MA 323 Geometric Modelling Course Notes: Day 21 Three Dimensional Bezier Curves, Projections and Rational Bezier Curves

In extending the planar methods to spatial bases, the algorithms require no modifications, as the algorithms are defined abstractly in terms of affine combinations with no regard to the location of the point as two dimensional or three dimensional (or even higher dimensional). In fact, the constructions of Bezier curves and Bezier splines work exactly as before. All that we need to do is supply three dimensional points instead of two dimensional points. The construction of higher dimensional curves (in dimensions other than two or three) will not be explicitly mentioned further. We limit ourselves to two dimensional curves and three dimensional curves as our concern is constructing mathematical models of physical objects. The higher dimensional versions of curves occasionally are needed in geometric modelling to model the characteristics of an object, that is position of the object plus color of the object, texture of the object, density of the object, specular nature of the object. The properties of the object are considered as functions on the object and thus become other coordinates for the object. However, our concern in this course is only with the shape of the object.