Here, for the record, are some technical papers I wrote in college and shortly thereafter.

**An Introduction To Knot Vectors**A tutorial on B-Spline curves, and in particular, on how to use the knot vector to control their shape. (PDF, 104K)

**Arc Length Parametrization of Spline Curves**. A useful mathematical exercise: How to use Chebyshev polynomials to create spline curves with arc-length parametrization (PDF, 226K)

**The Utah Raster Toolkit**Image processing in the days before Photoshop. Later made popular by Craig Kolb’s

*Rayshade*(PDF, 1.4MB)

**Ray Tracing Spline Surfaces with Motion Blur**A distillation of my MS thesis, it describes how to efficiently ray trace spline surfaces. Some of the techniques, such as the “pixel thresher” are still interesting. (PDF, 328K)

**Distributed Computation for Computer Animation**A survey of techniques for lashing together Unix workstations to compute animated movie frames. (PDF, 187K)

**Tessellation of NURB Surfaces**Originally published in

*Graphics Gems IV*, 1994. As of this writing it’s still in print (and I get a whopping $0.07 for each copy sold!). A great tutorial on how to turn surfaces into triangles for rendering or computation. (Amazon)

## Patents

More recent research efforts have been consumed by lawyers, further obfuscated, and fed to the US Patent office (a side effect of corporate patent filing bonuses). Out of twenty-odd patents, here are a few of the more interesting ones:

5,428,718:

**Tessellation System**– Some interesting tricks for speeding up B-Spline surface tessellation (w/Shaun Ho)6,911,980:

**Manipulation of Curves and Surfaces**– Direct manipulation UI techniques for 2D warping (w/Martin Newell)7,006,707:

**Projecting Images Onto a Surface**– Re-constructing camera parameters from a perspective-warped frame7,385,612:

**Distortion of Raster and Vector Artwork**– Applying the same Bezier warp to both vector and raster artwork simultaneously8,378,781:

**Animated Light String System**– Synchronized animations on a string of lights over a single daisy-chained serial line