Download A blossoming development of splines by Stephen Mann PDF

By Stephen Mann

During this lecture, we examine Bézier and B-spline curves and surfaces, mathematical representations for free-form curves and surfaces which are universal in CAD structures and are used to layout airplane and vehicles, in addition to in modeling applications utilized by the pc animation undefined. Bézier/B-splines characterize polynomials and piecewise polynomials in a geometrical demeanour utilizing units of keep an eye on issues that outline the form of the skin. the first research instrument utilized in this lecture is blossoming, which supplies a chic labeling of the keep watch over issues that enables us to investigate their houses geometrically. Blossoming is used to discover either Bézier and B-spline curves, and particularly to enquire continuity homes, switch of foundation algorithms, ahead differencing, B-spline knot multiplicity, and knot insertion algorithms. We additionally examine triangle diagrams (which are heavily relating to blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.

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Example text

Moving from right to left in Fig. 2), then we know the piecewise cubic curve is C 2 by construction. Note that I have left off some details: how many knots? How many control points? What about end conditions? But the idea should be clear: we can construct a C 2 piecewise, cubic curve by specifying roughly one point per cubic segment, with no constraints on the control points. The result is a cubic B-spline curve, with the gray points in the right subfigure of Fig. 2 being the cubic B-spline control points.

N f ∗ (0, n−i i (the proof is similar to that for uniqueness of f in the multiaffine blossoming theorem). Now we can show uniqueness as follows: ¯ u, ¯ . . , u) ¯ F(u) = f ∗ (u, ¯ ¯ . . , u) ¯ = f ∗ (0 + uδ, u, ¯ ¯ . . , u) ¯ + u f ∗ (δ, u, ¯ . . , u) ¯ = f ∗ (0, u, .. n n i ¯ u f ∗ (0, . . , 0¯ , δ, . . , δ ) = i i=0 n−i n = i=0 i n i ¯ u g ∗ (0, . . , 0¯ , δ, . . , δ ) i n−i ¯ . . , 0, ¯ δ, . . , δ) = g ∗ (0, ¯ . . , 0, ¯ δ, . . cls 22 September 25, 2006 16:36 A BLOSSOMING DEVELOPMENT OF SPLINES Proof of (2) follows from the construction of f and f ∗ .

In mathematics, a “smooth” curve usually refers to a C ∞ curve However, in geometric modeling, a “smooth” curve usually refers to a piecewise C ∞ curve, where the pieces meet with at least equal position and first derivatives. We will begin by looking at what is required for two B´ezier curves to meet with continuous position and first derivatives (or C 1 continuity). Suppose we wish to join two B´ezier curves together with continuous position and first derivatives. How do we do this? C 0 continuity is trivial: you set the first control point of the second curve equal to the last control point of the first curve.

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