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# Krzywe i powierzchnie Béziera

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q(u) p 0 p 1 p 2 p 3

Figure VII.1: A degree three Bezier urve q(u). The urve is parametri ally

de ned with 0  u  1, and it interpolates the rst and last ontrol points with

q(0) = p

0

and q(1) = p

3

. The urve is \pulled towards" the middle ontrol

points p 1 and p 2 . At p 0

, the urve is tangent to the line segment joining p

0

and p

1

. At p

3

, it is tangent to the line segment joining p

2

and p

3 .

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### Bryła obrotowa

p 0 p 1 p 2 p 3 p 0 p 1 p 2 p 3

Figure VII.2: Two degree three Bezier urves, ea h de ned by four ontrol

points. The urves interpolate only their rst and last ontrol points, p

0

and

p

3

. Note that, just as in gure VII.1, the urves start o , and end up, tangent

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B 0 B 1 B 2 B 3 1 1 0 y u

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### Bryła obrotowa

p 0 p 1 p 2 p 3 r 0 r 1 r 2 s 0 s 1 t 0

Figure VII.4: The de Casteljau method for omputing q(u) for q a degree

three Bezier urve. This illustrates the u = 1=3 ase.

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## u =

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p 0 p 1 p 2 p 3 r 0 r 1 r 2 s 0 s 1 t 0 q 1 (u) q 2 (u)

Figure VII.5: The de Casteljau method for omputing q(u) for q a degree

three Bezier urve is the basis for nding the new points needed for re ursive

subdivision. Shown here is the u = 1=2 ase. The points p

0 ;r 0 ;s 0 ;t 0 are the

ontrol points for the Bezier urve q

1

(u) whi h is equal to the rst half of the

urve q(u), i.e., starting at p

0 and ending at t 0 . The points t 0 ;s 1 ;r 2 ;p 3 are

the ontrol points for the urve q

2

(u) equal to the se ond half of q(u), i.e.,

starting at t 0 and ending at p 3 .

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### Bryła obrotowa

p 0 p 1 p 2 p 3 r 0 r 1 r 2 s 0 s 1 t 0 q 1 (u) q 2 (u)

Figure VII.5: The de Casteljau method for omputing q(u) for q a degree

three Bezier urve is the basis for nding the new points needed for re ursive

subdivision. Shown here is the u = 1=2 ase. The points p

0 ;r 0 ;s 0 ;t 0 are the

ontrol points for the Bezier urve q

1

(u) whi h is equal to the rst half of the

urve q(u), i.e., starting at p

0 and ending at t 0 . The points t 0 ;s 1 ;r 2 ;p 3 are

the ontrol points for the urve q

2

(u) equal to the se ond half of q(u), i.e.,

starting at t

0

and ending at p

3 .

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p 0 p 1 p 2 p 3 r 0 r 1 r 2 s 0 s 1 t 0

Figure VII.4: The de Casteljau method for omputing q(u) for q a degree

three Bezier urve. This illustrates the u = 1=3 ase.

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### kontrolnych

p 0 p 1 p 2 p 3 r 0 r 1 r 2 s 0 s 1 t 0 q 1 (u) q 2 (u)

Figure VII.6: The onvex hull of the ontrol points of the Bezier urves shrinks

rapidly during the pro ess of re ursive subdivision. The whole urve is inside

its onvex hull, i.e., inside the quadrilateral p

0 p 1 p 2 p 3

. After one round of

subdivision, the two sub urves are known to be onstrained in the two onvex

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### Bryła obrotowa

q 1 (u) q 2 (u) p 1;0 p 1;1 p 1;2 p 1;3 = p 2;0 p 2;1 p 2;2 p 2;3 (a) q 1 (u) q 2 (u) p 1;0 p 1;1 p 1;2 p 1;3 = p 2;0 p 2;1 p 2;2 p 2;3 (b)

Figure VII.7: Two urves, ea h formed from two Bezier urves, with ontrol

points as shown. The urve in part (a) is G 1

- ontinuous, but not C 1

- ontinuous. The urve in part (b) is neither C 1

- ontinuous nor G 1

- ontinuous.

Compare these urves to the urves of gures VII.5 and VII.6 whi h are both

C 1 - ontinuous and G 1 - ontinuous.

