An Approach to Conic Sections

Transcription

1 n pproach to Conic ections Jia. F. Weng Introduction. The study of conic sections can be traced back to ancient Greek atheaticians, usually to pplonious (ca bc) [2]. The nae conic section coes fro 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, ost odern textbooks on calculus depart fro this geoetric approach. Instead, conic sections are defined as soe types of loci and studied through analytic geoetry. 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 becoe the inherent foci of the conic section and a directrix exists associated with each of the inherent foci. ll 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 becoe the ones discussed in planar analytic geoetry. This new approach sees sipler and ore natural than the classical geoetric 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 ining industry. In ppendix 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 x P, y P, z P denote the Cartesian coordinates of a point P. uppose P and Q are two distinct points. The length of line segent P Q is denoted by l(p Q), and the gradient of P Q is denoted by g(p Q) which is defined as g(p Q) z Q z P (x Q x P ) 2 + (y Q y P ) 2. (1) Let C(P ; ) denote a (double-napped) right circular cone whose vertex is P and whose generating lines have gradient ( > 0). The angle β fored by the axis of C(P ; ) and the generating lines is referred to as the generating angle of the cone. Clearly, tan β 1/. Now suppose and are two distinct points in space. The intersection of cones C(; ) and C(; ) is denoted by C(, ; ). ssue the horizontal distance between and is 2u while the vertical distance between and is 2h. Then, after a transforation we ay assue (u, 0, h), ( u, 0, h), u 0, h 0 (Fig. 1). Hence the equations describing C(; ) and C(; ) are (x u) 2 + y 2 (z h)2 2, (x + u) 2 + y 2 (z + h)2 2. (2) If h 0, then the intersection C(, ; ) lies trivially in the Y Z-plane. ssue h 0. ubtracting the first equation fro the second, we have z 2 u h x 2 x, (3) k

2 V Z R 2 O R 1 Y α β V P ~ Z Y O V V R 1 O V V V V R2 (1) (2) Figure 1: Intersections of two cones. where k h/u g(). It follows that C(, ; ) lies on a plane P which contains the Y -axis and eets the Y -plane at an angle α, referred to as the intersecting angle of two cones. We call this planar curve C(, ; ) a conic section, or siply a conic. In particular, if g(), C(, ; ) is a closed curve and called an ellipse (Fig. 1(1)); if g(), C(, ; ) has two separate branches, called a hyperbola (Fig. 1(2)). Trivially, in their degenerate cases in which g(), an ellipse becoes a line segent, and a hyperbola becoes two half-lines that are the extensions of. Moreover, there are two special cases: 1. If, lie in a vertical line, then g() and the ellipse is a circle lying in a horizontal plane. 2. If, lie in a horizontal plane, then g() 0 and the hyperbola lies in a vertical plane. ubstituting z with the right side of (3), fro (2) we obtain Hence the paraetric expression of C(, ; ) is ( y ± 2 u 2 h 2 )( 2 x 2 h 2 ). (4) h ( ( x, ± 2 u 2 h 2 )( 2 x 2 h 2 ), h ) 2 u h x. (5) Let V and V be the intersections of C(, ; ) with the Z-plane that are close to and respectively (Fig. 1). Then by Equations (3) and (4) V ( h, 0, u), V ( h, 0, u). (6) 2

