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Chapter 6 Drawing a Circle and Conic Sections

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Chapter 6 Drawing a Circle and Conic Sections Chapter 6 Drawing a circle and conic sections Math 4520, Fall 2017 6.1 A Greek incidence theorem To properly understand projective geometry it is necessary to go beyond Desargues' Theorem and the geometry of lines alone. Greek geometry put a great deal of emphasis on the circle, and as anybody can see there are many other curves as well. However, a picture of a circle is not a circle. It is a conic section, the intersection of a cone over a circle with a plane. We look at what some of the incidence properties of a conic section are. Similar to the spirit of Desargues' Theorem we define a point hexagon as an ordered sequence of six distinct points in a projective plane (or projective 3-space), p1;:::; p6.A line hexagon will be a sequence of six distinct lines L1;:::;L6. Note that the lines through pi and pi+1, where i = 1; 2;:::; 6, define a line hexagon from a point hexagon. Conversely, the sequence of points incident to Li and Li−1 define a point hexagon from a line hexagon. Note that p1 = p7, and L1 = L7 in these definitions. We first consider a case of a \degenerate" conic section that was also considered by Pappus who lived about 300 A.D. Recall that a hyperbola degenerates into two distinct lines when the plane intersects the cone at its vertex. Theorem 6.1.1 (Pappus). If the alternate points of a (point) hexagon are incident to two fixed lines, then the opposite sides are incident to three collinear points. (See Figure 6.1.) One of the reasons for the importance of Pappus' Theorem is that it is stronger than Desargues' Theorem. In other words, by using only the notion of incidence in the plane, it is possible to give a formal proof Desargues' Theorem by three applications of the property in the statement of Pappus' Theorem. Another reason for the interest in Pappus' and Desargues' Theorems is that they can be used to provide coordinates for a formal \projective plane" over a field. Pappus' Theorem insures that the underlying field is commutative. There will be more about this later. 6.2 Cones and conic sections Let us recall the definition of a cone. A standard definition coming from Greek geometry defines a cone as set of lines from a given point p to a given circle C, where p (the vertex of 1 2CHAPTER 6. DRAWING A CIRCLE AND CONIC SECTIONSMATH 4520, FALL 2017 p p 5 p 3 1 2 CLASSICAL GEOMETRIES One of the reasons for the importance of Pappus' Theorem is that it is stronger than Desargues'Theorem. In other words, by using only the notion of incidence in the plane, it is possible to give a formal proof Desargues' Theorem by three applications of the property in the statement of Pappus' Theorem. p Another reason for the4 interest in Pappus'p and Desargues'p Theorems is that they can be used to provide coordinates for a formal "projective6 plane"2 over a field. Pappus'Theorem insures that the underlying field is commutative. There will be more about this later . Figure 6.1 7.2 Cones and conic sections Let us recall the definition of a cone. A standard definition coming from Greek geometry defines a cone as set of lines from a given point p to a given circle C, where p ( the vertex the cone) liesof the on cone) the lies line on perpendicularthe line perpendicular to theto the plane plane of the the circle circle C throughC through the center the center of C, but not inof the C Tbut plane. not in Thisthe plane. is calledThis is called a right a right circular circular cone.cone. The The intersection intersection of a plane of a plane with ".jth such a right circular cone is the geometric definition of a conic section. See Figure such a right7.2.1. circular cone is the geometric definition of a conic section. See Figure 6.2. Figure 6.2 It turns out that it is not necessary for the cone to be a right circular cone. Any cone over a conic section ( where the vertex of the cone is not necessarily on the line perpendicular to the plane through the "center" of the conic section), when intersected by a plane is again It turnsa conic out thatsection. it is not necessary for the cone to be a right circular cone. Any cone over a conic sectionLooking at (where things as the an vertexartist, this of says the that cone any is drawing not necessarily of a conic section on the is again line a perpendicular conic section. The picture plane is the intersecting plane. With this in mind we give an to the planealgebraic through definition the \center"of a conic section of the in conic the Euclidean section), plane when with Cartesian intersected coordinates. by a plane is again a conic section. Looking at things as an artist, this says that any drawing of a conic section is again a conic section. The picture plane is the intersecting plane. With this in mind we give an algebraic definition of a conic section in the Euclidean plane with Cartesian coordinates. We say that a conic section is any set S of the form S = f(x; y) j Ax2 + Bxy + Cy2 + Dx + Ey + F = 0g; where A; : : : ; F are constants. 