A geometric interpretation of the diagonal of a tensor-product Bézier volume

A geometric interpretation of the diagonal of a tensor-product trivariate Bézier volume using degree elevation of Bézier triangles is given. A brief discussion of the diagonal curve of a tensor-product bivariate Bézier surface will help to motivate our geometric interpretation for the diagonal curve of a Bézier volume. Suppose we have a tensor-product Bézier surface, x(u, v), given by x(u, v) = n i=0 n j =0 b ij B n i (u)B n j (v). The diagonal curve s(u) = x(u, u) is given by s(u) = 2n l=0 B 2n l (u) 1 2n l i+j =l b ij n i n j. (1) The control points s 0 , s 1 ,. .. , s 2n of s(u) can be interpreted in the following way. (Only those control points, s l , of s(u) where l n will be considered. The others follow by symmetry.) Treat the b ij , i + j = l, as a degree l Bézier curve. Degree elevate this curve 2n − l times (until it is of degree 2n). The middle control point on this degree-elevated curve will equal s l. The expression (1) is actually the formula for taking a curve of degree 2n − l and degree-elevating it l times (see Farin, 1996).