Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 70 / 4

A tensor-product surface basis combines two independent univariate bases:The coefficients $P_{ij}$ are three-dimensional control points. Fixing one parameter gives a curve in the other univariate basis; the coefficients of that curve are the corresponding weighted combinations of rows or columns of controls.

The control net must have a rectangular two-index topology, with row/column neighbors and quadrilateral mesh cells. “Rectangular” describes the index grid, not the geometry of the points: its faces need not be flat or rectangular in space. A surface requiring a triangular domain or arbitrary extraordinary mesh connectivity is not directly described by one such tensor-product patch.

Let $N_{ij}(u,v)=B_i(u)C_j(v)$. Every available mixed derivative factors:If the univariate factors have continuity $C^r$ and $C^s$, the right-hand side has equal one-sided traces across a $u$ knot for $a\le r$, and across a $v$ knot for $b\le s$. Their products therefore remain continuous, including where knot lines cross. The basis and its surface inherit coordinate-wise continuity orders $r,s$, and joint $C^{\min (r,s)}$ continuity. In particular, degree-$p$ and degree-$q$ B-splines at simple knots give $C^{p-1}$ and $C^{q-1}$ continuity in the respective directions. Higher continuity can occur through special coefficient cancellation, but is not generally guaranteed.

Nonnegativity passes to products. Their partition of unity follows fromThus the surface remains in the convex hull of its active control points and commutes with application of affine functions. The support is the Cartesian product of the two univariate supports, giving local rectangular influence regions and efficient local evaluation. Linear precision also passes independently in each coordinate. Clamped endpoint interpolation gives boundary curves determined by the outer rows/columns and corner interpolation; parameter-line curves inherit the corresponding univariate properties. These are the relevant inherited properties, rather than a claim of arbitrary global variation-diminishing behavior for surfaces.

A triangular Bézier patch is a non-tensor-product example. On barycentric parameters $u,v,w\ge 0$ with $u+v+w=1$, defineThe indices satisfy a total-degree constraint, and $w=1-u-v$ couples the two independent parameters. The control net is triangular with $(n+1)(n+2)/2$ points, rather than a rectangular product index set. The multinomial theorem gives nonnegative basis functions summing to one, but the entire basis is not a product of two independent univariate bases.