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Show 18 Note that, while each parameter in this form has a F(x,y,z,w) i L ai,j,k,l x i+j+k+l=n 1 w maximum exponent of three, the whole scheme is basically sixth order because of the U3V3 term. This fact is important in determining the maximum number of local maxima and minima to expect within a patch. This form (non-rational bicubic) is the one which was used to describe most of the objects pictured in this thesis. Rational Bicubic Surfaces The inclusion of a fourth bicubic function for the w component is sometimes done as a modelling technique, but its use is not widespr@ad. The importance of rational bicubics is that non-rationals turn into rationals after the application of a perspective transformation. Any algorithm that wishes to draw perspective pictures of bicubic patches must therefore be prepared to handle rational bicubics. This introduces some complications to some of the functions which must be evaluated but not much to the logical flow of the algorithm. General Algebraic Surfaces The general form of a homogeneous algebraic surface of order n s This polynomial contains a term for each combination of powers of x,y,z, w which add up to n. Each such term has |