3 Clearly, C(, ; ) is syetric with respect to two orthogonal lines: V V and the Y -axis. Therefore, O is the center of the conic section C(, ; ). When C(, ; ) is an ellipse, let R 1 and R 2 be the intersections of C(, ; ) with the positive and negative Y -axis respectively. Then again fro Equation (4) we find ( R 1 0, u ) ( k 2 2, 0, R 2 0, u ) k 2 2, 0. (7) When C(, ; ) is a hyperbola, (7) also defines two points on the Y -axis. In the case of an ellipse, V V is called the ajor axis while R 1 R 2 is called the inor axis of the ellipse. In the case of a hyperbola, V V is called the transverse axis while R 1 R 2 is called the conjugate axis of the hyperbola. Fro Equations (4) and (3), it is easy to derive that y x ±( 2 /k 2 1)x/y, z x 2 /k. Therefore, the tangent vector t of the conic section C(, ; ) is ( t 1, ±( 2 k 2 1)x y, 2 ). (8) k Reark 1 Note t is copletely deterined by and k, independent fro the coordinates of and. Now suppose C(, ; ) is a hyperbola. When x goes to infinity, by (4) y/x goes to 2 /k 2 1 and the tangent vector becoes t 0, ± 2 2 1,. k2 k The two lines through O in the directions of t are the asyptotes of the hyperbola, which lie in the plane P where the hyperbola lies. Conic sections have two iportant properties: constant su/difference property and reflective property. First we prove a lea [5]. Lea 2 uppose the endpoint of a line is perturbed in direction v. Let the angle between and v be θ. Then the directional derivative of l() with respect to v is ( cos θ). Proof: uppose oves to in direction v. Let l 0 l(), l l( ), ε l( ). Then l 2 l ε2 2εl 0 cos θ, and 2l dl 2(ε l 0 cos θ) dε. Note l l 0 when ɛ 0. Therefore The lea is proved. l v dl li cos θ. ε 0 dε Theore 3 (constant su/difference) For any point on an ellipse C(, ; ), the su of the distances fro to the vertices and is constant. For any point on a hyperbola C(, ; ), the difference of the distances fro to the vertices and is constant. 3

4 Proof: There is no loss of generality that we assue (u, 0, h), ( u, 0, h) as before. ecause g() g(), when k and C(, ; ) is an ellipse, l() + l() ( Z Z + Z Z ) 2h The arguent is siilar for the case of C(, ; ) being a hyperbola. This property copletely characterizes ellipses and hyperbolas, therefore, we can redefine ellipses/hyperbolas to be planar curves that satisfy the constant su/difference property. That is, an ellipse (or a hyperbola) is a planar curve such that the su (or difference respectively) of the distances between any point on the curve and two fixed distinct points is constant. This property iplies another property which is iportant in applications of conic sections. Corollary 4 (reflection) For any point on an ellipse or a hyperbola, the tangent line at eets and at the sae acute angle. Proof: Let the acute angle between t and, be θ, φ respectively. y Lea 2, l t() cos θ, l t() cos(π φ) cos φ. For an ellipse, since l() + l() is constant, l t() + l t() cos θ + cos φ 0. Hence θ φ. The arguent is siilar for hyperbolas. Reark 5 This corollary can also be proved by Equation (8). ecause of the reflective property and are called the inherent foci of C(, ; ). Let Ā, and C(Ā, ) be the projections of, and C(, ; ) on a horizontal plane respectively. Then any point on C(Ā, ) is the projection of certain point on C(, ; ). ecause l(ā ) ± l( ) (l() ± l()) sin β const, C(Ā, ) is also an ellipse or a hyperbola with foci Ā and. ince Ā and lie on the sae plane where the curve C(Ā, ) lies, we call the coplanar foci of the curve. Moreover, let V, V, R 1 and R 2 be the projections of V, V, R 1 and R 2 on the sae plane respectively. Then V V and R 1 R2 are the ajor and inor axes (or the transverse and conjugate axes) of the ellipse (or the hyperbola respectively) C(Ā, ). The equation of C(Ā, ) can be easily derived fro Equation (4) as follows: where x 2 ā 2 + y2 b2 1, (9) ā 2 h2 2, b 2 h2 2 u 2 2 (k2 2 )u 2 2. (10) Equation (9) is called the standard for of an ellipse or a hyperbola, depending on b 2 0 or b 2 0, i.e. on k or k. Note that 2ā l( V V ), 2 b l( R 1 R2 ). Let 2 c l(ā ) 2u. Then it is easy to see that c 2 ā 2 b 2, if C(Ā, ) is an ellipse, c 2 b 2 ā 2, if C(Ā, ) is a hyperbola. 4

