Making Smooth Curves
 | Cubic polynomials offer variety and flexibility |
| Goals when defining and editing, i.e., small changes to input cause small
changes in output
 | local control of shape |
| stability |
| smoothness and continuity when combining |
| ease of rendering |
|
| Degrees of smoothness, i.e., continuity
| parametric: differentiability of parametric representation (C0, C1,
C2, ...) |
| geometric: smoothness of resulting displayed shape (G0=C0,
G1=tangent-cont., G2=curvature-cont. ) |
|
Control Points
| Provide degrees of freedom, DOF, in specifying curve |
| Levels of flexibility
| must be points on curve |
| start and end of curve plus tangents at start and end |
| no points on curve, instead curve is interpolated through points using
blending functions |
|
Splines
| Originally used in ship building to bend wood into hull shape |
| Also used in model
railroads |
| Modeled in OpenGL using evaluators to generate vertices from array
of control points |
| Modeled in GLU using NURBS object |
Rendering Curves
| Can be expensive for high-quality approximation if simply evaluate at each
point |
| Evaluate using Horner's rule for factoring reduces multiples to O(n) |
| Approximate by sub-dividing control points based on de Casteljau's
geometric construction |
| Approximate using forward differencing reduces to adding pre-computed
constants each step |
Example
| teapot.cpp
rendering the Utah teapot using spline patches |
References
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