By Brian A. Barsky

Special effects and Geometric Modeling utilizing Beta-splines (Computer technology Workbench) [Hardcover] [May 03, 1988]

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3 Perturbation Due to the Movement of a Control Vertex 57 parameters can be modified in a manner that is more efficient than a complete recomputation. The following algorithm perturbs a general curve where the i 1h curve segment is computed at P; + 1 values of the domain parameter: for i := i- 1 to i + 2 do begin jump1 := alpha1;- alpha1;_ 1 ; jump2 := alpha2;- alpha2;_ 1 ; for each u in {u;klk = 0, 1, ... , p;} do begin s := (10 + (6 * u - 15) * u) * u * u * u; beta1 := alpha1;_ 1 + s *jump1; beta2 := alpha2;_ 1 + s *jump2; compute_d (beta1, beta2, d); compute_delta (d, delta); compute_currenLa (i- i, d, u, a); b;_; := a;_Jdelta; Qi'ew(u) := Qi''d(u) + b;-; * end end; vt where procedure compute_currenLa (r; d; u; a); begin (* compute_currenLa *) case r of -2: a_ 2 (beta1, beta2; u) := d0 , _ 2 * (1 - u) 3 ; -1: a_ 1 (beta1, beta2; u) := d0 , - 1 + u * (d 1 , - 1 + u * (d 2 , - 1 + u * d3 , -d); 0: a0 (beta1, beta2; u) := d0 , 0 + u * (d 1 , 0 + u * (d 2 , 0 + u * d3 , 0 )); 1: a 1 (beta1, beta2; u) := 2 * u 3 end(* case *) end (* compute_currenLa *); For comparison purposes, assume that the four curve segments affected were computed at the same set of p + 1 values of the domain parameters.

3) 6 IX1 2 Q~l(1) = _ _ m[Vm- Vm- 1] Ym An important property of an end condition is the curvature at the endpoints of the curve. It is sometimes desirable to have zero curvature at an endpoint to join the curve to a straight line, while other applications require nonzero curvature. 4, and required the evaluation of the first and second derivative vectors.

Fig. 3. The effects of moving a Beta-spline control vertex are confined to four segments. 7 Derivation of the Beta-spline Curve Representation 32 The Beta-spline formulation exploits the piecewise representation, in order to achieve local control, by defining each piece in terms of only a few nearby vertices. F or Beta-spline curves, each curve segment is controlled by only four of the control vertices and is completely unaffected by all the other control vertices. Equivalently, a given control vertex only influences four curve segments and has no effect whatsoever on the remaining segments.