A Conic Sections The ellipse, hyperbola, and parabola (and also the circle, which is a special case of the ellipse) are called the conic section curves (or the conic sections or just conics), since they can be obtained by cutting a cone with a plane (i.e., they are the intersections of a cone and a plane). The conics are easy to calculate and to display, so they are commonly used in applications where they can approximate the shape of other, more complex, geometric figures. Many natural motions occur along an ellipse, parabola, or hyperbola, making these curves especially useful. Planets move in ellipses; many comets move along a hyperbola (as do many colliding charged particles); objects thrown in a gravitational field follow a parabolic path. There are several ways to define and represent these curves and this section uses asimplegeometric definition that leads naturally to the parametric and the implicit representations of the conics. y F P D P D F x (a) (b) Figure A.1: Definition of Conic Sections. Definition: A conic is the locus of all the points P that satisfy the following: The distance of P from a fixed point F (the focus of the conic, Figure A.1a) is proportional 364 A. Conic Sections to its distance from a fixed line D (the directrix). Using set notation, we can write Conic = {P|PF = ePD}, where e is the eccentricity of the conic. It is easy to classify conics by means of their eccentricity: =1, parabola, e = < 1, ellipse (the circle is the special case e = 0), > 1, hyperbola. In the special case where the directrix is the y axis (x = 0, Figure A.1b) and the focus is point (k, 0), the definition results in (x − k)2 + y2 = e, or (1 − e2)x2 − 2kx + y2 + k2 =0. (A.1) |x| In this case, the conic is represented by a degree-2 equation. It can be shown that this is true for the general case, where the directrix and the focus can be located anywhere. It can also be shown that the inverse is also true, i.e., any degree-2 algebraic equation of the form ax2 + by2 +2hxy +2fx+2gy + c = 0 (A.2) represents a conic. Equation (A.2) can be used to classify the conics. If D is the determinant ahf D = hbg , fgc then Table A.2 provides a complete classification of the conics, including degenerate cases where the conic reduces to two lines (real or imaginary) or to a point. ab − h2 Conditions Conic =0 D =0 parabola D =0 b =0 g2 − bc > 0 2 parallel lines g2 − bc =0 2 parallel coincident lines g2 − bc < 0 2 parallel imaginary lines b = h =0 f 2 − ac > 0 2 parallel real lines f 2 − ac =0 2 parallel coincident lines f 2 − ac < 0 2 parallel imaginary lines > 0 D =0 point (degenerate ellipse) D =0 −bD > 0 real ellipse −bD < 0 imaginary ellipse < 0 D =0 2 intersecting lines D =0 hyperbola Table A.2: Classification of Conics. Conic Sections 365 Exercise A.1: Assume that the second-degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 (A.3) is given. Show how to use the six parameters to determine which conic is described by this equation. Equation (A.1) can be used to generate the familiar implicit representations of the conics. We first treat the case e = 1 by transforming x = x − k/(1 − e2). When this is substituted into Equation (A.1) (and the prime is eliminated), the result is x2 y2 + =1, (A.4) a2 b2 ke where a = and b2 = a2(1 − e2). 1 − e2 Case 1: Ellipse. The case e<1 implies that both a and b are positive and a>b.In this case, Equation (A.4) represents the canonical ellipse. This ellipse is centered on the origin with the x and y axes being the major and minor axes of the ellipse, respectively. The major radius is a and the minor one is b.Fora = b, this ellipse reduces to a circle. Hence, we can think of a circle as the limit of the ellipse when e → 0andk →∞. Case 2: Hyperbola. The case e>1 implies a negative a and a negative b2 (hence an imaginary b). If we use the absolute value of the imaginary b, Equation (A.4) becomes x2 y2 − =1,a,b>0. (A.5) a2 b2 This is a canonical hyperbola, where the x axis is the traverse axis and the y axis is called the semiconjugate or imaginary axis. The hyperbola consists of two distinct parts with the imaginary axis separating them. The two points (−a, 0) and (a, 0) are called the vertices of the hyperbola. Case 3: Parabola (e = 1). The simple transformation x = x − k/2 yields, when substituted into Equation (A.1), the canonical parabola y2 =4ax, where a = k/2 > 0, (A.6) with focus at (a, 0) (thus, a is the focal distance) and directrix x = −a. The origin is the vertex of the canonical parabola. All the conic sections can also be expressed (although not in their canonical forms) by K f(θ)= . 