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 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 P(0) P(1) P(.375) P(.625) P(.75) P(.875) P(.25) 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(.5)

Distance-Time Function of a Rigid Body Motion Along A Curve Given an initial point P 0 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 S(t 0 ) S(t 1 ) S(t 2 ) S(t 3 ) S(t 4 ) S(t 5 ) S(t 6 )

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

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:

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

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

On a unit interval (0,1), Given  Starting point p 0 at t=0  Ending point p 1 at t=1  Starting tangent m 0 at t=0  Ending tangent m 1 at t=2 For a natural cubic spline, m 0 =0, then m k can be chosen as Cubic Hermite Spline

On a unit interval (0,1), Given  Starting point p 0 at t=0  Point p 1 with vector p 0 p 1 be the curve tangent at t=0  Ending point p 3 at t=1  Point p 2 with vector p 2 p 3 be the curve tangent at t=1 General Bezier curve Cubic Bezier Spline