By Hans Hagen
This choice of rules and effects on subject matters of curve and floor layout is meant for learn within the educational setting in addition to for functional use in business functions. major emphasis is on minimum strength splines and geometric spline curves, and on ideas past tensor product surfaces.
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2 Except for r = 2, curvature continuous cubics are equivalent to v-splines. 3 Curves holding higher order geometic continuity properties as well as being equipped with shape parameters which have tension character are described in . But this is at the expense of the polynomial nature, for Pottmann's curves are built up by a polynomial and an exponential part. Interval Weighted Tau-Splines 43 in . The second, independent of us, in , for curve and surface definition, illustrates very nicely the influence of the tension parameters on the curve and surface shape.
4. Open minimal-energy splines. 3b. The reference circle is obtained with ao = 0, 0:3 = —lit, and /i = / 2 = ^3 = 27T/3. 045457T has 0 curvatures at end points P0 and P3. Note that the four-point fixed-length closed natural minimal-energy spline with the constraint set Hn(P0, P1? P2, PS', ^1,^2,^3)? ,^). Finally, we compute four-point various-length open clamped, natural, and mixed minimal-energy splines with the constraint sets Hc(P0,Pi,P2,P3; a 0 , a3; /i,/ 2 ,/ 3 ), ^n(^o, PI, ^2, ^3! 6832. The three open minimal-energy splines are shown in Fig.
3. 19 Turning points in free elastic curves. If the scale invariance property is used to normalize the given data, the parameters of the scaled problem can be obtained by scaling the parameters of the corresponding normalized problem. 1. 31) can be constructed using the relations: Proof. 33) for the curvature functions of y and x. 1). 8. The Shape of Minimal-Energy Splines The behaviour of minimal-energy curves is guided by the principle of avoiding regions with extreme curvature because large curvature results in a large energy value.
Curve and surface design by Hans Hagen