6.3. INCIDENCE PROPERTIES OF CONICS, PAPPUS' AND PASCAL'S THEOREMS3 If S is empty, one point, oneDRAWING line, or A two CIRCLE, lines, CONIC then SECTIONS we say that S is degenerate3 . Otherwise we say S is non-degenerateWe say that a conic. section Note is thatany set a S non-degenerateof the form conic is either an ellipse, parabola, hyperbola, or circle. If we wish, we can regard S as a subset of the (points of the) projective plane and the parabola ands = hyperbola{(x, y) Ax2 +Bxy are + distinguished Cy2 + Dx + Ey +F by = their0}, having one or two points, respectively, onwhere the A, ..., line F are at constants. infinity. For a parabola, the equivalence class of lines parallel If S is empty, one point, one line, or two lines, then we say that S is degenerate. to the axis ofOtherwise symmetry we say corresponds S is non-degenerate. to Note a point that a non-degenerate at infinity conic that is either lies onan ellipse, the parabola. For a hyperbola, theparabola, two equivalencehyperbola, or circle. classes If we of wish lines we can parallel regard S toas a the subset two of the asymptotes (points of correspond to the) projective plane and the parabola and hyperbola are distinguished by their having two points atone infinity or two points, that lierespectively, on the on hyperbola. the line at infinity. For a parabola, the equivalence It turns outclass that of lines the parallel algebraic to the axis definition of symmetry andcorresponds the geometric to a point at definitioninfinity that lies of a conic section on the parabola. For a hyperbola, the two equivalence classes of lines parallel to the two are equivalent,as~rmptotes and the correspond projection to two of points a conic that lie section on the hyperbola from one at infinity. plane to another plane is a conic section in the otherIt turns plane.out that the For algebraic instance, definition if and one the looks geometric at definition a parabola of a conic receding section off to infinity, are equi'\'alent, and the projection of conic section from one plane to another plane is a then a pictureconic of section this willin the beother an plane. ellipse For instance, tangent if one to looks the at horizon a parabola inreceding the pictureoff to plane. Note that this impliesinfinity, another then a picture symmetry of this will on be thean ellipse conic tangent section to the thanhorizon onein themight picture plane. not first realize. See Note that this implies another s)rmmetry on the conic section than one might first realize. Figure 6.3 See Figure 7.2.2. Figure 6.3 The previous defmition of a conic is more accurately called a point conic, since it is a set of points. A dual notion is a line conic which consists of the collection of lines tangent to a non-degeneratepoint conic. For instan.ce,if the line conic is a parabola, the line at The previousinfinity definition is the only line of aon conicthe line isconic. more Note accurately tp,~t a line is tangent called to aa conicpoint if and conic only, since it is a set if the line intersects the (point) conic at exactly one point. Note that this definition of of points. A dualtangency notion is not so is good a line for other conic curves,which of course. consists SeeFigure of the7.2.3. collection of lines tangent to a non-degenerate point conic. For instance, if the line conic is a parabola, the line at infinity is on the line conic. The other line conics do not have the line at infinity. Note that a line is tangent to a conic if and only if the line intersects the (point) conic at exactly one point. Note that this definition of tangency is not so good for other curves, of course. See Figure 6.4. 6.3 Incidence properties of conics, Pappus' and Pas- cal's Theorems The \generalization" of Pappus Theorem to include the case of a non-degenerate conic was first discovered by a sixteen-year-old boy in the the seventeenth century.
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