1cs426-winter-2008 Notes  Ian Mitchell is running a MATLAB tutorial, Tuesday January 15, 5pm-7pm, DMP 110 We won’t be directly using MATLAB in this course, but it can be extremely useful for plotting curves, testing formulas, debugging mathematical expressions, etc.

2cs426-winter-2008 Example: piecewise linear spline  Restrict all polynomials to be linear f(t)=a i (t-t i )+b i in [t i, t i+1 ]  Enforce continuity: make each line segment interpolate the control point on either side a i (t i -t i )+b i =y i and a i (t i+1 -t i )+b i =y i+1  Solve to get a i =(y i+1 -y i )/(t i+1 -t i ) b i =y i  End result: straight line segments connecting the control points  C 0 but not C 1

3cs426-winter-2008 Choices in Animation  Piecewise linear usually not smooth enough: need higher degree polynomials  For motion curves, cubic splines basically always used  Three main choices: Hermite splines: interpolating, up to C 1 Catmull-Rom: interpolating C 1 B-splines: approximating C 2

4cs426-winter-2008 Cubic Hermite Splines  Our generic cubic in an interval [t i,t i+1 ] is q i (t) = a i (t-t i ) 3 +b i (t-t i ) 2 +c i (t-t i )+d i  Make it interpolate endpoints: q i (t i )=y i and q i (t i+1 )=y i+1  And make it match given slopes: q i ’(t i )=s i and q i ’(t i+1 )=s i+1  Work it out to get

5cs426-winter-2008 Hermite Basis  Rearrange the solution to get  That is, we’re taking a linear combination of four basis functions Note the functions and their slopes are either 0 or 1 at the start and end of the interval

6cs426-winter-2008 Breaking Hermite Splines  Usually specify one slope at each knot  But a useful capability: use a different slope on each side of a knot We break C 1 smoothness, but gain control Can create motions that abruptly change, like collisions  Aside: artists like to break things! Animation systems should have as much flexibility as possible

7cs426-winter-2008 Catmull-Rom Splines  This is really just a C 1 Hermite spline with an automatic choice of slopes Use a 2nd order finite difference formula to estimate slope from values For equally spaced knots, simplifies to