p 3 0 2 p 3 1 in .NET Draw Code 128 Code Set C in .NET p 3 0 2 p 3 1

1 p 3 0 2 p 3 1 using none toassign none on web,windows application Web app + 2 p1 3 + 1 p2 , 3. Team LRN p1 p1 p2 B zier Curves e p2 = p3 p0 = p0 Figure VII.10. The curve q(u) = q(u) is both a degree two B zier curve with control points p0 , p1 , and p2 e and a degree three B zier curve with control points p0 , p1 , p2 , and p3 .

e. as shown in Figure none for none VII.10. These choices for control points give q(u) the right starting and ending derivatives.

Since q(u) and q(u) both are polynomials of degree 3, it follows that q(u) is equal to q(u). Now, we turn to the general case of degree elevation. Suppose q(u) is a degree k curve with control points p0 , .

. . , pk : we wish to nd k + 1 control points p0 , .

. . , pk+1 which de ne the degree k + 1 B zier curve q(u) that is identical to q(u).

For this, the following de nitions e work: p0 = p0 pi = pk+1 = pk i k i +1 pi 1 + pi . k+1 k+1. Note that the rst two equations, for p0 and pk+1 , can be viewed as special cases of the third by de ning p 1 and pk+1 to be arbitrary points. Theorem VII.8 Let q(u), q(u), pi , and pi be as above.

Then q(u) = q(u) for all u. Proof We need to show that. k+1 i=0 k+1 i u (1 u)k i+1 pi = i k i=0 k i u (1 u)k i pi . i VII.10 The left-hand side of this equation is also equal to k+1 i=0 k+1 i u (1 u)k i+1 i i k i +1 pi 1 + pi . k+1 k+1 Regrouping the sum mation, we calculate the coef cient of pi in this last equation to be equal to k + 1 i + 1 i+1 k +1 k i +1 i u (1 u)k i + u (1 u)k i+1 . i +1 k+1 i k+1 Using the identities. k+1 i+1 i+1 k+1 k+1 k i+1 , k+1 i we nd this is further equal to k (u + (1 u))u i (1 u)k i = i k i u (1 u)k i . i Thus, we have show n that pi has the same coef cient on both sides of Equation VII.10, which proves the desired equality..

Team LRN VII.10 B zier Surface Patches e p0,3. p3,3. p0,0. p3,0. Figure VII.11. A d none for none egree three B zier patch and its control points.

The control points are shown joined e by straight line segments.. VII.10 B zier Surface Patches e This section exten none for none ds the notion of B zier curves to de ne B zier patches. A B zier curve is e e e a one-dimensional curve; a B zier patch is a two-dimensional parametric surface. Typically, a e B zier patch is parameterized by variables u and v, which both range over the interval [0, 1].

e The patch is then the parametric surface q(u, v), where q is a vector-valued function de ned on the unit square [0, 1]2 .. VII.10.1 Basic Properties of B zier Patches e B zier patches of none none degree three are de ned using a 4 4 array of control points pi, j , where i, j e take on values 0, 1, 2, 3. The B zier patch with these control points is given by the formula e. q(u, v) =. i=0 j=0 Bi (u)B j (v)pi, j . VII.11 An example is show n in Figure VII.11. Intuitively, the control points act similarly to the control points used for B zier curves.

The four corner control points, p0,0 , p3,0 , p0,3 , and p3,3 form the e four corners of the B zier patch, and the remaining twelve control points in uence the patch e by pulling the patch towards them. Equation VII.11 can be equivalently written in either of the forms.

q(u, v) =. i=0 3. Bi (u) . j=0 3. B j (v)pi, j VII.12 q(u, v) =. B j (v) . Bi (u)pi, j . VII.13 Consider the cross none for none sections of q(u, v) obtained by holding the value of v xed and varying u. Some of these cross sections are shown going from left to right in Figure VII.12.

Equation VII.12 shows that each such cross section is a degree three B zier curve with control points ri equal e to the inner summation, that is,.
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