A method of representing n-dimensional parametric surfaces (animated
shapes) is provided that expresses all shapes in a consistent manner to
reduce storage requirements, support deformation and simplify interaction
between shapes. The method, a version of sweeps, represents shapes using
a unique combination of three discrete types of (piecewise polynomial)
curves: spine (sweeping) curves, slice (section) curves, and lathe
(plane) curves, which are combined to create surfaces. The curves
required to make simple 3D primitives (i.e. torus, sphere, cube and
pyramid) are themselves simple 2D primitives (i.e. line, circle, square,
triangle). The storage size of this system's shapes is exponentially
smaller than the size of polygonal versions of the same shapes (as a
function of the number of polygons). Complex models can be broken into
multiple shapes, which are arranged in a tree hierarchy. The shapes of
this invention can be tiled with other shapes of this invention (i.e. a
row of smokestacks made of bricks). The shapes of this invention can
smoothly travel on other shapes of this invention (i.e. a football
rolling over arbitrary terrain). This invention supports fast, intuitive
creation of shapes such as hand-drawn shapes. This invention is a closed
system, providing a suite of operations that can occur in arbitrary order
sans approximation errors. When scenes are constructed of parametric
building blocks, each represented with the same universal formula, a
suite of advanced operations becomes available, providing support
critical to advanced software simulations.