5 Figure 2: Confocal ellipses and hyperbolas. 3 Confocal and iilar Conics, Parabolas. The shape of a conic section is copletely deterined by two paraeters and k. investigate how the curve varies as or k changes. Now we (1) If and are fixed, i.e. k is fixed, but changes, we obtain a faily of conics that lie on different planes but share, as their coon foci. These conics are called confocal. s argued above, when these curves are projected to the sae horizontal plane, their projections are also confocal with Ā, as their coon foci (Fig. 2, based on a figure in [3]). (2) Now we study how C(, ; ) varies if is fixed but k changes. Without loss of generality we assue is fixed and oves to that lies in the sae vertical plane. Let (u+ε, 0, h+ δ), ε 0, δ 0. Then the equations of cones C( ; ) and C(; ) becoe (x u ε) 2 + y 2 (z h δ)2 2, (x + u) 2 + y 2 (z + h)2 2. (11) Of all possible oves of we discuss the oves in three special directions: along, horizontal and along V. (2.1) ove of either preserves the gradient between two cone vertices or does not. If it does, then δ/ε h/u k. olving the syste (11) with δ kε we have z 2 k x + (h2 2 u 2 )ε. 2hu Therefore, the new conic section C(, ; ) lies in a plane P that is parallel to the plane P where C(, ; ) lies. esides, since both curves lie on the sae cone C(; ), it follows that the two conics C(, ; ) and C(, ; ) are siilar relative to their coon focus. When projected on the sae horizontal plane, their projections are also siilar relative to (Fig.??). On the other hand, if the ove of does not preserve the gradient between the vertices, then P is not parallel to P, and C(, ; ) is not siilar to C(, ; ) by the definition of siilarity. 5

6 Z Y * P ~ L V O D D L V Y Figure 3: parabola as an extree ellipse. Reark 6 curve C 1 (x 1 (t), y 1 (t), z 1 (t)) is said to be siilar to another curve C 2 (x 2 (t), y 2 (t), z 2 (t)) relative to point (x 0, y 0, z 0 ) if x 2 (t) x 0 x 1 (t) x 0 y 2(t) y 0 y 1 (t) y 0 z 2(t) z 0 z 1 (t) z 0. (2.2) The vertex oves in the direction of the positive -axis, that is, δ 0 and ε onotone increases. ssue we start with g() k > and the conic is an ellipse. In this ove, the ellipse C(, ; ) becoes narrower and narrower, and finally becoes a line segent, a degenerate ellipse with k (h + δ)/(u + ε). If the increase of ε continues, then as stated in ection 2, the line segent first suddenly jups into two half-lines as a degenerate hyperbola, and then becoes a noral hyperbola with k <. (2.3) s we have discussed in (2.2), in the degenerate case, an ellipse or a hyperbola overlaps the line through and. One ay ask if there is a non-degenerate coon extree of ellipses and hyperbolas. Look at the ove of along V. In such a ove δ ε. olving the syste (11) with δ ε and then let ε go to infinity, we have y ± 2 (h u)(x + h), z (x u) + h. (12) Therefore the paraetric expression of the resulting curve is ( x, ± 2 ) (h u)(x + h), (x u) + h. (13) 6

7 If starting with a hyperbola (h/k < ), we obtain the sae result. Therefore, the resulting planar curve, called a parabola, is the non-degenerate coon extree of ellipses and hyperbolas. The parabola has a vertex V and is syetric about the line V, the intersection of P and the Z-plane. Let L {( 2h + u, y, h)} be the point set that is the intersection of the plane P and the horizontal plane through (Fig. 3). For any point on the parabola, let D be the foot of the perpendicular fro to L. Then the gradient of D is the slope of the plane P, which is equal to the gradient of any generating line of C(; ). Therefore, siilarly to the arguent for ellipses and hyperbolas, we have l() l(d). Theore 7 (equidistance) For any point on the parabola, the distance fro to equals the distance fro to the line L. For this reason, is called the inherent focus of the hyperbola, and L is called the directrix associated with the inherent focus. s ellipses and hyperbolas, this equidistance property copletely characterizes parabolas and we can redefine a parabola to be a planar curve that satisfies the equidistance property. The ratio of the distance fro to the focus to the distance fro to the directrix is called the eccentricity of the parabola. Thus, a parabola can also be defined as a planar curve whose eccentricity is one. gain, the equidistance property iplies the reflective property: Corollary 8 (reflection of parabolas) For any point on a parabola, the tangent line t at eets two lines at the sae acute angle: the line joining and the focus and the perpendicular fro to the directrix. Now denote the parabola by C(, L; ). Let C(, L), V,,, L and D be the projections of C(, L; ), V,,, L and D on a horizontal plane respectively. ecause and D have the sae gradient, l() l(d) iplies l( ) l( D). Therefore, the projection C(, L) is also a parabola, whose coplanar focus is and the directrix associated with is L. Fro (12) the equation of C(, L) is y 2 4 p(x + h/), where p h/ u is the distance fro the vertex V to the coplanar focus or to the directrix L. This equation is the standard for of a parabola with vertex at ( h/, 0). If the origin oves to the vertex V, then the equation is further siplified to y 2 4px. 4 Discussions (1) It is easy to see that any ellipse (or hyperbola) C obtained by cutting a cone with a plane can be generated by two cones. Factually, we ay assue that the cone C is C(; ). Then ake a copy of C(; ) and turn the copy 180 around the Y -axis, the copy becoes the required C(; ). (2) s parabolas, an ellipse or a hyperbolas has two directrixes associated with its inherent foci and : They are the intersections of P and the horizontal planes through and. Figure 4(1) shows the directrix L of an ellipse C(, ; ) associated with. For any point on the ellipse, let D be the foot of the perpendicular fro to L. The eccentricity of the conic section C(, ; ) is still defined to be the ratio of l() to l(d). Let H be the foot of the perpendicular 7

