# 7.1: Prelude to Analytic Geometry

The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone. Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry. It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular. He presumed that the planets moved in circular orbits around Earth, and for nearly (2000) years this was the commonly held belief.

It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path. In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems.

## An Introduction to Analytic Geometry and Calculus

An Introduction to Analytic Geometry and Calculus covers the basic concepts of analytic geometry and the elementary operations of calculus. This book is composed of 14 chapters and begins with an overview of the fundamental relations of the coordinate system. The next chapters deal with the fundamentals of straight line, nonlinear equations and graphs, functions and limits, and derivatives. These topics are followed by a discussion of some applications of previously covered mathematical subjects. This text also considers the fundamentals of the integrals, trigonometric functions, exponential and logarithm functions, and methods of integration. The final chapters look into the concepts of parametric equations, polar coordinates, and infinite series. This book will prove useful to mathematicians and undergraduate and graduate mathematics students.

## 8 Introduction to Analytic Geometry

The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone. Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry.

It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular. He presumed that the planets moved in circular orbits around Earth, and for nearly 2000 years this was the commonly held belief.

It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path.

In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems.

## Introduction to Analytic Geometry

The Greek mathematician Menaechmus (c. 380–c. 320 BCE) is generally credited with discovering the shapes formed by the intersection of a plane and a right circular cone. Depending on how he tilted the plane when it intersected the cone, he formed different shapes at the intersection–beautiful shapes with near-perfect symmetry.

It was also said that Aristotle may have had an intuitive understanding of these shapes, as he observed the orbit of the planet to be circular. He presumed that the planets moved in circular orbits around Earth, and for nearly 2000 years this was the commonly held belief.

It was not until the Renaissance movement that Johannes Kepler noticed that the orbits of the planet were not circular in nature. His published law of planetary motion in the 1600s changed our view of the solar system forever. He claimed that the sun was at one end of the orbits, and the planets revolved around the sun in an oval-shaped path.

In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems.

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Let $(a,b)$ be the center of the circle. Since the center lies on the line $y=3-x$, then we have $b=3-a$. The equation of the circle if the center on $(a,b)$ and radius $r$ is $(x-a)^2+(y-b)^2=r^2 ag1$ and the equation of its tangent line is $(x_c-a)(x-a)+(y_c-b)(y-b)=r^2, ag2$ where $(x_c,y_c)$ is point of contact. Equation $(2)$ can be written as $(x_c-a)x+(y_c-b)y=r^2+a(x_c-a)+b(y_c-b). ag3$ The tangent lines are $-x+2y=22 ag4$ and $2x+y=11. ag5$ Using $b=3-a$, comparing $(3)$ and $(4)$ yields $x_1-a=-1$, $y_1-b=2$, and $egin r^2+a(x_1-a)+b(y_1-b)&=22 r^2+a(-1)+b(2)&=22 r^2-a+2(3-a)&=22 r^2-3a&=16. ag6 end$ Similarly, comparing $(3)$ and $(5)$ yields $x_2-a=2$, $y_2-b=1$, and $egin r^2+a(x_2-a)+b(y_2-b)&=11 r^2+a&=8. ag7 end$ Solving $(6)$ and $(7)$ yields $a=-2$, $b=5$, and $r^2=10$. Thus, using $(1)$, the equation of the circle is $Largecolor<(x+2)^2+(y-5)^2=10>.$

Hint: Can you use the two tangent lines you have to find a line which must pass through the centre of the circle?

I suggest sketching a diagram.

And I suggest you give more detail of what you know about tangents and circles, and what you have tried.

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## Key Equations

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## 3010tangents

Analytic geometry is the study of geometry using a coordinate system. Basically it’s the idea of expressing geometric objects such a as a line or a plane as an algebraic equation, think y=mx+b or ax+by+cz=k. This may be done by use of the more familiar Cartesian coordinates, by something such as polar coordinates or by just about any system for defining coordinates in a Euclidean space. The Common Core has the concept of graphing introduced in 5 th grade, and graphing simple functions in the 8 th grade. It’s quite interesting that something which took brilliant men so long to develop is now introduced to ten year olds.

The earliest evidence of anything resembling analytic geometry was by the Geek mathematician Menaechmus (380–320 BC), who was a student of Eudoxus and a tutor of Alexander the Great. Proclus and Eutocius both report that Menaechmus discovered the ellipse, hyperbola and parabola and that these were initially called the “Menaechmian triad”. These were used along with something resembling analytic geometry to solve the Delian problem, which is to, given the edge of a cube to construct the edge of a cube with double the volume. Though most of what we know of Menaechmus and his exact solution is second hand as his original work was lost, it appears as though he argued his solution for doubling the cube with proportions of a side length to the area of a side which fairly quickly leads to conics.

Another early manifestation of analytic geometry was by Omar Khayyám, whom we have mentioned in class. He drew a connection between algebra and geometry in his solution of general cubic equations. His idea to do this was to create a geometrical construction of a cubic equation by considering the variable to be the edge of a cube and constructing a set of curves from which a solution could be discerned. While it might seem far flung from Cartesian coordinates it was a significant leap in connecting the separate concepts of algebra and geometry.

Analytic geometry was more or less formalized in the early 17 th century independently by René Descartes and Pierre de Fermat. Descartes published first and so he is commonly credited as the sole creator which leads to analytic geometry often being call Cartesian geometry. As Fermat has already been much discussed, I’ll skip his background and instead jump to Descartes. René Descartes was a French mathematician and philosopher who is most well known as the (co-)creator of analytic geometry and as the father of modern philosophy. He is the origin of the well-known quote “Je pense, donc je suis” or “I think, therefore I am” which appeared in in Discours de la methode (Discourse on the Method).

