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.