8 Z O Y α β F V P ~ Z Y * P ~ D L α L ~ V F C β H L L ~ F O C (1) (2) Figure 4: Coplanar foci and their directrixes. fro to the horizontal plane through. ince H equals the generating angle β, and HD equals the intersecting angle α (Fig. 4(1)), When projected to a horizontal plane, l( ) l( l() l(h)/ cos β l(d) l(h)/ sin α sin α cos β D) l(h) l(dh) const < 1. l(h)/ cot β l(h)/ tan α tan α const < 1. cot β Thus, the third definition of ellipses is that an ellipse is a planar curve whose eccentricity is a constant less than one. These arguent can be siilarly apply to hyperbolas. Hence, the third definition of hyperbolas is that a hyperbola is a planar curve whose eccentricity is a constant greater than one. (3) Finally we should point out that an ellipse or a hyperbola C(, ; ) also has its own coplanar foci lying on the plane P. Take an ellipse as an exaple. ccording to Quetelet-Dandelin s construction [2], suppose the sphere inscribing C(; ) and P touches P at a point F. Then F is a coplanar focus of C(, ; ). nother focus F can be found in a siilar way. ssociated with these coplanar foci, there are two directrixes. For instance, suppose the sphere defined above touches the cone C(; ) at a horizontal circle C (Fig. 4(1)). Then the intersection of P and the plane through C is the directrix L associated with the focus F. In a siilar way we can find the coplanar focus F and its associated directrix L for a parabola C(, L; ) (Fig. 4(2)). The coordinates of the coplanar foci and the standard equations of conic sections with respect to its coplanar foci and directrixes are derived in ppendix. 5 n application. Given a point set, the iniu network proble asks for a network of shortest total length interconnecting all given points. To shorten the network, probably soe points not in the given 8

9 deposit ground deposit C Figure 5: ining network. point set are added. In the literature this iniu network is called a teiner inial tree and the additional points are called teiner points [4]. The teiner tree proble was first posed by Ferat who asked how to find the point in a triangle C so that the su of the distances fro to the vertices,, and C is inial [4]. There are any generalizations of the teiner tree proble. recently studied one is the gradient-constrained teiner tree proble [1]. Figure 5 depicts a siple exaple of an underground ining network in which the ore in two deposits is extracted through tunnels to a vertical shaft and then hauled up to the ground. In the figure and are two prescribed access points in deposits. In practice, the gradient of any tunnel cannot be very steep. The typical axiu gradient is about 1:7. In this exaple we need to find a point so that with this gradient constraint the total length of the network f l()+l()+l(c) is iniized, where C is the access point to be deterined on the vertical shaft. Clearly, to iniize f, C should be perpendicular to the vertical shaft. Hence, C is horizontal and autoatically satisfies the gradient constraint. It can be shown that if g() >, then and ust have the axial gradient in order to iniize f [6]. It follows that lies on the ellipse C(, ; ), and that C C because of the reflective property. These conditions copletely deterine and C. References [1] M.razil, D..Thoas and J.F.Weng, Gradient-constrained inial teiner trees, DIMC eries in Discrete Matheatics and Theoretical Coputer cience, Vol. 40 (1998), pp [2] J.L.Coolidge, history of the conic sections and quadric surfaces, Oxford University Press, London, [3] R.Courant and H.Robbis, What is atheatics? Oxford University Press, London, [4] F.K.Hwang, D..Richard and P.Winter, The teiner tree proble, North-Holland, [5] J.H.Rubinstein and D..Thoas, variational approach to the teiner network proble, nn. Oper. Res., Vol. 33 (1991), pp [6] J.F.Weng, note on constrained shortest connections of two points to a vertical line, preprint. 9

