Affine Invariance

Convex Hull Property

Variation diminishing properties

Continuity

Locality property

 


Affine Invariance --

Affine Invariance, namely, translation invariance, rotation invariance, and scaling invariance

Convex Hull Property -- Junk

Variation Diminishing Property --

It is difficult to formulate a variation diminishing property for tensor product B-spline surfaces. Here are some examples that show how a naive approach fails. This is a linear by quadratic B-spline surface. A straight line can intersect the surface but never touches its control mesh. Here is another example, a linear by quadratic B-spline surface. It is partitioned by this plane into three components and yet its control mesh is partitioned into two components.

Continuity --

The next property is the continuity. If we maintain the knots in V direction, but create a double knot in U direction, we can clearly see a vertical seam. Bear in mind that it is of order 3, quadratic, in U direction. Therefore, if we further increase its multiplicity to 3 at this knot, we have a jump discontinuity. The two edges of the vertical trench interpolate the boundary curves defined by these two columns of vertices. Let us make it even more interesting. If we close up portions of these two columns, we have a contiguous control mesh that generated a surface with a hole.

Locality Property --

Let us demonstrate the locality property for surfaces. This vertex on the control mesh corresponds to the basis function in yellow when we consider in U direction, it has the local influence over the support of the yellow basis function which is extended over the middle three intervals. The same vertex corresponds to the basis function in purple when we consider in V direction. It has the local influence over the support of the purple basis function which is extended over the middle four intervals. Therefore, this point has local influence over 3 by 4, that is, 12 surface patches in the middle area.