 8.4 we saw that there are non-intersecting lines that are not parallel transports of each other on a hyperbolic plane and on any cone with cone angle larger than 360 ° .

    Figure 10.6 is an attempt to represent the relationships among parallel transport, non-intersecting lines, EFP, and Playfair's Postulate. Can you fit your postulate into the diagram?

    Figure 10.7. Parallelism.

    Problem 10.4. Sum of the Angles of a Planar Triangle

    a. What is the sum of the angles of a triangle on the plane ?

    b. Show that the postulates in Problem 10.2 are equivalent to " The sum of the angles of a triangle on the plane is a constant ."

    c. What happens to your proof on spheres and hyperbolic planes ?

    There are many approaches to this problem. It can be done using only results from this chapter or results from other chapters may also be used. Be sure to draw pictures and to be careful about what previous results you are using.

    Remember: If you assumed facts about the sum of the angles of a triangle on the plane in a previous problem, then you can not use the results of that problem here.


    10.1: Plane - Mathematics

    MATH 21C (SECTIONS C01,C02), 212 Viehmeyer, 9-9:50 MWF, Thursday Discussions

    Last Updated: December 10, 2017


    Text: Thomas' Calculus: Early Transcendentals (13th edition) by Weir, Hass, Giordano

    My Office Hours: Variable (Announced in class or sent via UC Davis e-mail)

    TA OFFICE HOURS Monday 5-7 p.m. Arturo Palomino Calculus Room-- 1118 MSB
    Tuesday 4:30-5:30 p.m. Arturo Palomino 1011 Ghausi
    Thursday 4-6 p.m. Douglas Sherman 3240 MSB

        EXAM 1-- MONDAY, October 16, 2017

      The course will likely cover the following sections in our textbook : 10.1-10.10, 12.1-12.5, and 14.1-14.8.

      Here you can find all Math 21C Homework Solutions and Exam Solutions .

      Here are Math 21C Practice Exams .

      This is an OPTIONAL EXTRA CREDIT survey

      Click here for additional optional PRACTICE PROBLEMS with SOLUTIONS found at

      Here are some TIPS for doing well on my exams.

      The following homework assignments are subject to minor changes.

          • 6-8 -- Determine convergence or divergence of series using various series tests
          • 1 -- Alternating series
          • 1 -- Epsilon,N Proof
          • 1 or 2 -- (*) or (*)(*) problem
          • 1 or 2 -- Others
          • 1 -- OPTIONAL EXTRA CREDIT
            • 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
            • 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
            • 2.) You will be graded on proper use of limit notation.
            • 3.) You will be graded on proper use of derivative and integral notation.
            • 4.) Put units on answers where units are appropriate.
            • 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

            SOLUTIONS TO EXAM 1 ARE POSTED ON THIS WEBPAGE.

            THE GRADING SCALE FOR FALL 2017 EXAM 1 IS :

            SCANNED PROBLEMS for Chapter 12 (Sections 12.1-12.5)

                • 1 -- Find interval of convergence for power series
                • 2 -- Find 1st 3 nonzero terms of Taylor Series centered at x=a
                • 1 -- Lagrange form of the Taylor remainder.
                • 1 -- Use Taylor Polynomial to compute an estimate
                • 5 -- Problems involving lines, planes, angles, normal vectors, parallel vectors, points of intersection, etc.
                • 1 -- Other
                • 1 -- OPTIONAL EXTRA CREDIT
                  • 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
                  • 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
                  • 2.) You will be graded on proper use of derivative and integral notation.
                  • 3.) Put units on answers where units are appropriate.
                  • 4.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

                  SOLUTIONS TO EXAM 2 ARE POSTED ON THIS WEBPAGE.

