The natural domains of interest of tensor product implicit patches are boxes. The space can be divided in a natural and symmetrical way in such boxes. We can for example use the division in the cubes of the same size. For the tensor-product functions the theory of B-splines basis functions, assuring the desired continuity, is simple and efficient.

In order to obtain the knot vectors, we consider a bounding box of the data, and subdivide a slightly enlarged area in cubic cells of constant size s. This subdivision induces an equidistant grid on the x, y and z axis. Adding additional d knots at the beginning and at the end of the grid, we obtain the knot vectors

The domain of interest Ω does not in fact need to contain all the cells within this grid. Obviously it has to contain the cells containing points. Additionally we consider neighboring cells. The reason for this is that the resulting surface is likely to pass through such a cell, and otherwise might be cut away. The basis B-spline functions vanish on the great number of knot intervals. Due to the restriction of F on Ω, we only need to take the products of basis functions M_{p}(x) N_{q}(y) O_{p}(z) into account that do not vanish on Ω. The indices are:

J ={(i,j,k) | ∃ l∈{1,...,n}, r ∈{1,...,m}, s∈{1,...,n}, t∈{1,...,σ}: M_{r}(p_{l,x}) N_{s}(p_{l,y}) O_{t}(p_{l,z}) ≠ 0 ∧ max{|i-r|,|j-s|,|k-t|} ≤ 1}We will therefore find the minimum of the functional only for the functions of the type

F = ∑_{(i,j,k)∈J } c_{i,j,k }M_{i}(x) N_{j}(y) O_{k}(z).

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Published June 24, 2005

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