1cs426-winter-2008 Notes  Assignment 0 is due today  MATLAB tutorial tomorrow 5-7 if you’re interested (see web-page for link)

2cs426-winter-2008 Refinability  Cubic Hermite splines have an additional important feature: you can easily refine them  You can add a new knot and control point in between existing ones without changing the curve Just use the value and derivative from the original spline  Other splines (including Catmull-Rom) don’t necessarily have this property

3cs426-winter-2008 Catmull-Rom Boundaries  Recall: Catmull-Rom is a cubic Hermite with an automatic choice for slopes based on control points  Need to use slightly different formulas for the boundaries  For example, 2nd order accurate finite difference at the start of the interval: Symmetric formula for end of interval  Which simplifies for equal spaced knots:

4cs426-winter-2008 Catmull-Rom Basis  Catmull-Rom also implies a set of basis functions…  Can you work out the basis function for equally spaced knots?

5cs426-winter-2008 B-Splines  We’ll drop the interpolating condition, and instead design a basis that is C 2 smooth So control points say how much of each basis function to use, not exactly where the curve goes  This time a basis function overlaps more than one interval  Want to be able to interpolate constants  We won’t cover full derivation

6cs426-winter-2008 B-Spline Basis  Define recursively: each order is a linear blend of previous order  Note: this construction allows nonuniform knots, unlike assignment 0 version, but the 0 and 2 order splines are shifted…

7cs426-winter-2008 Looking at B-splines  B i,3 (t) peaks at (or near) knot t i, but is nonzero on the interval [t i-2, t i+2 ]  Always ≥ 0, Always < 1, Basis functions add up to 1 everywhere Any point on the spline curve is a weighted average of nearby control points

8cs426-winter-2008 Control  Local control: adjusting a control point only changes curve locally Far enough away, curve is exactly the same  Global control: adjusting one control point changes entire curve Not as desirable - working on one part of the curve can perturb the parts you already worked out to perfection But, for decent splines, effect is small--- decays quickly away from adjustment

9cs426-winter-2008 Controlling Cubics  All three of the cubic splines we saw have local control  But if we enforce C 2 smoothness and make it interpolating, we end up with global control Have to solve a big linear system to determine cubic coefficients…

10cs426-winter-2008 Summary  Cubic Hermite Spline: the standard tool for animation. C 1, interpolating, local control. Smoothness easy to break if needed: flexible!  Catmull-Rom: a good default choice for the slopes, based on finite difference formulas  Cubic B-Spline: C 2, approximating, local control. Not so useful for animating in time, very useful for defining geometry