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 )

Question

Want to see more full solutions like this?

Answer 1

Textbook Question

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

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)=6t2−5t3y(t)=6t−12t2+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(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−0)=0cx=3(x2−x1)−bx=3(2−0)−0=6dx=x4−x1−bx−cx=1−0−0−6=−5

by=3(y2−y1)=3(2−0)=6cy=3(y3−y2)−bx=3(0−2)−6=−12dy=y4−y1−by−cy=0−0−6+12=6

⇒x(t)=x1+bxt+cxt2+dyt3=0+0+6t2−5t3=6t2−5t3⇒y(t)=y1+byt+cyt2+dyt3=0+6t−12t2+6t3+0=6t−12t2+6t3

Therefore, the Bezier curve equations are

x(t)=6t2−5t3y(t)=6t−12t2+6t3

b)

Expert Solution

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)=1−3t−3t2+3t3y(t)=1−3t+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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−1)=−3cx=3(x2−x1)−bx=3(−2−0)−0=−3dx=x4−x1−bx−cx=−2−1+3+3=3

by=3(y2−y1)=3(0−1)=−3cy=3(y3−y2)−bx=3(0−0)+3=3dy=y4−y1−by−cy=1−1+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(−3)t+(−3)t2+3t3=1−3t−3t2+3t3⇒y(t)=y1+byt+cyt2+dyt3=1−3t+3t2+0=1−3t+3t2

Therefore, the Bezier curve equations are

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

c)

Expert Solution

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+3t2−2t3y(t)=2+3t−3t2

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(1−1)=0cx=3(x2−x1)−bx=3(2−1)−0=3dx=x4−x1−bx−cx=2−1−0−3=−2

by=3(y2−y1)=3(3−2)=3cy=3(y3−y2)−bx=3(3−3)−3=−3dy=y4−y1−by−cy=2−2+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(−2)t3=1+3t2−2t3⇒y(t)=y1+byt+cyt2+dyt3=2+3t−3t2

Therefore, the Bezier curve equations are

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Students have asked these similar questions

Two functions ƒ and g are defined in the figure below. g 6 6 7 8 8 8 9 Domain of f Range of f Domain of g Range of g Find the domain and range of the composition g.f. Write your answers in set notation. (a) Domain of gof: (b) Range of gof: ☐ ☑ 0,0,...

The table below shows the estimated effects for a logistic regression model with squamous cell esophageal cancer (Y = 1, yes; Y = 0, no) as the response. Smoking status (S) equals 1 for at least one pack per day and 0 otherwise, alcohol consumption (A) equals the average number of alcohoic drinks consumed per day, and race (R) equals 1 for blacks and 0 for whites. Variable Effect (β) P-value Intercept -7.00 <0.01 Alcohol use 0.10 0.03 Smoking 1.20 <0.01 Race 0.30 0.02 Race × smoking 0.20 0.04 Write-out the prediction equation (i.e., the logistic regression model) when R = 0 and again when R = 1. Find the fitted Y S conditional odds ratio in each case. Next, write-out the logistic regression model when S = 0 and again when S = 1. Find the fitted Y R conditional odds ratio in each case.

The chi-squared goodness-of-fit test can be used to test if data comes from a specific continuous distribution by binning the data to make it categorical. Using the OpenIntro Statistics county_complete dataset, test the hypothesis that the persons_per_household 2019 values come from a normal distribution with mean and standard deviation equal to that variable's mean and standard deviation. Use signficance level a = 0.01. In your solution you should 1. Formulate the hypotheses 2. Fill in this table Range (-⁰⁰, 2.34] (2.34, 2.81] (2.81, 3.27] (3.27,00) Observed 802 Expected 854.2 The first row has been filled in. That should give you a hint for how to calculate the expected frequencies. Remember that the expected frequencies are calculated under the assumption that the null hypothesis is true. FYI, the bounderies for each range were obtained using JASP's drag-and-drop cut function with 8 levels. Then some of the groups were merged. 3. Check any conditions required by the chi-squared…

Answer 2

Textbook Question

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

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)=6t2−5t3y(t)=6t−12t2+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(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−0)=0cx=3(x2−x1)−bx=3(2−0)−0=6dx=x4−x1−bx−cx=1−0−0−6=−5

by=3(y2−y1)=3(2−0)=6cy=3(y3−y2)−bx=3(0−2)−6=−12dy=y4−y1−by−cy=0−0−6+12=6

⇒x(t)=x1+bxt+cxt2+dyt3=0+0+6t2−5t3=6t2−5t3⇒y(t)=y1+byt+cyt2+dyt3=0+6t−12t2+6t3+0=6t−12t2+6t3

