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CSCI480/582 Lecture 9 Chap.2.2 Cubic Splines – Hermit and Bezier Feb, 11, 2009

CSCI480/582 Lecture 9 Chap.2.2 Cubic Splines – Hermit and Bezier Feb, 11, 2009. Outline. Parametric curve and arc length Distance-time function in general Spline curve General polynomial spline Hermite spline Bezier spline. Rigid Body Motion – Parametrization and Arc Length along a curve.

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CSCI480/582 Lecture 9 Chap.2.2 Cubic Splines – Hermit and Bezier Feb, 11, 2009

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  1. CSCI480/582 Lecture 9 Chap.2.2 Cubic Splines – Hermit and Bezier Feb, 11, 2009

  2. Outline • Parametric curve and arc length • Distance-time function in general • Spline curve • General polynomial spline • Hermite spline • Bezier spline

  3. Rigid Body Motion – Parametrization and Arc Length along a curve P(.125) Curve Parameterization: Each coordinate of an arbitrary point on the curve is a function of a unique parameterization variable, denoted as u. P(.25) P(0) P(.625) P(.375) P(.75) P(.5) P(1) P(.875) • Arc length: the Euclidean length between two points on a curve, or the distance along a curve, denoted as s • Arc length is usually non-linear with parameterization variable

  4. Distance-Time Function of a Rigid Body Motion Along A Curve S(t1) S(t2) S(t0) S(t5) S(t3) S(t4) S(t6) • Given an initial point P0 on a curve, a point moving from the initial point for an arc length at a time instance of t forms the distance-time function S(t) • The point location at a time instance can be written as

  5. Non-analytic methods to calculate U(s) and S(u) • Sampling and linear approximation

  6. Spline curve in general • Spline: a wide class of functions used in applications requiring data interpolation and/or smoothing • In the polynomial case, a spline is a piecewise polynomial function • Each interval boundary point is termed a knot • Knot vector of a spline:

  7. Spline curve in general – Cont’d • Uniform spline: • Degreen spline: • Smoothness: • Smoothness vector of a spline: A Polynomial Spline Space can be defined by Knot vector: t Smoothness vector: r Degree: n

  8. Polynomials in animation – Order v.s. Motion • Step function: n=0 • Linear Splines: n=1, ri=0 • Natural Cubic Splines: n=3, ri=2, and S’’(a)=S’’(b)=0 Closed/ Polygon Open/ Piece-wise linear lines

  9. Cubic Hermite Spline • On a unit interval (0,1), Given • Starting point p0 at t=0 • Ending point p1 at t=1 • Starting tangent m0 at t=0 • Ending tangent m1 at t=2 • For a natural cubic spline, m0=0, then mk can be chosen as

  10. Cubic Bezier Spline • On a unit interval (0,1), Given • Starting point p0 at t=0 • Point p1 with vector p0p1 be the curve tangent at t=0 • Ending point p3 at t=1 • Point p2 with vector p2p3 be the curve tangent at t=1 • General Bezier curve

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