Numerical Analysis
Numerical Analysis
3rd Edition
ISBN: 9780134696454
Author: Sauer, Tim
Publisher: Pearson,
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Textbook Question
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Chapter 3.5, Problem 1E

Find the one-piece BĂ©zier curve ( x ( t ) , y ( t ) ) defined by the given four points.

(a) ( 0 , 0 ) , ( 0 , 2 ) , ( 2 , 0 ) , ( 1 , 0 )

(b) ( 1 , 1 ) , ( 0 , 0 ) , ( 2 , 0 ) , ( 2 , 1 )

(c) ( 1 , 2 ) , ( 1 , 3 ) , ( 2 , 3 ) , ( 2 , 2 )

a)

Expert Solution
Check Mark
To determine

To find:The one- piece Bezier curve defined by the given four points.

Answer to Problem 1E

The Bezier curve defined by the given points is;

   x(t)=6t25t3y(t)=6t12t2+6t3

Explanation of Solution

Given:

Four points are (0,0),(0,2),(2,0),(1,0).

Formula used:

Bezier curve equations are

   x(t)=x1+bxt+cxt2+dyt3y(t)=y1+byt+cyt2+dyt3co-efficients arebx=3(x2x1)cx=3(x2x1)bxdx=x4x1bxcxby=3(y2y1)cy=3(y3y2)bxdy=y4y1bycy

Calculation:

From the given data,

We have,

   (x1,y1)=(0,0)(x2,y2)=(0,2)(x3,y3)=(2,0),(x4,y4)=(1,0)

The coefficients of Bezier curve are determined as follows:

   bx=3(x2x1)=3(00)=0cx=3(x2x1)bx=3(20)0=6dx=x4x1bxcx=1006=5

   by=3(y2y1)=3(20)=6cy=3(y3y2)bx=3(02)6=12dy=y4y1bycy=006+12=6

   x(t)=x1+bxt+cxt2+dyt3=0+0+6t25t3=6t25t3y(t)=y1+byt+cyt2+dyt3=0+6t12t2+6t3+0=6t12t2+6t3

Therefore, the Bezier curve equations are

   x(t)=6t25t3y(t)=6t12t2+6t3

b)

Expert Solution
Check Mark
To determine

To find:The one- piece Bezier curve defined by the given four points.

Answer to Problem 1E

The Bezier curve defined by the equations

   x(t)=13t3t2+3t3y(t)=13t+3t2

Explanation of Solution

Given:

Four points are (1,1),(0,0),(2,0),(2,1).

Formula used:

Bezier curve equations are

   x(t)=x1+bxt+cxt2+dyt3y(t)=y1+byt+cyt2+dyt3coefficients arebx=3(x2x1)cx=3(x2x1)bxdx=x4x1bxcxby=3(y2y1)cy=3(y3y2)bxdy=y4y1bycy

Calculation:

From the given data,

We have,

   (x1,y1)=(1,1(x2,y2)=(0,0)(x3,y3)=(2,0),(x4,y4)=(2,1)

The coefficients of Bezier curve are determined as follows:

   bx=3(x2x1)=3(01)=3cx=3(x2x1)bx=3(20)0=3dx=x4x1bxcx=21+3+3=3

   by=3(y2y1)=3(01)=3cy=3(y3y2)bx=3(00)+3=3dy=y4y1bycy=11+33=0

   x(t)=x1+bxt+cxt2+dyt3=1+(3)t+(3)t2+3t3=13t3t2+3t3y(t)=y1+byt+cyt2+dyt3=13t+3t2+0=13t+3t2

Therefore, the Bezier curve equations are

   x(t)=13t3t2+3t3y(t)=13t+3t2

c)

Expert Solution
Check Mark
To determine

To find:The one- piece Bezier curve defined by the given four points.

Answer to Problem 1E

The Bezier curve defined by the equations

   x(t)=1+3t22t3y(t)=2+3t3t2

Explanation of Solution

Given:

Four points are (1,2),(1,3),(2,3),(2,2).

Formula used:

Bezier curve equations are

   x(t)=x1+bxt+cxt2+dyt3y(t)=y1+byt+cyt2+dyt3coefficients arebx=3(x2x1)cx=3(x2x1)bxdx=x4x1bxcxby=3(y2y1)cy=3(y3y2)bxdy=y4y1bycy

Calculation:

From the given data,

We have,

   (x1,y1)=(1,2)(x2,y2)=(1,3)(x3,y3)=(2,3),(x4,y4)=(2,2)

The coefficients of Bezier curve are determined as follows:

   bx=3(x2x1)=3(11)=0cx=3(x2x1)bx=3(21)0=3dx=x4x1bxcx=2103=2

   by=3(y2y1)=3(32)=3cy=3(y3y2)bx=3(33)3=3dy=y4y1bycy=22+33=0

   x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(2)t3=1+3t22t3y(t)=y1+byt+cyt2+dyt3=2+3t3t2

Therefore, the Bezier curve equations are

   x(t)=1+3t22t3y(t)=2+3t3t2

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