While the Fermat and Descartes constructions are equivalent, they did differ in several ways which primarily stem from which direction their creator worked. Fermat started with the algebraic equation and described the analogous geometric curve while Descartes worked in reverse, starting with the curve and finding the equation. To contrast the methods, the way most of us learn analytic geometry is much more similar to Fermat than to Descartes, where we learn to recognize that a degree 1 polynomial will represent a straight line then we learn how to find that line, next that quadratic function represents a parabola and so on. Whereas if we were to learn as Descartes’ work, we would take a straight line then learn that it represented a degree 1 polynomial which is similar to Fermat. But then working further in this direction, it doesn’t make sense to jump to parabolas and instead to talk about conics and all degree 2 polynomials with no reason to talk specifically about parabolas.

In 1637, Descartes published his method of connecting arithmetic, algebra, and geometry in the appendix La géométrie (The Geometry) of Discourse on the Method. However, given Descartes’s opaque writing style (to discourage “dabblers”) as well as The Geometry being written in French rather than in the more common (for academic purposes) Latin, the book was not very well received until it was translated into Latin in 1649, by Frans van Schooten, with the addition of commentary clarifying certain arguments. Interestingly, though Descartes is credited with the invention of the coordinate plane, since he describes all necessary concepts, no equations are in fact graphed in The Geometry and his examples used only one axis. It was not until its translation into Latin that the concept of 2 axes was introduced in Schooten’s commentary.

One of the most important early uses for analytic geometry was to help prove the validity of the heliocentric theory of planetary motion, the (then) theory that the planets orbited around the Sun. As analytic geometry was one of the first methods one could use to actually make computations about curves, it was used to model elliptical orbits so as to demonstrate the correctness of this theory. Analytical geometry, and particularly Cartesian coordinates, were instrumental in the creation of calculus. Just consider how you might calculate something like the “area under the curve” without the concept of the curve being described by some algebraic equation. Similarly, the idea of rate of change of as function of time at a particular time becomes much clearer when thought of as the slope of the tangent line, but to do this, we need to think of the function as having some representation in the plane for which we need analytic geometry.

Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics) Jul 7, 1999

Of course, there is an area called Real Analytic Geometry, dealing with real analytic spaces. It can be traced back to a seminal paper by H. Cartan (Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85, 1957, 77-99). One main difference in the real setting is the lack of coherence, a basic result in complex analytic spaces. A serious difficulty coming from this lack is that a set that is locally described by real analytic equations need not have global analytic equations. This lead to the introduction of global analytic or C-analytic sets by H. Whitney and F. Bruhat in Quelques propriétés fondamentales des ensembles analytiques réels (Comment. Math. Helv. 33,1959, 132-160), a fundamental paper. Another difficulty settled in this paper is the notion of irreducibility. In the complex setting this amounts to connectedness of the singular locus, while this topological condition is not enough for real analytic sets also striking is that a real irreducible analytic set can consists of pieces of different dimensions. Thus we see from the very basis of the theory the peculiarities of the real category.

A somehow naive difference comes from the fact that reals have an order structure. Thus we can consider $ge0$ and not only $=0$ to describe sets. This remark gives way to a whole new notion: semianalytic sets, introduced by S. Lojasiewicz (Ensembles semi-analytiques. I.H.E.S. Bures-sur-Yvette, 1964) in his studies of distributions. There is nothing like this in the complex realm! In the same vein, note that any system of real equations $f_1=cdots=f_r=0$ can be replaced by the single equation $f_1^2+dots+f_r^2=0$. Alas, is everything hypersurface? Not, the complex fact that one equation gives always a codimension 1 set fails over the reals: even the empty set can be described by a single equation ($x^2+1=0$). What is behind is the radical Nullstellensatz, again failing over the reals.

Also important, in the real setting there is no proper mapping theorem dealing with images of analytic sets. The failure of this important complex tool gives rise to a new class of sets, called subanalytic. They were introduced by H. Hironaka at the beginning of the 1970s he studied them systematically using his desingularization theorems.

Third, it is worth remarking that in the real category everything is affine. Real projective spaces and real grassmannians can be analytically embedded in some $mathbb^n$, in fact, algebraically embedded. As a consequence, in the real category everything is Stein, that is, there are a plenty of analytic functions to do things. For instance, to represent objects from Algebraic Topology (homology, cohomology, homotopy classes) using analytic data.

One can consider also singularities of real analytic functions and maps as part of the field. Real Algebraic Geometry is an included area, but this is more formal than practical. In any case all these areas named REAL have a very strong connection with differential topology. as the Nash(-Tognoli) theorem quoted by @Matt E shows. And Nash functions are also a relevant subarea since M. Artin and B. Mazur called attention to them.

One point is that complex analytic structures are much more rigid than the corresponding real ones. On the other hand real analytic (or algebraic) structures include always the tool of complexification (as real numbers are the real part of complex numbers), through which one always analyses matters. Some people even say that real means just complex plus an involution (conjugation, so to say). In this sense all is a part of Complex Analytic Geometry, and any real expert will agree that seeing real objects as part of complex objects is always essential. All in all, there is a wealth of research literature in what we can call Real Analytic Geometry.

## Tangents and normals

### Tangent lines and planes

<<#invoke:main|main>> In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope fTemplate:'(c) where fTemplate:' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.

As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.

Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized see Tangent space.

### Normal line and vector

<<#invoke:main|main>> In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the three-dimensional case a surface normal, or simply normal, to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality.