1. 08/30/00 Dinesh Manocha, COMP258 Hermite Curves A mathematical representation as a link between the algebraic & geometric form Defined by specifying the end points and tangent vectors at the end points Use of control points –Geometric points that control the shape –Algebraically: used for linear combination of basis functions

2. 08/30/00 Dinesh Manocha, COMP258 Cubic Parametric Curves Power basis: X(u) = a x u 3 + b x u 2 + c x u + d x Y(u) = a y u 3 + b y u 2 + c y u + d y Z(u) = a z u 3 + b z u 2 + c z u + d z P(u) = (X(u) Y(u) Z(u)), u [0,1] Cubic curve defined by 12 parameters Hermite curve: Specified using endpoints and tangent directions at these points

3. 08/30/00 Dinesh Manocha, COMP258 Hermite Cubic Curves P(u) = F 1 (u) P(0) + F 2 (u) P(1) + F 3 (u) P u (0) + F 4 (u) P u (1) where F 1 (u) = 2u 3 – 3u 2 + 1 F 2 (u) = -2u 3 + 3u 2 F 3 (u) = u 3 – 2u 2 + u F 4 (u) = u 3 – u 2, Hermite basis functions The F i (u) are the Hermite basis functions and P(0), P(1), P u (0) and P u (1) are the geometric coefficients The coefficients are specified to maintain continuity between different segments

4. 08/30/00 Dinesh Manocha, COMP258 Hermite Basis Functions Important Characteristics Universality – hold for all cubic Hermite curves Dimensional independence: extend to higher dimension Separation of Boundary Condition Effects: constituent boundary condition coefficients are decoupled from each other (i.e P(0) & P(1)) –Local Control: can modify a single specific boundary condition to alter the shape of the curve locally Can be extended to higher degree curves

5. 08/30/00 Dinesh Manocha, COMP258 Cubic Hermite Curve: Matrix Representation Let B = [P(0) P(1) P u (0) P u (1)] F = [F 1 (u) F 2 (u) F 3 (u) F 4 (u)] or F = [u 3 u 2 u 1] 2-2 1 1 -3 3 -2 -1 0 0 1 0 1 0 0 0 This is the 4 X 4 Hermite basis transformation matrix. P(u) = U M f B, where U = [u 3 u 2 u 1]

6. 08/30/00 Dinesh Manocha, COMP258 Composing Parametric Curves Given a large collection of data points, compute a curve representation that approximates or interpolates Higher degree curves (say more than 4 or 5) can result in numerical problems (evaluation, intersection, subdivision etc.) Need to multiple segments and compose them with appropriate continuity

7. 08/30/00 Dinesh Manocha, COMP258 Parametric & Geometric Continuity Parametric Continuity (or C n ): Two curves have nth order parametric continuity, C n, if their 0 th to n th derivatives match at the end points Geometric Continuity (or G n ): Less restrictive than parametric continuity. Two curves have nth order geometric continuity, G n, if there is a reparametrization of the curve, so that the reparametrized curves have C n continuity. –G 1: Unit tangent vectors at the end point are continuous –G 2: Relates the curvature of the curves at the endpoints –Geometric continuity results in more degrees of freedom