Matrix weighted rational curves and surfaces

• We extend rational curves and surfaces that have real weights to rational curves and surfaces permitting matrix weights.
• Matrix weighted rational curves and surfaces maintain many similar properties or algorithms of traditional rational curves and surfaces.
• The matrix weights permit novel geometric means for shape control of rational curves and surfaces.
• Matrix weighted NURBS curves and surfaces are suitable for curve or surface reconstruction with no need of solving large systems.

Rational curves and surfaces are powerful tools for shape representation and geometric modeling. However, the real weights are generally difficult to choose except for a few special cases such as representing conics. This paper presents an extension of rational curves and surfaces by replacing the real weights with matrices. The matrix weighted rational curves and surfaces have the same structures as the traditional rational curves and surfaces but the matrix weights can be defined in geometric ways. In particular, the weight matrices for the extended rational Bézier, NURBS or the generalized subdivision curves and surfaces are computed using the normal vectors specified at the control points. Similar to the effects of control points, the specified normals can be used to control the curve or the surface's shape efficiently. It is also shown that matrix weighted NURBS curves and surfaces can pass through their control points, thus curve or surface reconstruction by the extended NURBS model no longer needs solving any large system but just choosing control points and control normals from the input data.