By Stephen Mann

During this lecture, we learn 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 programs utilized by the pc animation undefined. Bézier/B-splines signify polynomials and piecewise polynomials in a geometrical demeanour utilizing units of keep watch over issues that outline the form of the outside. the first research software utilized in this lecture is blossoming, which provides a sublime labeling of the regulate issues that enables us to investigate their houses 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 regarding blossoming), direct manipulation of B-spline curves, NURBS curves, and triangular and tensor product surfaces.

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**Extra resources for A blossoming development of splines**

**Example text**

But since the two segments yield a different location for f (4, 5, 6, 7), the two segments do not meet C 3 at t = 4. Note that we need two segments to talk about continuity. This need for two segments motivates the conditions on which knots have multiplicity greater than one in the following theorem. 1. Given a degree n B-spline F with knot vector t0 ≤ · · · ≤ ti−1 < ti = · · · = ti+k−1 < ti+k ≤ · · · ≤ tL+2n−2 , where 0 < k ≤ n, i ≥ n, and i + k < L + n − 3, and where no knot occurs with multiplicity greater than n.

3. For F and G to meet with C k continuity at u, we must have f ∗ (u¯ n−i , u¯ 1 , . . , u¯ i ) = g ∗ (u¯ n−i , u¯ 1 , . . , u¯ i ), n−k , u¯ 1 , . . , u¯ k ), n−k , δ k ). for i = 0 . . k. Note that it is sufficient to show that f ∗ (u¯ n−k , u¯ 1 , . . , u¯ k ) = g ∗ (u¯ Proof. cls September 25, 2006 16:36 POLYNOMIAL CURVES 29 Assume that we know f ∗ (u¯ n−k , u¯ 1 , . . , u¯ , δ k− ) = g ∗ (u¯ n−k , u¯ 1 , . . , u¯ , δ k− ) Then f ∗ (u¯ n−k , u¯ 1 , . . , u¯ , δ k− ) = g ∗ (u¯ n−k , u¯ 1 , .

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. This is seen in the definition of the B-spline, since some of our intervals will be of zero length.