A blossoming development of splines by Stephen Mann

By Stephen Mann

During this lecture, we research Bézier and B-spline curves and surfaces, mathematical representations for free-form curves and surfaces which are universal in CAD platforms and are used to layout plane and vehicles, in addition to in modeling applications utilized by the pc animation undefined. Bézier/B-splines signify polynomials and piecewise polynomials in a geometrical demeanour utilizing units of keep an eye on issues that outline the form of the outside. the first research device utilized in this lecture is blossoming, which provides a sublime labeling of the regulate issues that permits us to investigate their houses geometrically. Blossoming is used to discover either Bézier and B-spline curves, and particularly to enquire continuity houses, swap of foundation algorithms, ahead differencing, B-spline knot multiplicity, and knot insertion algorithms. We additionally examine triangle diagrams (which are heavily with regards to blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.

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F ∗ (α u, Now let us evaluate our multilinear blossom at f ∗ (u¯ 1 , u¯ 2 , δ). Then F(u) = 3u 3 + 2u 2 + 6u + 1 f ∗ (u¯ 1 , u¯ 2 , u¯ 3 ) = 3u 1 u 2 u 3 + 2(u 1 u 2 w3 + u 2 u 3 w1 + u 3 u 1 w2 )/3 +2(u 1 w2 w3 + u 2 w3 w1 + u 3 w1 w2 ) + w1 w2 w3 f ∗ (u¯ 1 , u¯ 2 , δ) = 3u 1 u 2 + 2(0 + u 2 w1 + u 1 w2 )/3 + 2(0 + 0 + w1 w2 ) + 0 = 3u 1 u 2 + 2(u 2 w1 + u 1 w2 )/3 + 2w1 w2 ¯ u, ¯ δ) = 3u 2 + 4u/3 + 2 f ∗ (u, = F (1) (u)/3 By computing the derivative of F in the usual fashion, we see that the last step is true.

And finally, while speed is the primary advantage of forward differencing, realize that de Casteljau’s algorithm is easily fast enough to draw curves at interactive rates. Thus, the complicated fast forward differencing algorithm is unlikely to be chosen over de Casteljau’s algorithm for drawing curves. However, we will briefly return to forward differencing when we look at surfaces, where the additional number of evaluation points may again make forward differencing an algorithm worth considering.

U n−k = g ti , u 1 , . . , u n−k This is seen by noting that the segment defined over [ti−1 , ti ] is defined by the control points k f (ti−n , . . , ti−1 ), . . , f (ti , . . , ti+n−1 ) = f ti , ti+k , ti+k+1 , . . cls T1: IML September 25, 2006 16:37 B-SPLINES 45 while the segment defined over [ti = ti+k−1 , ti+k ] is defined by the control points f ti−n+k , . . , ti−1 , ti k , . . , f (ti+k , . . , ti+k+n−1 ) (we consider the interval [ti , ti+k ] because it is the next nonzero length interval after [ti−1 , ti ]).

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