                  THE GRADING SCALE FOR FALL 2017 EXAM 2 IS :

                  SCANNED PROBLEMS for Chapter 14 (Sections 14.1-14.6)

                      • 1 -- 3D Graphing (intercepts, traces, level curves)
                      • 1 -- Domain and Range
                      • 2 or 3 -- Limits
                      • 1 -- Compute various partial derivatives
                      • 1 or 2 -- Chain Rule
                      • 1 or 2 -- Directional Derivative
                      • 1 -- Epsilon,Delta Proof
                      • 1 -- Other
                      • 1 -- OPTIONAL EXTRA CREDIT
                        • 0.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
                        • 00.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT's EXAM.
                        • 1.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
                        • 2.) You will be graded on proper use of derivative and integral notation.
                        • 3.) Put units on answers where units are appropriate.
                        • 4.) Do not use any shortcuts from the book when using the method of integration by parts.
                        • 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

                        SOLUTIONS TO EXAM 3 ARE POSTED ON THIS WEBPAGE.

                        THE GRADING SCALE FOR FALL 2017 EXAM 3 IS :

                          SCANNED PROBLEMS for Chapter 14 (Sections 14.7. 14.8)

                        The FINAL EXAM is Monday, December 11, 2017

                        BRING A PICTURE ID TO THE EXAM
                        AND BE PREPARED TO SHOW IT TO KOUBA OR THE TEACHING ASSISTANT !!

                        The final exam will cover handouts, lecture notes, and examples from class, homework assignments 1 through 23 (Omit problems 31, 34, and 41 and omit HW #24.), and material from sections 10.1-10.10, 12.1-12.5, 14.1-14.7 (Omit Section 14.8.), and discusssion sheets 1-10.

                        TYPES OF QUESTIONS FOR THE FALL 2017 FINAL EXAM (THIS IS SUBJECT TO UNANNOUNCED CHANGES.). The following topics will NOT BE COVERED on this final exam -- Taylor Error (Remainder), 3D-graphing, epsilon/delta proofs, epsilon/N proofs, and 3D-limits.


                        Application of Derivatives

                        Or, y + $Delta $y = $frac<1><2>$(x + $Delta $x) 2 + 3(x + $Delta $x).

                        Or, $Delta $y = $frac<1><2>$(x + $Delta $x) 2 + 3(x + $Delta $x) &ndash ($frac<<<< m>^2>>><2>$ + 3x).

                        dy = f&rsquo(x).dx = (x + 3).dx = (2 + 3) * 0.5 = 2.5

                        or, $Delta $y &ndash dy = 2.625 &ndash 2.5 = 0.125.

                        $Delta $x = dx = 1% of x = 0.01x.

                        Approximate increase in volume,

                        dv = 3x 2 .dx = 3 * x 2 * 0.01x = 0.03x 3 m 2 .

                        x = 5, $Delta $x = dx = 5.01 &ndash 5 = 0.01

                        dy = 3x 2 .dx = 3 * (5) 2 * 0.01 = 0.75.

                        Or, $Delta $x = dx = 0.98 &ndash 1 = - 0.02.

                        dA = 2&pir.dr = 2&pi * 5 * 0.06 = 0.6&picm 2 .

                        So, approximate increase in area = 0.6&pi cm 2 .

                        Or, $Delta $A = &pi(r 2 + 2r.$< m<: >>Delta $r + ($Delta $r) 2 ) &ndash &pir 2 = &pi.$< m<: >>Delta $r(2r + $Delta $r)

                        = &pi * 0.06(2 * 5 + 0.06) = &pi * 0.06 * 10.06 = 0.6036 &pi cm 2 .

                        So, actual increase in area = 0.6036 &pi cm 2 .

                        $Delta $x = dx = 10.01 &ndash 10 = 0.01 cms

                        dA = 12x.dx = 12 * 10 * 0.01 = 1.2cm 2 .

                        So approximate increase in area = 1.2 cm 2 .

                        A + $Delta $A = 6(x + $Delta $x) 2 .

                        $Delta $A = 6[x 2 + 2x$Delta $x + ($Delta $x) 2 ] &ndash 6x 2 = 6$Delta $x(2x + $Delta $x)

                        = 6 * 0.01 (2 * 10 + 0.01) = 6 * 0.01 * 20.01 = 1.2006

                        Actual increase in area = 1.2006

                        r = radius = 2 r + $Delta $r = 2.1

                        or, $Delta $r = 2.1 &ndash r = 2.1 &ndash 2 = 0.01.

