Given the implicit equation for degree two curves (conics) and degree two surfaces (conicoids), algorithms are described here, which obtain their corresponding rational parametric equations (a polynomial divided by another). These rational parameterizaUons are considered over the fields of rationals, reals and complex numbers. In doing so, solutions are given to important subprob/ems of finding rational and real points on the given conic curve or conicoid surface. Further polynomial parameterizations are obtained whenever they exist for the conics or conicoids. These algorithms have been implemented on a VAX-780 using VA XIMA.
geometry, surfaces, curves, conics, conicoids
General curves and surfaces can be represented by implicit or parametric equations. A general (degree two) conic implicit equation is given by C(x,,V) = ax 2 + by 2 + cx,v + dx + e,v + f = 0, and rational parametric equations given by x = u(t)/w(t) and y = v(t)lw(t), where u, v and w are no more than quadratic polynomials. Further, a general (degree two) conicoid implicit equation is given by C(x,y,z) = ax 2 +by 2 +cz 2 +dxy+exz+fyz+gx+hy+iz+j=0, with corresponding rational parametric equations x = u(s, t)/ q(s, t), y = v(s, t)/q(s, t), andz = w(s, t)lq(s, t), where again u, v, w and q are no more than quadratic polynomials. The rational parametric form of representing a surface allows greater ease for transformation and shape control b2. The implicit form is preferred for testing whether a point is above, on, or below the surface, where above and below is determined relative to the direction of the surface normal.
As both forms have their inherent advantages, it becomes crucial to be able to go efficiently from one form to the other, especially when surfaces of an object are automatically generated in one of the two representations.
Both conics and conicoids always have a rational parameterization. In the second and third sections of this paper algorithms are described which can obtain rational parametric equations for the conics and conicoids, given the implicit equations. Polynomial parameterizations are also obtained whenever they exist for the conics and conicoids. These parameterizations are at most degree 2 and are over the field of reals, or the field of complex numbers when real solutions do not exist. In the fourth section, consideration is given to obtaining rational parameterizations over Q, the fields of rationals. Computations over Q are exact and hence give rise to stable computational algorithms, as opposed to finite precision calculations with real numbers.