Real-time GPU rendering of piecewise algebraic surfaces

A graphics system for generating a visual representation of a three-dimensional surface, the surface being described by reference to one or more polynomials bounded by one or more Bézier tetrahedra having vertices, the graphics system comprising:

• a vertex processor configured to:
• defining a bounding Bézier tetrahedron as a set of coordinates and a symmetric weight tensor containing Bézier weights which define a shape of the surface within the bounding Bézier tetrahedron,
• project the bounding Bézier tetrahedron onto a planar screen space,
• divide the projected bounding Bézier tetrahedron into triangles having three vertices,
• associate with each vertex of a triangle in the projected bounding Bézier tetrahedron a minimum and maximum height value as vertex attribute data,
• compute a partially collapsed symmetric tensor at each vertex of the triangle, the partially collapsed symmetric tensor having a lower rank than the symmetric weight tensor and representing a common factor in coefficients of a univariate polynomial bounded by the bounding Bézier tetrahedron, and
• associate with each vertex of the triangle unique elements of the partially collapsed tensor as vertex attribute data to be interpolated when rasterizing the triangle;
• an interpolator configured to interpolate the vertices and vertex attribute data across the triangle;
• and
• a pixel processor configured to:
• reconstruct the partially collapsed symmetric tensor for a pixel within the triangle using the vertex attribute data,
• compute the coefficients of the univariate polynomial bounded by the bounding Bézier tetrahedron,
• determine if the three-dimensional surface is visible at the pixel within the triangle in the projected bounding Bézier tetrahedron based on the computed coefficients of the univariate polynomial, and
• determine color and z-depth information if the three-dimensional surface is visible at the pixel within the triangle.

A computer-readable storage medium that does not consist of a signal, said computer-readable storage medium storing computer-executable instructions that, when executed by a graphics processing unit of a computer, cause the computer to perform a computer-implemented method for rendering an algebraic surface defined by one or more polynomials bounded by one or more Bézier tetrahedra, the computer-executable instructions comprising:

• a pixel shader program configured to compute coefficients of a univariate polynomial at each pixel within a triangle formed by a projection onto planar screen space of one or more of the bounding Bézier tetrahedra and to solve for polynomial roots of the univariate polynomial at each pixel based on the computed coefficients for determining if the algebraic surface at a particular pixel within the triangle exists, wherein:
• the pixel shader skips solving for polynomial roots corresponding to a pixel if all coefficients of the univariant polynomial have equivalent signs,
• the univariant polynomial results from evaluating a multivariant polynomial at the pixel, and
• the multivariant polynomial describes the algebraic surface within at least one Bézier tetrahedra.