“Resolution independent curve rendering using programmable graphics hardware” by Loop and Blinn

We present a method for resolution independent rendering of paths and bounded regions, defined by quadratic and cubic spline curves, that leverages the parallelism of programmable graphics hardware to achieve high performance. A simple implicit equation for a parametric curve is found in a space that can be thought of as an analog to texture space. The image of a curve’s Bézier control points are found in this space and assigned to the control points as texture coordinates. When the triangle(s) corresponding to the Bézier curve control hull are rendered, a pixel shader program evaluates the implicit equation for a pixel’s interpolated texture coordinates to determine an inside/outside test for the curve. We extend our technique to handle anti-aliasing of boundaries. We also construct a vector image from mosaics of triangulated Bézier control points and show how to deform such images to create resolution independent texture on three dimensional objects.

1. Arnon, D. 1983. Topologically reliable display of algebraic curves. Siggraph 1983 Conference Proceeding 17, 3 (July), 219–227. Google ScholarDigital Library
2. Blinn, J. 2003. Jim Blinn’s Corner Notation, notation, notation. Morgan Kaufmann. Chap. 14,15,16, and 19. Google ScholarDigital Library
3. Gumhold, S. 2003. Splatting illuminated ellipsoids with depth correction. In Proceedings of 8th International Fall Workshop on Vision, Modelling and Visualization 2003, 245–252.Google Scholar
4. Gupta, S., and Sproull, R. 1981. Filtering edges for gray-scale displays. Siggraph 1981 Conference Proceeding 15, 3, 1–5. Google ScholarDigital Library
5. Prasad, L. 1997. Morphological analysis of shapes. CNLS Newsletter 139 (July), 1–18.Google Scholar
6. Press, W., Teukolsky, S., Vetterling, W., and Flannery, B. 1992. Numerical Recipes in C. Cambridge Press. Google ScholarDigital Library
7. Ramanarayanan, G., Bala, K., and Walter, B. 2004. Feature-based textures. In Eurographics Symposium on Rendering, Eurographics Association. Google ScholarDigital Library
8. Salmon, G. 1852. A Treatise on the Higher Plane Curves. Dublin, Hodges & Smith. available online at http://name.umdl.umich.edu/ABQ9497.Google Scholar
9. Sederberg, T. 1983. Implicit and Parametric Curves and Surfaces for Computer Aided Geometric Design. PhD thesis, Purdue University. Mechanical Engineering Department. Google ScholarDigital Library
10. Sen, P. 2004. Silhouette maps for improved texture magnification. In Proceedings of the ACM SIGGRAPH/EUROGRAPHICS Conference on Graphics Hardware, Eurographics Association. Google ScholarDigital Library
11. Stone, M., and DeRose, T. 1989. A geometric characterization of parametric cubic curves. ACM Transactions on Graphics 8, 4 (July), 147–163. Google ScholarDigital Library
12. Taubin, G. 1994. Distance approximations for rasterizing implicit curves. ACM Transactions on Graphics 13, 1 (January), 3–42. Google ScholarDigital Library
13. Tumblin, J., and Choudhury, P. 2004. Bixels: Picture samples with sharp embedded boundaries. In Eurographics Symposium on Rendering, Eurographics Association. Google ScholarDigital Library
14. Tupper, J. 2001. Reliable two-dimensional graphing methods for mathematical formulae with two free variables. Siggraph 2001 Conference Proceeding, 77–86. Google ScholarDigital Library
15. Turkowski, K. 1982. Anti-aliasing through the use of coordinate transformations. ACM Transactions on Graphics 1, 3 (July), 215–234. Google ScholarDigital Library