Therefore, the Bezier curve equations are

x(t)=6t2−5t3y(t)=6t−12t2+6t3

b)

Expert Solution

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)=1−3t−3t2+3t3y(t)=1−3t+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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−1)=−3cx=3(x2−x1)−bx=3(−2−0)−0=−3dx=x4−x1−bx−cx=−2−1+3+3=3

by=3(y2−y1)=3(0−1)=−3cy=3(y3−y2)−bx=3(0−0)+3=3dy=y4−y1−by−cy=1−1+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(−3)t+(−3)t2+3t3=1−3t−3t2+3t3⇒y(t)=y1+byt+cyt2+dyt3=1−3t+3t2+0=1−3t+3t2

Therefore, the Bezier curve equations are

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

c)

Expert Solution

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+3t2−2t3y(t)=2+3t−3t2

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(1−1)=0cx=3(x2−x1)−bx=3(2−1)−0=3dx=x4−x1−bx−cx=2−1−0−3=−2

by=3(y2−y1)=3(3−2)=3cy=3(y3−y2)−bx=3(3−3)−3=−3dy=y4−y1−by−cy=2−2+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(−2)t3=1+3t2−2t3⇒y(t)=y1+byt+cyt2+dyt3=2+3t−3t2

Therefore, the Bezier curve equations are

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 3

Textbook Question

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

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)=6t2−5t3y(t)=6t−12t2+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(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−0)=0cx=3(x2−x1)−bx=3(2−0)−0=6dx=x4−x1−bx−cx=1−0−0−6=−5

by=3(y2−y1)=3(2−0)=6cy=3(y3−y2)−bx=3(0−2)−6=−12dy=y4−y1−by−cy=0−0−6+12=6

⇒x(t)=x1+bxt+cxt2+dyt3=0+0+6t2−5t3=6t2−5t3⇒y(t)=y1+byt+cyt2+dyt3=0+6t−12t2+6t3+0=6t−12t2+6t3

Therefore, the Bezier curve equations are

x(t)=6t2−5t3y(t)=6t−12t2+6t3

b)

Expert Solution

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)=1−3t−3t2+3t3y(t)=1−3t+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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−1)=−3cx=3(x2−x1)−bx=3(−2−0)−0=−3dx=x4−x1−bx−cx=−2−1+3+3=3

by=3(y2−y1)=3(0−1)=−3cy=3(y3−y2)−bx=3(0−0)+3=3dy=y4−y1−by−cy=1−1+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(−3)t+(−3)t2+3t3=1−3t−3t2+3t3⇒y(t)=y1+byt+cyt2+dyt3=1−3t+3t2+0=1−3t+3t2

Therefore, the Bezier curve equations are

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

c)

Expert Solution

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+3t2−2t3y(t)=2+3t−3t2

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(1−1)=0cx=3(x2−x1)−bx=3(2−1)−0=3dx=x4−x1−bx−cx=2−1−0−3=−2

by=3(y2−y1)=3(3−2)=3cy=3(y3−y2)−bx=3(3−3)−3=−3dy=y4−y1−by−cy=2−2+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(−2)t3=1+3t2−2t3⇒y(t)=y1+byt+cyt2+dyt3=2+3t−3t2

Therefore, the Bezier curve equations are

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 4

Textbook Question

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

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)=6t2−5t3y(t)=6t−12t2+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(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−0)=0cx=3(x2−x1)−bx=3(2−0)−0=6dx=x4−x1−bx−cx=1−0−0−6=−5

by=3(y2−y1)=3(2−0)=6cy=3(y3−y2)−bx=3(0−2)−6=−12dy=y4−y1−by−cy=0−0−6+12=6

⇒x(t)=x1+bxt+cxt2+dyt3=0+0+6t2−5t3=6t2−5t3⇒y(t)=y1+byt+cyt2+dyt3=0+6t−12t2+6t3+0=6t−12t2+6t3

Therefore, the Bezier curve equations are

x(t)=6t2−5t3y(t)=6t−12t2+6t3

b)

