By Stephen Mann

ISBN-10: 1598291165

ISBN-13: 9781598291162

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 platforms and are used to layout plane and cars, in addition to in modeling programs utilized by the pc animation undefined. Bézier/B-splines symbolize polynomials and piecewise polynomials in a geometrical demeanour utilizing units of regulate issues that outline the form of the outside. the first research instrument utilized in this lecture is blossoming, which provides a sublime labeling of the keep an eye on issues that enables us to research their houses geometrically. Blossoming is used to discover either Bézier and B-spline curves, and specifically 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 concerning blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.

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Write an interactive 2D cubic B-spline editor with the following functionality: • The left mouse button adds a new control point. • The middle mouse button is used to move control points. , when adding a new control point, assume the value of any new knot to be one more than the last knot in the knot sequence). • There are two display modes: – Just the curve. – The curve and the control polygon. • There should be a reset key/menu option that clears all the control points. 2 KNOT MULTIPLICITY If a knot has multiplicity greater than 1, then some of the B-spline segments are of zero length.

0¯ , δ, . . , δ ) i n−i i F (i) (0)u i /i! by Taylor expansion, and since the monomials form a basis, we have F (i) (0) = n! ¯ . . , 0¯ , δ, . . , δ ) f ∗ (0, (n − i)! n−i i Now, we have the following: n! n! ¯ . . , u¯ , δ, . . , δ ) = f ∗ (u, (n − j )! (n − j )! n− j n! (n − j )! n− j n! (n − j )! n− j n− j j = = n− j = k=0 ( j) F k=0 k n− j −k j j +k (n − j )! (n − j − k)! (n − j − k)! n! (0) k! n− j −k n− j ¯ . . , 0¯ , δ, . . , δ )u k f ∗ (0, k k=0 k=0 ( j +k) = F (u) n− j ¯ . .

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). ” We begin with a brief review of continuity. 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.

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