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An Approach to Conic Sections 1 Introduction. 2 Intersections of Two

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An Approach to Conic Sections 1 Introduction. 2 Intersections of Two An Approach to Conic Sections Jia. F. Weng 1998 1 Introduction. The study of conic sections can be traced back to ancient Greek mathematicians, usually to Applo- nious (ca. 220-190 bc) [2]. The name ‘conic section’ comes from the fact that the principle types of conic sections, known as ellipses, hyperbolas and parabolas, are generated by cutting a cone with a plane. However, most modern textbooks on calculus depart from this geometric approach. Instead, conic sections are defined as some types of loci and studied through analytic geometry. In this paper we show a new approach to conic sections which are defined as the intersections of two cones. Then the vertices of two cones become the inherent foci of the conic section and a directrix exists associated with each of the inherent foci. All known properties of conic sections still hold for the inherent foci and their associated directrixes in this new approach. Moreover, when a conic section and its foci and directrixes in space are projected to a horizontal plane, they become the ones discussed in planar analytic geometry. This new approach seems simpler and more natural than the classical geometric and analytic approaches in defining conic sections and proving their properties. In the last section we show an application of the new approach in the network design in mining industry. In Appendix we derive the standard equation of a conic section with respect to the foci, lying on the cutting plane and referred to as the coplanar foci of the conic section. The author does not know any textbook that gives such a derivation. 2 Intersections of Two Right Circular Cones. Let xP ; yP ; zP denote the Cartesian coordinates of a point P . Suppose P and Q are two distinct points. The length of line segment PQ is denoted by l(PQ), and the gradient of PQ is denoted by g(PQ) which is defined as jz ¡ z j g(PQ) = q Q P : (1) 2 2 (xQ ¡ xP ) + (yQ ¡ yP ) Let C(P ; m) denote a (double-napped) right circular cone whose vertex is P and whose generating lines have gradient m (m > 0). The angle ¯ formed by the axis of C(P ; m) and the generating lines is referred to as the generating angle of the cone. Clearly, tan ¯ = 1=m. Now suppose A and B are two distinct points in space. The intersection of cones C(A; m) and C(B; m) is denoted by C(A; B; m). Assume the horizontal distance between A and B is 2u while the vertical distance between A and B is 2h. Then, after a transformation we may assume A = (u; 0; h);B = (¡u; 0; ¡h); u ¸ 0; h ¸ 0 (Fig. 1). Hence the equations describing C(A; m) and C(B; m) are (z ¡ h)2 (z + h)2 (x ¡ u)2 + y2 = ; (x + u)2 + y2 = : (2) m2 m2 If h = 0, then the intersection C(A; B; m) lies trivially in the YZ-plane. Assume h 6= 0. Subtracting the first equation from the second, we have u m2 z = m2 x = x; (3) h k Z ~ Z Y Y A P β VA R1 VA A α X X O O S R2 B VB VB B R1 B O A V VB A B VB VA A R2 S (1) (2) Figure 1: Intersections of two cones. where k = h=u = g(AB): It follows that C(A; B; m) lies on a plane P˜ which contains the Y -axis and meets the XY -plane at an angle ®, referred to as the intersecting angle of two cones. We call this planar curve C(A; B; m) a conic section, or simply a conic. In particular, if g(AB) ¸ m, C(A; B; m) is a closed curve and called an ellipse (Fig. 1(1)); if g(AB) · m, C(A; B; m) has two separate branches, called a hyperbola (Fig. 1(2)). Trivially, in their degenerate cases in which g(AB) = m, an ellipse becomes a line segment AB, and a hyperbola becomes two half-lines that are the extensions of AB. Moreover, there are two special cases: 1. If A; B lie in a vertical line, then g(AB) = 1 and the ellipse is a circle lying in a horizontal plane. 