Expert Solution

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)=1−3t−3t2+3t3y(t)=1−3t+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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−1)=−3cx=3(x2−x1)−bx=3(−2−0)−0=−3dx=x4−x1−bx−cx=−2−1+3+3=3

by=3(y2−y1)=3(0−1)=−3cy=3(y3−y2)−bx=3(0−0)+3=3dy=y4−y1−by−cy=1−1+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(−3)t+(−3)t2+3t3=1−3t−3t2+3t3⇒y(t)=y1+byt+cyt2+dyt3=1−3t+3t2+0=1−3t+3t2

Therefore, the Bezier curve equations are

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

c)

Expert Solution

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+3t2−2t3y(t)=2+3t−3t2

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(1−1)=0cx=3(x2−x1)−bx=3(2−1)−0=3dx=x4−x1−bx−cx=2−1−0−3=−2

by=3(y2−y1)=3(3−2)=3cy=3(y3−y2)−bx=3(3−3)−3=−3dy=y4−y1−by−cy=2−2+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(−2)t3=1+3t2−2t3⇒y(t)=y1+byt+cyt2+dyt3=2+3t−3t2

Therefore, the Bezier curve equations are

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!

Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

a)

Expert Solution

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)=6t2−5t3y(t)=6t−12t2+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(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−0)=0cx=3(x2−x1)−bx=3(2−0)−0=6dx=x4−x1−bx−cx=1−0−0−6=−5

by=3(y2−y1)=3(2−0)=6cy=3(y3−y2)−bx=3(0−2)−6=−12dy=y4−y1−by−cy=0−0−6+12=6

⇒x(t)=x1+bxt+cxt2+dyt3=0+0+6t2−5t3=6t2−5t3⇒y(t)=y1+byt+cyt2+dyt3=0+6t−12t2+6t3+0=6t−12t2+6t3

Therefore, the Bezier curve equations are

x(t)=6t2−5t3y(t)=6t−12t2+6t3

b)

Expert Solution

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)=1−3t−3t2+3t3y(t)=1−3t+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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−1)=−3cx=3(x2−x1)−bx=3(−2−0)−0=−3dx=x4−x1−bx−cx=−2−1+3+3=3

by=3(y2−y1)=3(0−1)=−3cy=3(y3−y2)−bx=3(0−0)+3=3dy=y4−y1−by−cy=1−1+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(−3)t+(−3)t2+3t3=1−3t−3t2+3t3⇒y(t)=y1+byt+cyt2+dyt3=1−3t+3t2+0=1−3t+3t2

Therefore, the Bezier curve equations are

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

c)

Expert Solution

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+3t2−2t3y(t)=2+3t−3t2

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(1−1)=0cx=3(x2−x1)−bx=3(2−1)−0=3dx=x4−x1−bx−cx=2−1−0−3=−2

by=3(y2−y1)=3(3−2)=3cy=3(y3−y2)−bx=3(3−3)−3=−3dy=y4−y1−by−cy=2−2+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(−2)t3=1+3t2−2t3⇒y(t)=y1+byt+cyt2+dyt3=2+3t−3t2

Therefore, the Bezier curve equations are

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Answer 8

a)

Expert Solution

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)=6t2−5t3y(t)=6t−12t2+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(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−0)=0cx=3(x2−x1)−bx=3(2−0)−0=6dx=x4−x1−bx−cx=1−0−0−6=−5

by=3(y2−y1)=3(2−0)=6cy=3(y3−y2)−bx=3(0−2)−6=−12dy=y4−y1−by−cy=0−0−6+12=6

⇒x(t)=x1+bxt+cxt2+dyt3=0+0+6t2−5t3=6t2−5t3⇒y(t)=y1+byt+cyt2+dyt3=0+6t−12t2+6t3+0=6t−12t2+6t3

Therefore, the Bezier curve equations are

x(t)=6t2−5t3y(t)=6t−12t2+6t3

Answer 9

a)

Expert Solution

Answer 10

a)

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

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

Answer 13

Answer to Problem 1E

The Bezier curve defined by the given points is;

x(t)=6t2−5t3y(t)=6t−12t2+6t3

Answer 14

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(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−0)=0cx=3(x2−x1)−bx=3(2−0)−0=6dx=x4−x1−bx−cx=1−0−0−6=−5

by=3(y2−y1)=3(2−0)=6cy=3(y3−y2)−bx=3(0−2)−6=−12dy=y4−y1−by−cy=0−0−6+12=6

⇒x(t)=x1+bxt+cxt2+dyt3=0+0+6t2−5t3=6t2−5t3⇒y(t)=y1+byt+cyt2+dyt3=0+6t−12t2+6t3+0=6t−12t2+6t3

