CompSci 124 : Fall 2009

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. )

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

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

Different spline types will have different tradeoffs and capabilities:

Hermite splines: Require tangent vectors at each keyframe.
Bézier splines: Require additional information for each segment and do not automatically maintain continuity.
Catmull-Rom splines: Provide only first derivative continuity.
B-splines: Give automatic continuity control, but only approximate their control points.
Non-uniform Rational B-splines (NURBS): Provide additional local control over the shape of a curve.
Non-uniform B-splines: Are needed to break continuity.