2. If A; B lie in a horizontal plane, then g(AB) = 0 and the hyperbola lies in a vertical plane. Substituting z with the right side of (3), from (2) we obtain p (m2u2 ¡ h2)(m2x2 ¡ h2) y = § : (4) mh Hence the parametric expression of C(A; B; m) is à p ! (m2u2 ¡ h2)(m2x2 ¡ h2) m2u x; § ; x : (5) mh h Let VA and VB be the intersections of C(A; B; m) with the XZ-plane that are close to A and B respectively (Fig. 1). Then by Equations (3) and (4) h h V = ( ; 0; mu);V = (¡ ; 0; ¡mu): (6) A m B m 2 Clearly, C(A; B; m) is symmetric with respect to two orthogonal lines: VAVB and the Y -axis. Therefore, O is the center of the conic section C(A; B; m). When C(A; B; m) is an ellipse, let R1 and R2 be the intersections of C(A; B; m) with the positive and negative Y -axis respectively. Then again from Equation (4) we find µ ¶ µ ¶ u q ¡uq R = 0; jk2 ¡ m2j; 0 ;R = 0; jk2 ¡ m2j; 0 : (7) 1 m 2 m When C(A; B; m) is a hyperbola, (7) also defines two points on the Y -axis. In the case of an ellipse, VAVB is called the major axis while R1R2 is called the minor axis of the ellipse. In the case of a hyperbola, VAVB is called the transverse axis while R1R2 is called the conjugate axis of the hyperbola. From Equations (4) and (3), it is easy to derive that 0 2 2 0 2 yx = §(m =k ¡ 1)x=y; zx = m =k: Therefore, the tangent vector t of the conic section C(A; B; m) is à ! m2 x m2 t = 1; §( ¡ 1) ; : (8) k2 y k Remark 1 Note t is completely determined by m and k, independent from the coordinates of A and B. p Now suppose C(A; B; m) is a hyperbola. When x goes to infinity, by (4) y=x goes to m2=k2 ¡ 1 and the tangent vector becomes 0 s 1 m2 m2 t = @0; § ¡ 1; A : 1 k2 k The two lines through O in the directions of t1 are the asymptotes of the hyperbola, which lie in the plane P˜ where the hyperbola lies. Conic sections have two important properties: constant sum/difference property and reflective property. First we prove a lemma [5]. Lemma 2 Suppose the endpoint S of a line SA is perturbed in direction v. Let the angle between ¡! SA and v be θ. Then the directional derivative of l(SA) with respect to v is (¡ cos θ). ¤ ¤ ¤ Proof: Suppose S moves to S in direction v. Let l0 = l(SA); l = l(S A);" = l(SS ). Then 2 2 2 l = l0 + " ¡ 2"l0 cos θ; and 2l ¢ dl = 2(" ¡ l0 cos θ) ¢ d": Note l ! l0 when ² ! 0. Therefore 0 dl lv = lim = ¡ cos θ: "!0 d" The lemma is proved. Theorem 3 (constant sum/difference) For any point S on an ellipse C(A; B; m), the sum of the distances from S to the vertices A and B is constant. For any point S on a hyperbola C(A; B; m), the difference of the distances from S to the vertices A and B is constant. 3 Proof: There is no loss of generality that we assume A = (u; 0; h);B = (¡u; 0; ¡h) as before. Because g(AS) = g(BS) = m, when k ¸ m and C(A; B; m) is an ellipse, p p ¡2 ¡2 l(AS) + l(SB) = 1 + m (jZA ¡ ZSj + jZS ¡ ZBj) = 2h 1 + m : The argument is similar for the case of C(A; B; m) being a hyperbola. This property completely characterizes ellipses and hyperbolas, therefore, we can redefine el- lipses/hyperbolas to be planar curves that satisfy the constant sum/difference property. That is, an ellipse (or a hyperbola) is a planar curve such that the sum (or difference respectively) of the distances between any point on the curve and two fixed distinct points is constant. This property implies another property which is important in applications of conic sections. Corollary 4 (reflection) For any point S on an ellipse or a hyperbola, the tangent line at S meets SA and SB at the same acute angle. 0 Proof: Let the acute angle between t and SA; SB be θ; Á respectively. By Lemma 2, lt(SA) = 0 0 ¡ cos θ; lt(SB) = ¡ cos(¼ ¡ Á) = cos Á. For an ellipse, since l(SA) + l(SB) is constant, lt(SA) + 0 lt(SB) = ¡ cos θ + cos Á = 0. Hence θ = Á. The argument is similar for hyperbolas. Remark 5 This corollary can also be proved by Equation (8). Because of the reflective property A and B are called the inherent foci of C(A; B; m). Let A;¯ B¯ and C¯(A;¯ B¯) be the projections of A; B and C(A; B; m) on a horizontal plane respectively. Then any point S¯ on C¯(A;¯ B¯) is the projection of certain point S on C(A; B; m). Because l(A¯S¯) § l(S¯B¯) = (l(AS) § l(SB)) sin ¯ = const; C¯(A;¯ B¯) is also an ellipse or a hyperbola with foci A¯ and B¯.
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