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Conic splines are a useful graphic primitive. They exactly represent any conic
section: line, circle, ellipse, parabola, or hyperbola. Lines and circles are
of obvious importance, and parabolic splines are a primitive building block for
shapes in QuickDraw GX. To maintain closure under the full set of perspective
transformations allowed by QuickDraw GX, the full set of conic sections must be
used.
This Technote gives a derivation of some of the mathematical formulas
associated with conic splines. It defines a quadratic rational spline as a
weighted mean of three control points whose weights vary quadratically in the
parameter t. A canonical form is derived for the most general form of the
weighted mean. Then the effect of perspective transforms on the weights and
control points is explained. Finally, a method is derived for determining which
conic section contains a given conic spline.
This Technote is directed primarily at developers working with the paths and
perspective transforms defined in QuickDraw GX. A firm grasp of those concepts
is necessary to understanding this Technote.
Updated: [June 1 1996]
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This Technote is heavily dependent upon mathematical derivation, which HTML does not yet adequately support. In order to ensure the mathematical integrity of the text, we are not publishing the body of the Technote as an HTML file. You can download the Technote, in its entirety, as a PDF document, by clicking here.
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Technote 1052
Technote 1052 - QuickDraw GX ConicLibrary.c in Detail: Description, and Derivations also addresses the concept of conic splines, and approaches it from
a different perspective. See Inside Macintosh: QuickDraw GX Graphics and Inside Macintosh: QuickDraw GX Objects for further
documentation.
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References
Technote 1052 - QuickDraw GX ConicLibrary.c in Detail: Description, and Derivations
Inside Macintosh: QuickDraw GX Graphics
Inside Macintosh: QuickDraw GX Objects
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