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Samantha Klein 2009 the Conic Sections Samantha Klein 2009 The Conic Sections Have you ever heard someone say, “ I was great at algebra but terrible at geometry” or visa versa? In mathematics there are topics taught in a geometry class or an algebra class. The conics sections are one of the topics in mathematics that can be taught in either a geometry or algebra class. Each of the conics has both a geometric definitions and an algebraic definition. In this paper we will look at the definitions of each conic section and proofs of how the definitions are related to each other. Geometric definitions: A conic section is a curve created by the intersection of a plane and a right conical surface. There are four curves that can be created when intersecting a plane with a cone (conical surface), a circle, ellipse, parabola and hyperbola. Some argue that the circle is a distinct fourth conic section, other argue that the circle is a special case of the ellipse; for this paper the circle will be considered a special case of the ellipse. Changing the angle of the intersecting plane with the right cone creates the conic sections. Let’s look at this definition in more detail. First start by creating a cone. To create a cone you need to start with a vertical line called an axis. Then a second line called a generator will intersect the axis at some angle, called the vertex angle or . Let the generator be attached to the axis such that if you were to grab the top of the axis and spin it around 360o, the generator will also rotate with the axis, thus creating a conical surface or cone with two nappes and a vertex where the two nappes meet. QuickTime™ and a decompressor are needed to see this picture. GENERATOR Once the cone is formed we can then cut it with a flat plane at certain angles to create the conic sections. A plane that is parallel to the generator (side of the cone) will produce an open curve called a parabola. If the plane cuts through the cone at an angle with the axis that is between perpendicular to the axis and parallel to the generator then the result is a closed curve known as an ellipse. The special case of the ellipse, the circle, is formed when the plane’s angle is perpendicular to the axis. In all the above cases the plane will only cut through one nappe of the cone. The last conic, the hyperbola, is the one conic that is produced by cutting through both nappes of a cone. In order to cut through both nappes of the cone the angle between the plane and the axis must be smaller than the vertex angle. (See Figure 1) - 1 - Samantha Klein Figure 1 2009 Parabola Ellipse Circle Hyperbola Angle: Angle: Special case of Angle: Parallel to Between ellipse Smaller than generator (side Perpendicular to Angle: vertex angle of cone) axis and parallel Perpendicular to generator to axis The parabola, ellipse, circle and hyperbola are all curves that are formed by planes cutting through the nappes of a cone. These planes never go through the vertex of the cone. If the plane does go through the vertex of a cone then we create what are called the degenerate conics. The degenerate conics consist of a point, line and two intersecting lines. If the plane is perpendicular to the axis and goes through the vertex we will create a point. If the plane is parallel to the side of the cone and goes through the vertex we will get a line. Last if the plane is not parallel to the side or perpendicular to the axis and goes through the vertex then the resulting degenerate conic is two intersecting lines. (See Figure 2) Figure 2 QuickTime™ and a QuickTime™ and a QuickTime™ and a decompressor decompressor decompressor are needed to see this picture. are needed to see this picture. are needed to see this picture. Point Line Two intersecting lines According to “The History of Greek mathematics, Volume II “ by Sir Thomas Little Heath the discovery of the conics is said to have been by a Greek mathematician Menaechmus in about 360 or 350 B.C. (page 116). He discovered the conics in his attempts to solve one of the three problems of antiquity, doubling the cube (page 110). The construction of the conics that Menaechmus studied were a little different than what - 2 - Samantha Klein 2009 we think of as the conics today. In order for Menaechmus to get a desired conic section he would leave the plane at a right angle to one of the generators and then change the angle of the cone, where as today we change the angle of the plane and use a right cone to create the conic sections. Apollonius of Perga known as “The Great Geometer” published his great work Conics in eight volumes. The first four books of the Conics is said to be a compilation of all the work done before him. Although according to Apollonius himself the books are “... worked out more fully and generally than in the writings of others.” It is in this work Conics that Apollonius introduces the terms we use today of parabola, ellipse and hyperbola. He was also credited with changing the idea that conic sections could be formed by changing the angle of the plane as opposed to changing the angle of the cone. That is he states what we use today as a geometric definition of the conic sections. Among other famous Greek mathematician to study the conic sections were Aristaeus and Euclid. Euclid made a compilation or rearrangement of all the work on conics known to him in his time (Heath, 116). According to “Selections Illustrating the History of Greek mathematics, Volume II” by Ivor Bulmer-Thomas, “Aristaeus was obviously more original and more specialized; that of Euclid was admittedly a compilation largely based on Aristaeus” (page 281). According the Pappus the first four books of Apollonius’s Conics were based on the first four books of Euclid’s work (Heath, 119). Interestingly, it is noted by Pappus that the focus – directrix property must have been known to Euclid, and probably to Aristaeus, but is never mentioned in Apollonius’s Conics. The focus- directrix property is what we think of today as the “plane geometry” or locus definition of the conics. “ Pappus’s enunciation of the theorem is to the effect that the locus of a point such that its distance from a fixed point is a given ratio to its distance from a fixed straight line is a conic” (Heath, 119). Today we refer to this as the property of eccentricity. Before we look at the idea of eccentricity let’s first look at each individual conic and its “plane geometry” definition. In plane geometry the ellipse is defined as the locus of a point on a plane, in which the sum of the distance from two fixed points, called foci, remains a constant 2a. The circle, the special case, is defined as the locus of a point on a plane that is a fixed distance from a fixed point (its center). A hyperbola is defined as the locus of a point in a plane where the difference in the distance from any point to two fixed points, called foci, remains a constant 2a. The parabola is defined as the locus of a point on a plane equidistant from a fixed point (focus) and a fixed line (directrix). Although these locus definitions seem to have no connection to the definitions in terms of the intersection of a plane and a cone that we mentioned earlier. We know that Menaechmus knew that the conic sections were formed by a plane intersecting a cone, and according to Pappus, Euclid must have know the focus- directrix property (locus definition). Therefore one can only assume that the connection between these two definitions was known prior to the 19th century. However, it was not until 1822 when the Belgian mathematician Germinal Pierre Dandelin was accredited with finding a simple proof using tangents and spheres in a cone to prove the connections between the locus definitions and the intersections of a plane and cone for the conics. Before we can do this proof we must get some background information out of the way. We must first show the proof of why the length of two tangents drawn from an exterior point to a circle or sphere are equal. This is a key idea used by Dandelin in his proof. - 3 - Samantha Klein 2009 Proof that the length of two tangents drawn from an exterior point to a circle or sphere are equal. Given that AB and AC are tangent to circle O at points from common exterior point A. Figure 3 Then we can prove that AB = AC. First construct segment AO. Then construct BO and CO, which by definition are both radii r of circle O, so OB r OC (See Figure 3). Since a tangent is defined as a being r perpendicular to the radius then AB OB and AC OC. Now we have two right triangles ABO and ACO, which share a side AO. Since ABO and ACO are 2 2 right triangles we can use Pythagorean theorem to show that AB r 2 AO so 2 2 2 2 AB AO r 2 and AC r 2 AO so AC AO r 2 . Since 2 2 AB AO r 2 and AC AO r 2 , then AB = AC by the transitive property of equality.
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