![](\"..\/..\/writing\/graphics\/KnotVectorsThumb.jpg\")
<\/a><\/div>\nAn Introduction To Knot Vectors<\/b> A tutorial on B-Spline curves, and in particular, on how to use the knot vector to control their shape. (PDF<\/a>, 104K)<\/div>\n
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<\/a><\/div>\nArc Length Parametrization of Spline Curves<\/b>. A useful mathematical exercise: How to use Chebyshev polynomials to create spline curves with arc-length parametrization (PDF<\/a>, 226K)<\/div>\n
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<\/a><\/div>\nThe Utah Raster Toolkit<\/b> Image processing in the days before Photoshop. Later made popular by Craig Kolb’s Rayshade<\/i> (PDF<\/a>, 1.4MB)<\/div>\n
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<\/a><\/div>\nRay Tracing Spline Surfaces with Motion Blur<\/b> 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<\/a>, 328K)<\/div>\n
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<\/a><\/div>\nDistributed Computation for Computer Animation<\/b> A survey of techniques for lashing together Unix workstations to compute animated movie frames. (PDF<\/a>, 187K)<\/div>\n
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<\/a><\/div>\nTessellation of NURB Surfaces<\/b> Originally published in Graphics Gems IV<\/i>, 1994. As of this writing it’s still in print (and I get a whopping $0.07 for each copy sold!)<\/span>.\n A great tutorial on how to turn surfaces into triangles for rendering or computation. (Amazon<\/a>\n <\/div>\n
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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). \u00a0Out of thirty-odd patents, here are a few of the more interesting ones:<\/p>\n\n\n\n
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<\/a><\/div>\n5,428,718<\/a>: Tessellation System<\/b> – Some interesting tricks for speeding up B-Spline surface tessellation (w\/Shaun Ho)<\/div>\n
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\n![](\"..\/..\/writing\/graphics\/US6911980.png\")
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<\/a><\/div>\n6,911,980<\/a>: Manipulation of Curves and Surfaces<\/b> – Direct manipulation UI techniques for 2D warping (w\/Martin Newell)<\/div>\n
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\n![](\"..\/..\/writing\/graphics\/US7006707.png\")
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<\/div>\n7,006,707<\/a>: Projecting Images Onto a Surface<\/b> – Re-constructing camera parameters from a perspective-warped frame<\/div>\n
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\n![](\"..\/..\/writing\/graphics\/US7385612.png\")
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<\/a><\/div>\n7,385,612<\/a>: Distortion of Raster and Vector Artwork<\/b> – Applying the same Bezier warp to both vector and raster artwork simultaneously<\/div>\n
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\n![](\"..\/..\/writing\/graphics\/US8378781.png\")
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<\/a><\/div>\n8,378,781<\/a>: Animated Light String System<\/b> – Synchronized animations on a string of lights over a single daisy-chained serial line<\/div>\n
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\n![](\"..\/..\/writing\/graphics\/US11069028.png\")
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<\/a><\/div>\n11,069,028<\/a>: Automated Generation of Anamorphic Images for Catoptric Anamorphosis<\/b> – Automatically generate distorted images that look correct when viewed reflected in a cylinder<\/div>\n
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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 read more