Geometric manipulation of tensor product surfaces

Tensor product surfaces are now widely used in application areas such as industrial design and computer animation and thus the quest for more effective design methods continues. Although several methods exist for applying high-level operators such as bends, twists and free-form deformations (FFD's), much less effort has been applied to improving direct and precise free-form shaping which is often desired. The dominant form of free-form manipulation has been control-point based. Here we offer a manipulative method that presents geometric properties (e.g. points on the surface, normal vectors, etc.), rather than control vertices or deformation lattices, and allows direct manipulation of these properties at any selected point on the surface. The difficulties of interacting with these threedimensionalgeometric entities usingboth two-and three-dimensional input devices are discussed, as are possible interactive schemes using several such devices.