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Descriptions of Bezier Curves for the Typo-L Website
by Rodger Whitlock



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Method:

These curves were generated using a Lotus spreadsheet. Control points were generated randomly. Some curves were generated by starting with a random control point and adding random X and Y increments to obtain the next control point. Other curves were generated by directly generating random X and Y coordinates.

From the random curves thus generated, a number considered "typographically interesting" were selected. Generally, those selected are not self-intersecting, but one self-intersecting fifth-order curve BEZ_5_H.GIF was retained.

The quadratic Bezier curves were generated with random off-curve control points, on-curve control points being taken as halfway between successive off-curve points. This mimics a methodology often used in TrueType fonts. Note, however, that TrueType fonts can explicitly position an on-curve control point.

The complex hull of each curve is shown in blue with control points successively designated P1, P2, etc. The resultant Bezier curve is shown in red, and was evaluated at parametric intervals of 0.05 (per segment in multi-segment quadratics); the points evaluated are marked with squares.

Because these curves are not oriented with extrema at horizontal and vertical tangents, their use in actual glyph design might prove problematic.

The quartic and quintic Beziers shown do not correspond to either the TrueType or the Type 1 font formats. In practice, if one desired to use one of these curves to describe a glyph shape, one would dissect it into segments and approximate each segment by an appropriate quadratic or cubic Bezier. In some cases, mulitple segments might be required to attain a sufficiently accurate approximation.

All files are named Bez_(n)_(l).gif where (n) is the order of the curve (2, 3, 4, or 5) and (l) is a simple one-letter identifier.


Bez_2_a through Bez_2_e are two-segment quadratic Bezier curves.
Double Quadratic Curves at 0.05

Double Quadratic Curves at 0.05
BEZ_2_B.gif
Double Quadratic Curves at 0.05
BEZ_2_C.gif
Double Quadratic Curves at 0.05
BEZ_2_D.gif
Double Quadratic Curves at 0.05
BEZ_2_E.gif



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ButtonRodger Whitlock : totototo@pacificcoast.net