                        dv = $frac<4><3>$.&pi.3r 2 .dr = 4&pi * (2) 2 * 0.1 = 1.6&pi

                        So, approximate increase in volume = 1.6&pi.

                        or, v + $Delta $v = $frac<4><3>$.&pi(r + $Delta $r) 3 .

                        Actual increase in volume = $frac<4><3>pi * 1.261 = $frac<<5.044>><3>pi.


                        10.1: Plane - Mathematics

                        Final Exam Sunday May 7
                        7:45am - 9:45am
                        B115 Van Vleck Hall

                        T 1-17 8.1 Quiz: draw cylindar, cube,sphere,torus, square pyramid, plane intersecting a sphere

                        R 1-19 8.1 intersecting plane and cube
                        Hmwk 8.1 - 4,5,6

                        T 1-24 8.2 angles, law optics
                        Hmwk 8.2 - 4,8,9
                        Quiz: sketch the intersection of vertical and horizontal planes with a solid torus

                        R 1-26 8.3 circles and spheres
                        Hmwk 8.3 - 1,2,3

                        T 1-31 8.4, 8.5 triangles and polygons
                        class activities: 8.1, 8.2

                        R 2-2 8.6 constructions with straight edge and compass
                        Hmwk 8.4 - 4,5,8

                        T 2-7 8.6 constructing paraellel lines, trisecting a line segment
                        class activities 8.3, 8.4
                        Hmwk p.365 8.5 - 4
                        Hmwk p.371 8.6 - 4,6

                        R 2-9 8.7 polyhedron - assigned class activities
                        Here are some paper models for the group doing activity 8-OO
                        pyr . hex . oct . tri
                        They come from the web site www.korthalsaltes.com

                        T 2-14 9.1 Isometries of the plane
                        Hmwk p.381-2 . 8.7 - 1,9,12

                        R 2-16 class activities 8.5-6-7, glide reflections - foot prints in the snow.

                        Exam 1 will be in class on Thursday Feb 23. It will cover Chapter 8.

                        T 2-28 9.2 plane isometries, symmetry
                        Hmwk p.391 9.1-5
                        Hmwk p.407 9.2-6,16

                        R 3-2 9.3 congruence SSS SAS
                        class activities assigned - new groups formed

                        T 3-7 9.4 similar triangles
                        Hmwk p.413 9.3-3
                        Hmwk p.427 9.4-6,14

                        R 3-9 symmetry groups and symmetric designs
                        class activities 9.3,4
                        Hmwk p.407 9.2-2,5,8

                        T 3-21 9.4 similar triangle, pythagorus, ancient greek astronomy
                        Hmwk p.428 9.4- 7,12,17
                        Euclid's proof of Pythagorus using similar triangles ,
                        this paper appeared in American Math Monthly, March 2006.
                        How high is the moon?

                        R 3-23 10.1 units of measure
                        assigned class activities

                        T 3-28 10.1,2,3 error, dimension
                        Hmwk p. 442 10.1-2
                        Hmwk p. 447 10.2-4,6

                        R 3-30 10.4,5 perimeters, comparisons
                        Class activities 10.1-2
                        Hmwk p. 453 10.3-1
                        Hmwk p. 458 10.4-5
                        Hmwk p. 460 10.5-1

                        T 4-4 10.6 converting units
                        Hmwk p. 468 10.6-6,8,11

                        R 4-6 Pick's formula, area of parallelogram, triangle, circle

                        T 4-11 Review for exam
                        class activities 10.3-6

                        T 4-18 cutting and moving 11.1, Pythagorus 11.2
                        Hmwk p.481 11.1-3
                        Hmwk p.494 11.2-2,3

                        R 4-20 cutting areas into little rectangles, Cavalieri's principle
                        Hmwk p. 498 11.3-3
                        Hmwk p. 504 11.4-6
                        Hmwk p. 511 11.5-1

                        T 4-25 11.10 volumes of cylindars and cones
                        class projects chapter 11 due Tue May 2
                        Hmwk p. 519 11.6-3 due last day of class Thur May 4
                        Hmwk p. 525 11.7-1
                        Hmwk p. 540 11.10-6,8


                        Watch the video: Intersecting Planes, Lines, and Solutions Part 2 (November 2021).