10 ppendix. If C(, ; ) is an ellipse, then let F and F be a pair of points lying on V V and syetric to O such that 0 ρ l(of )/l(ov ) 1. iilarly, if C(, ; ) is a hyperbola, then define F and F be the points lying on the extension of V V and syetric to O such that ρ l(of )/l(ov ) > 1. y the definition ( ) ρh F, 0, ρu, F ( ρh ), 0, ρu. (14) Let (x, y, z) be a point on C(, ; ), then l 1 l(f ) l 2 l(f ) (ρh (ρh x ) 2 + y 2 + (ρu z) 2, + x ) 2 + y 2 + (ρu + z) 2. Let l 1 ± l 2 2ã, ã 0. Then l 2 1 (2ã l 2) 2, l 2 1 l2 2 4ã2 4ãl 2, ±ãl 2 ã 2 l2 1 l2 2 4 quaring both sides, after a siplification we obtain ã 2 + ρhx + ρuz. (h 2 ρ 2 2 ã 2 )x huρ 2 xz + ( 2 u 2 ρ 2 ã 2 ) 2 z 2 2 ã 2 y 2 ( 4 u 2 + h 2 )ã 2 ρ 2 + ã ubstituting y and z with (4) and (3) respectively, finally we have an equation containing only one variable x: f 1 x 2 + f 2 0, (15) where f 1 4 u 2 (1 + 2 )ã 2 (h u 2 ) 2 ρ 2, f 2 h 2 (h 2 2 u 2 )ã 2 2 h 2 ã 4 + h 2 (h u 2 )ã 2 ρ 2. Equation (15) will hold for any x if both f 1 0 and f 2 0. olving this syste we obtain h ã u 2, ρ u ã Thus, the constant su/difference of distances and the coordinates of the coplanar foci F, F are both deterined. y the definition of the intersecting angle α, we have cos α h/(ã), sin α u/ã. fter an anticlockwise rotation of the coordinate syste around the Y -axis by α, let the new axes be O, OỸ and O Z. Then the Ỹ -plane is the plane P where the conic section lies. ince the curve lies on the Ỹ -plane, we have z 0, and the the transforation is x x cos α, y ỹ, z x sin α. Fro (4) we can derive the standard for of the conic section on the plane P: ( x) 2 (ã) 2 + (ỹ)2 1, (16) ( b) 2 10

11 where b 2 (h 2 / 2 ) u 2. Note b 2 0 when g() and the conic section is an ellipse, and b2 0 when g() and the conic section is a hyperbola. It is easy to check ã l(ov ), b l(or 1 ). Let c l(of ). Then as in the projection of C(, ; ), the following equations hold: c 2 ã 2 b 2, if C(, ; ) is an ellipse, c 2 b 2 ã 2, if C(, ; ) is a hyperbola. In the sae way we can deterine the coplanar focus F and its directrix L for a parabola. s shown in Figure 4(2), let ρ be the horizontal distance between V and F. ince the gradient of V F is and since V has the sae distance to F and to L, we have F (ρ h ) (, 0, (ρ u), L ρ h ), y, (ρ + u). uppose (x, y, z) is a point on the parabola. Then the square of l(f ) equals the square of the distance fro to L, ( x ρ + ) h ( ) 2 2 ( + y 2 + z ρ2 h + u x + ρ + ) h ( ) z + ρ2 h + u. ecause y and z satisfy (12), this equation always holds if ρ h u (1 + 2 ). Note now cos α 1/ s argued in the case of ellipses and hyperbolas, after an anticlockwise rotation of the coordinate syste by α, fro (12) we obtain the standard for of the parabola on the plane P: ỹ 2 4 p( x + h ), (17) where p 4(h u)/( ) and h / l(v F ). If the origin oves to V, then it can be further siplified to ỹ 4 p x. 11