A blossoming development of splines - download pdf or read online
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 autos, in addition to in modeling programs utilized by the pc animation undefined. Bézier/B-splines characterize polynomials and piecewise polynomials in a geometrical demeanour utilizing units of regulate 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 homes geometrically. Blossoming is used to discover either Bézier and B-spline curves, and specifically to enquire continuity houses, switch of foundation algorithms, ahead differencing, B-spline knot multiplicity, and knot insertion algorithms. We additionally examine triangle diagrams (which are heavily on the topic of blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.
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Extra info for A blossoming development of splines
Two curves F(t) and G(t) are said to meet with C k continuity at t0 if F (i) (t0 ) = G (i) (t0 ) for 0 ≤ i ≤ k. If there is a discontinuity in the position of a curve at parameter value t0 , then in geometric modeling, we commonly say that the curve is “C −1 ” at t0 . A curve is said to be C ∞ if all its derivatives are continuous everywhere. 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.
Our goal is to compute F(0 + ih) for i running from 0 to S, where S + 1 is the number of samples of F that we want and h = 1/S. Starting with a linear function F(u), consider computing F(u + h) when we already know F(u): F(u + h) = = = = f (u + h) = f ∗ (u¯ + hδ) ¯ + h f ∗ (δ) f ∗ (u) f (u) + h f ∗ (δ) F(u) + 1 , where 1 = h f ∗ (δ). If we precompute 1 = h f ∗ (δ), then we can compute f (u + h) from f (u) with a single addition (albeit an addition of an n-dimensional point with an n-dimensional vector) and in turn we can compute f (u + 2h) from f (u + h) with a second addition reusing the same precomputed 1 .
Find the tangent line to B(0) for the portion of the curve parameterized over [0, 1]. Prove your result. 3. Given a two-space quadratic polynomial in B´ezier form over the interval [0, 1] (this specifies the control points; the domain of the curve is the entire real line) and its biaffine blossom f , is there a blossom value of f for every point in the range? If so, give a formula/algorithm for determining a range point’s blossom arguments. , given a point (x, y) in the plane, find u, v such that f (u, v) = (x, y).
A blossoming development of splines by Stephen Mann