Therefore, the Bezier curve equations are

x(t)=6t2−5t3y(t)=6t−12t2+6t3

Answer 15

b)

Expert Solution

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)=1−3t−3t2+3t3y(t)=1−3t+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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−1)=−3cx=3(x2−x1)−bx=3(−2−0)−0=−3dx=x4−x1−bx−cx=−2−1+3+3=3

by=3(y2−y1)=3(0−1)=−3cy=3(y3−y2)−bx=3(0−0)+3=3dy=y4−y1−by−cy=1−1+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(−3)t+(−3)t2+3t3=1−3t−3t2+3t3⇒y(t)=y1+byt+cyt2+dyt3=1−3t+3t2+0=1−3t+3t2

Therefore, the Bezier curve equations are

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

Answer 16

b)

Expert Solution

Answer 17

b)

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

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

Answer 20

Answer to Problem 1E

The Bezier curve defined by the equations

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

Answer 21

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(0−1)=−3cx=3(x2−x1)−bx=3(−2−0)−0=−3dx=x4−x1−bx−cx=−2−1+3+3=3

by=3(y2−y1)=3(0−1)=−3cy=3(y3−y2)−bx=3(0−0)+3=3dy=y4−y1−by−cy=1−1+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(−3)t+(−3)t2+3t3=1−3t−3t2+3t3⇒y(t)=y1+byt+cyt2+dyt3=1−3t+3t2+0=1−3t+3t2

Therefore, the Bezier curve equations are

x(t)=1−3t−3t2+3t3y(t)=1−3t+3t2

Answer 22

c)

Expert Solution

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+3t2−2t3y(t)=2+3t−3t2

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(1−1)=0cx=3(x2−x1)−bx=3(2−1)−0=3dx=x4−x1−bx−cx=2−1−0−3=−2

by=3(y2−y1)=3(3−2)=3cy=3(y3−y2)−bx=3(3−3)−3=−3dy=y4−y1−by−cy=2−2+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(−2)t3=1+3t2−2t3⇒y(t)=y1+byt+cyt2+dyt3=2+3t−3t2

Therefore, the Bezier curve equations are

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Answer 23

c)

Expert Solution

Answer 24

c)

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

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

Answer 27

Answer to Problem 1E

The Bezier curve defined by the equations

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Answer 28

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+dyt3co−efficients arebx=3(x2−x1)cx=3(x2−x1)−bxdx=x4−x1−bx−cxby=3(y2−y1)cy=3(y3−y2)−bxdy=y4−y1−by−cy

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(x2−x1)=3(1−1)=0cx=3(x2−x1)−bx=3(2−1)−0=3dx=x4−x1−bx−cx=2−1−0−3=−2

by=3(y2−y1)=3(3−2)=3cy=3(y3−y2)−bx=3(3−3)−3=−3dy=y4−y1−by−cy=2−2+3−3=0

⇒x(t)=x1+bxt+cxt2+dyt3=1+(0)t+(3)t2+(−2)t3=1+3t2−2t3⇒y(t)=y1+byt+cyt2+dyt3=2+3t−3t2

Therefore, the Bezier curve equations are

x(t)=1+3t2−2t3y(t)=2+3t−3t2

Answer 29

Ch. 3.1 - Use Lagrange interpolation to find a polynomial...Ch. 3.1 - Use Newtons divided differences to find the...Ch. 3.1 - How many degree d polynomials pass through the...Ch. 3.1 - (a) Find a polynomial P(x) of degree 3 or less...Ch. 3.1 - (a) Find a polynomial P(x) of degree 3 or less...Ch. 3.1 - Write down a polynomial of degree exactly 5 that...Ch. 3.1 - Find P(0), where P(x) is the degree 10 polynomial...Ch. 3.1 - Let P(x) be the degree 9 polynomial that takes the...Ch. 3.1 - Give an example of the following, or explain why...Ch. 3.1 - Let P(x) be the degree 5 polynomial that takes the...

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 )

Concept explainers

Videos

Answer to Problem 1E

Explanation of Solution

Answer to Problem 1E

Explanation of Solution

Answer to Problem 1E

Explanation of Solution

Want to see more full solutions like this?

Chapter 3 Solutions