CS 551/651 Advanced Graphics

1. CS 551/651Advanced Graphics Arc Length

2. Assignment 1 • Due Week from Thursday • Building a Cubic Bézier curve • OpenGL/Glut • Insert up to 100 points • Render Bézier curve using recursive subdivision • Left mouse button to add, middle to move, and right to remove • FLTK is extra credit

3. Papers for Next TuesdayPresenters Needed • Tour Into the Picture • William T. Reeves, Particle systems - a technique for modelling a class of fuzzy objects. In SIGGRAPH '83, Computer graphics 17, 3, July 1983. • " Animation of plant development," Przemyslaw Prusinkiewicz, Mark S.Hammel, and Eric Mjolsness. Siggraph 1993.

4. Arc Length • Arc length is the distance along the curve • To ensure constant speed of curve evaluation, perform arc length parameterization of curve • Compute a mapping offline of arc length values to parameter values • Analytically compute arc length

5. Arc Length • Given parameters u1 and u2, find LENGTH(u1, u2) • Given an arc with length s and a parameter u1, find u2 s.t. LENGTH(u1, u2) is s • Can we compute s = G(u) = distance from start of curve to point at u? • If so, G-1 is used to build arc length parameterized curve: P(G-1(s))

6. Analytic Computation Cubic curve example:

7. Forward Differencing • Sample curve a many parameter values • Create piecewise linear representation of curve from parameter evaluations • Compute linear distance between samples • Store distances in table • Limitations/Shortcomings?

8. Adaptive Approach • Adaptively subdivide • First step, compare: • length (start, middle) + length (middle, end) • length (start, end) • If error is too large, subdivide • Use link list to store data

9. Numerical Computation • Numerical methods exist to approximate integral of curve given sample points/derivatives • Instead of using sum of linear segments, use numerical method to compute sum of curved segments

10. Gaussian Quadrature • Commonly used to integrate function between –1 and +1 • Function to be integrated is evaluated at fixed points within [-1, 1] and multiplied by a precalculated weight • More sample points leadsto better accuracy

11. Adaptive Gaussian Integration • Gaussian quadrature uses sampling to preseve accuracy • How many samples are necessary for given curve? • Adaptive gaussian integration monitors errors to subsample when necessary • Start trying few samples and add more as necessary • Recursive like spline subdivision

12. Adaptive Gaussian Integration • Build a table that maps parameter values to arc length values • To solve arc length at u, find ui and ui+1that bound u • Use gaussian quadrature to compute intermediate arc length between that at uiand ui+1

13. Finding u given s • Must compute u s.t. length (u1, u) = s • Solve: s – length(u1, u) = 0 • Find the zeros (roots) of the function • Newton-Raphson does this for us: • f = s – length(u1, pn-1) f’= dp/du evaluated at pn-1

14. Newton-Raphson • Be aware that Newton-Raphson could set pn to a value not defined by curve • f’(pn-1) could be 0 (cannot divide by 0)

15. s(t) t Speed Control • Given arc-length parameterized curve • Let s(t) define amount the curve has traveled given t • For simplicity, normalize total distance to 1 • Monotonic (can’t go backwards) and continuous • To ease in/out, make s(t) = ease (t) that looks like:

16. Speed Control • Easing in/out: • -pi/2 shifts curve rightward • +1 shifts curve upward • Divide by 2 scales from 0 to 1

17. vel(t) v0 t t1 t2 Speed Control • Transcendental functions are expensive • Table lookups are one possibility\ • Roll your own • User specifies acceleration period, deceleration period, and max velocity

18. v0 t t1 t2 Speed Control • Distance traveled is area under curve • User selects two of three unknowns and distance traveled constraint determines third • Integrate velocity function to find s(t)

19. SLERPing • Quaternions are points on the unit sphere in four dimensions • Unit sphere in three dimensions plus a fourth dimension representing the rotation about the normal at a point on the sphere • Interpolating quats • Linearly interpolate each of four components • Doesn’t quite work right

20. SLERPing • Linear interpolation doesn’t produce constant velocity interpolation of quats • Instead, interpolate along the arc on sphere between two quats

21. SLERPing • Which way do we go? • Long way or short way around sphere • q = [s,v] = [-s,-v] = -q • Compute angle between q1 and q2 • Four dimensional dot product • cos(Q)= q1qq2 = s1*s2 + v1q v2 • If positive, this is shortest, • Else, interpolate between q1 and -q2

22. SLERPing • SLERP: • Unit quaternion is not guaranteed • What about continuity of interpolation?

23. Continuity of SLERP • Interpolate between points • [p0, …, pn-1, pn, pn+1, …] • Bezier interpolation would compute intermediate points for each pn • To define tangent at pn • Define bn to be point before pn • Define an to be point after pn

24. Continuity of SLERP • an = (pn – pn+1 + pn * ½) + (½ * pn+1) • pn + pn – an = bn pn pn+ pn – pn+1 an pn-1 pn+1

25. an bn+1 qn+1 qn Continuity of SLERP • To compute midway (average) points between quats, SLERP halfway • Instead of adding vectors, concatenate rotations P1 = slerp(qn, qn, 1/3) P2 = slerp(an, bn+1, 1/3) P3 = slerp (bn+1, qn+1, 1/3) P12 = slerp (p1, p2, 1/3) P23 = slerp (p2, p3, 1/3) P = slerp (p12, p23, 1/3)