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The rational wedge basis functions are not defined at points where the denominator polynomials vanish. Eq. 18) applies over the polycon where, by property (6), the wedges are well defined. Much of the analysis applies to polynomials with coefficients in the complex field. Boundaries of algebraic elements such as polycons are defined by irreducible polynomials with real coefficients. We cannot restrict the analysis to polynomials over the reals because this field is not algebraically closed. Points of intersection of curves of polynomials with only real coefficients are in general points with complex coordinates which may reduce in specific cases to real coordinates.
Function f is linear on curve P if there are any constants a(P) , b(P) , c(P) for which f(x,y) [a(P)x + b(P)y +c(P)] mod P . ) Hence, linearity of the patchwork approximation on a straight line side together with fitting of vertex values ensures continuity across the side. On any curved side of a polycon,however, a linear form has three degrees of freedom. Continuity is achieved by fitting the function value at 15 PATCHWORK APPROXIMATION another point on each conic side. This point is called a "side node".
These extensions are noted but not pursued in any depth. 8 RJ3MARKS AND FEFERENCES An excellent reference book on the theory of finite element methods is that of Strang and Fix (1973). Applications oriented texts containing some of the underlying theory and written more for engineers are Zienkiewicz and Cheung (1967) and Zienkiewicz (1971). The role of pyramids and wedges in patchwork approximation and some of the geometric implications is described by Synge (1957). Some of the definitions and symbols introduced in this chapter are new.
A Rational Finite Element Basis by Wachspress