1 ± e cos(θ) For e = 0 this is a circle. For 0 <e<1 this is an ellipse. For e = 1 this is a parabola and for e>1itisahyperbola. 366 A. Conic Sections The parametric representations of the conics are simple. We start with the ellipse. In order to show that the expression 1 − t2 2t a ,b , −∞ <t<∞, (A.7) 1+t2 1+t2 traces out an ellipse we show that it satisfies Equation (A.4): 2 2 2 2 1−t 2 2t a 1+t2 b 1+t2 1 − 2t2 + t4 +4t2 + = =1. a2 b2 1+2t2 + t4 The first quadrant is obtained for 0 ≤ t ≤ 1. To get the second quadrant, however, t has to vary from 1 to ∞. Quadrants 4 and 3 are obtained for −∞ ≤ t ≤ 0. The canonical hyperbola is represented parametrically by 1+t2 2t a ,b , −∞ <t<∞. (A.8) 1 − t2 1 − t2 The right branch is traced out when −1 ≤ t ≤ 1, and the left branch is obtained when −∞ ≤ t ≤−1and1≤ t ≤∞. Thus, the two values t = ±1 represent hyperbola points at infinity. The simple expression (at2, 2at), −∞ <t<∞, (A.9) traces out the canonical parabola. Equations (A.7) and (A.8) are called rational parametrics since they contain the parameter t in the denominator. Rational parametric curves are generally complex but can represent more shapes and are therefore more general than the nonrational ones. One disadvantage of the rational parametrics is variable velocity. Varying t in equal increments generally results in traveling along the curve in unequal steps. In practice, it is sometimes necessary to have conics placed anywhere in three- dimensional space, not just on the xy plane. This is done by taking a general two- dimensional conic P(t) [one of Equations (A.7), (A.8), or (A.9)], adding a third co- ordinate z = 0 and transforming it with the general 4 × 4 transformation matrix T [Equation (9.1)]. Normally, such a curve is translated and rotated. It may also be scaled and sheared. The result is a three-dimensional curve of the form 2 2 2 ∗ a0 + a1t + a2t b0 + b1t + b2t c0 + c1t + c2t P (t)= 2 , 2 , 2 w0 + w1t + w2t w0 + w1t + w2t w0 + w1t + w2t 2 a ti 2 b ti 2 c ti = i=0 i , i=0 i , i=0 i . 2 i 2 i 2 i i=0 wit i=0 wit i=0 wit Denoting xi = ai/wi, yi = bi/wi, zi = ci/wi,andai =(xi,yi,zi), we can write this as w a + w a t + w a t2 2 w a ti P∗(t)= 0 0 1 1 2 2 = i=0 i i . (A.10) w + w t + w t2 2 i 0 1 2 i=0 wit Conic Sections 367 This is the general rational form of the conic sections. It can also be shown that any rational parametric expression of the form (A.10) represents a conic. She could see at once by his degenerate conic and dissipative terms that he was bent on no good, “Arcsinh,” she gasped. “Ho, Ho,” he said. “What a symmetric little asymptote you have. I can see your angles have a lit of secs.” Richard Woodman, Impure Mathematics (1981) B Approximate Circles Parametric curves are general and can take many shapes. In principle, such a curve can be based on any functions, but in practice polynomials are virtually always used. It is well known, however, that one common, important curve, namely the circle, cannot be precisely represented by a polynomial. This short appendix discusses ways to compute approximate, albeit high precision, circles with B´ezier and B-spline methods. B.1 Circles and B´ezier Curves √ The equation of a circle is x2 +y2 = r2 or y = ± r2 − x2. This is not a polynomial and, in fact, Exercise B.1 proves that a polynomial cannot represent a circle. Applying B´ezier methods to circles can be done either by using rational B´ezier curves (Section 6.15) or by deriving an approximation to the circle. This section discusses the latter approach. We start with a three-point example. We select the three points P0 =(1, 0), P1 =(k, k), and P2 =(0, 1) and attempt to find the value of k such that the quadratic B´ezier curve defined by the points will best approximate a quarter circle of radius 1 (Figure B.1).