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p i 1 p i p i p + i p i+1 p i+1 p + i+1 p i+2 2l i+1 2l i

Figure VII.22: De ning the Catmull-Rom spline segment from the point p

i to the point p i+1 . The points p i , p i , and p + i

are ollinear and parallel to

p i+1 p i 1 . The points p i , p + i , p i+1 , and p i+1

form the ontrol points of a

degree three Bezier urve, whi h is shown asa dotted urve.

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### Bryła obrotowa

p 0 p 1 p 2 p 3 p 4 p 5 p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7

Figure VII.23: Two examples of Catmull-Rom splines with uniformly spa ed

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p 0 p 1

(a) Degree one

p 0 p 2 p 1 (b) Degree two p 0 p 1 p 2 p 3 p 4 p 5 p 6 p 7 p 8 ( ) Degree eight

Figure VII.9: (a) A degree one Bezier urve is just a straight line interpolating

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### Bryła obrotowa

p 0;0 p 3;0 p 0;3 p 3;3

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### Bryła obrotowa

Figure VII.12: A degree three Bezier pat h and some ross se tions. The ross

se tions are Bezier urves.

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p 0;0 p 3;0 p 0;3 p 3;3

Figure VII.11: A degree three Bezier pat h and its ontrol points. The ontrol

points are shown joined by straight line segments.

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p 0;0 p 0;3 p 3;0 = r 0;0 p 3;3 = r 0;3 r 3;0 r 3;3 q 1 q 2

Figure VII.13: Two Bezier pat hes join to form a single smooth surfa e. The

two pat hes q

1

and q

2

ea h have sixteen ontrol points. The four rightmost

ontrol points of q

1

are the same as the four leftmost ontrol points of q

2

. The

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### i

hp 0 ; 1i h3p 1 ;3i h 1 3 p 2 ; 1 3 i hp 3 ; 1i

Figure VII.16: A degree three, rational Bezier urve. The ontrol points are

the same as in the left-hand side of gure VII.2 on page 156, but now the

ontrol point p

1

is weighted 3, and the ontrol point p

2

is weighted only 1=3.

The other two ontrol points have weight 1. In omparison with the urve of

gure VII.2, this urve more losely approa hes p

1

, but does not approa h p

2

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### Bryła obrotowa

p 0 = h0;1;1i p 2 = h0; 1;1i p 1 = h1; 0;0i q(u)

Figure VII.17: The situation of Theorem VII.9. The middle ontrol point is

a tually a point at in nity, and the dotted lines joining it to the other ontrol

points are a tually straight and are tangent to the ir le at p

0

and p

2 .

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### .

p 0 p 1 p 2 T 0 T 2

Figure VII.18: A portion of a bran h of a oni se tion C is equal to a rational

quadrati Bezier urve. Control points p

0

and p

2

have weight 1 and p

1

gets

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### Bryła obrotowa

p 2 = h0;1i; w 2 = 1 p 1 = h1;1i; w 1 = p 2 2 p 0 = h1;0i; w 0 = 1 p 2 = h p 3 2 ; 1 2 i; w 2 = 1 p 0 = h p 3 2 ; 1 2 i; w 0 = 1 p 1 = h0;2i; w 1 = 1 2

Figure VII.19: Two ways to de ne ir ular ar s with rational Bezier urves

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p 0 = h0;1i; w 0 = 1 p 3 = h0; 1i; w 3 = 1 p 1 = h2;1i; w 1 = 1 3 p 2 = h2; 1i; w 2 = 1 3

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### )

p 0 = h0;1;1i p 2 = h0; 1;1i p 1 = h1; 0;0i q(u)

Figure VII.17: The situation of Theorem VII.9. The middle ontrol point is

a tually a point at in nity, and the dotted lines joining it to the other ontrol

points are a tually straight and are tangent to the ir le at p

0

and p

2 .

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h2; 1; 0i h3;0;0i h 3 2 ; 1 2 ;0i h2;1; 0i (a) (b)

Figure VII.21: (a) A silhouette of a surfa e of revolution (the ontrol points

are in x;y;z- oordinates). (b) The front half of the surfa e of revolution. This

example is implemented in the SimpleNurbs progam.

### (−2 : −1 : 0 : 1) (0 : 0 : 2 : 0) (2 : −1 : 0 : 1)

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