CompSci 344 : Spring 2015

Cubic polynomials offer variety and flexibility
Invariant under Affine transformations
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: smoothness of movement along curve (C0, C1, C2: differentiability)
- geometric: smoothness of displayed curve (G0=C0, G1=tangent-cont., G2=curvature-cont.)
- parametric continuity is more expensive to arrange (uses up more degrees of freedom), but is mathematically simpler

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
Provide convex hull, want the curve to stay within the shape defined by the control points

Originally used in ship building to bend wood into hull shape, with strips would be held in place at key points (using lead weights called "ducks")
Also used in model railroads
Standard for modeling curved surfaces: defined piecewise by low-order polynomial functions
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.