EXERCISE 3.2 A parametric surface is P(u, w) = [0.5(1 − u)w + u, w, (1 − u)(1 − w)]. 0 ≤u, w ≤ 1. Compute the corner points, boundary curves, and diagonals of the bilinear surface patch.
EXERCISE 3.2 A parametric surface is P(u, w) = [0.5(1 − u)w + u, w, (1 − u)(1 − w)]. 0 ≤u, w ≤ 1. Compute the corner points, boundary curves, and diagonals of the bilinear surface patch.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Exercise 3.2: Parametric Surface Analysis
**Problem Statement:**
A parametric surface is defined as \( \mathbf{P}(u, w) = \left[ 0.5(1 - u)w + u, \; w, \; (1 - u)(1 - w) \right] \) for \( 0 \leq u, w \leq 1 \).
You are tasked to:
1. Compute the corner points.
2. Determine the boundary curves.
3. Find the diagonals of the bilinear surface patch.
#### 1. Corner Points
To find the corner points of the bilinear surface patch, evaluate \( \mathbf{P}(u, w) \) at the vertices of the unit square in the parameter space \( (u, w) \).
- \( \mathbf{P}(0, 0) \)
- \( \mathbf{P}(0, 1) \)
- \( \mathbf{P}(1, 0) \)
- \( \mathbf{P}(1, 1) \)
#### 2. Boundary Curves
The boundary curves can be found by setting \( u = 0 \), \( u = 1 \), \( w = 0 \), and \( w = 1 \):
- For \( u = 0 \): \( \mathbf{P}(0, w) \)
- For \( u = 1 \): \( \mathbf{P}(1, w) \)
- For \( w = 0 \): \( \mathbf{P}(u, 0) \)
- For \( w = 1 \): \( \mathbf{P}(u, 1) \)
#### 3. Diagonals
To find the diagonals, evaluate the parametric surface along lines where \( u = w \) and \( u = 1 - w \).
- For \( u = w \): \( \mathbf{P}(u, u) \)
- For \( u = 1 - w \): \( \mathbf{P}(u, 1 - u) \)
Lastly, visualize these computations to better understand the geometric properties of the bilinear surface patch.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F780f9839-f737-4aba-91a2-6210989911b1%2Ff06af002-3dc7-4ea9-b8b4-73b4b20340e7%2F6s1w4x_processed.png&w=3840&q=75)
Transcribed Image Text:### Exercise 3.2: Parametric Surface Analysis
**Problem Statement:**
A parametric surface is defined as \( \mathbf{P}(u, w) = \left[ 0.5(1 - u)w + u, \; w, \; (1 - u)(1 - w) \right] \) for \( 0 \leq u, w \leq 1 \).
You are tasked to:
1. Compute the corner points.
2. Determine the boundary curves.
3. Find the diagonals of the bilinear surface patch.
#### 1. Corner Points
To find the corner points of the bilinear surface patch, evaluate \( \mathbf{P}(u, w) \) at the vertices of the unit square in the parameter space \( (u, w) \).
- \( \mathbf{P}(0, 0) \)
- \( \mathbf{P}(0, 1) \)
- \( \mathbf{P}(1, 0) \)
- \( \mathbf{P}(1, 1) \)
#### 2. Boundary Curves
The boundary curves can be found by setting \( u = 0 \), \( u = 1 \), \( w = 0 \), and \( w = 1 \):
- For \( u = 0 \): \( \mathbf{P}(0, w) \)
- For \( u = 1 \): \( \mathbf{P}(1, w) \)
- For \( w = 0 \): \( \mathbf{P}(u, 0) \)
- For \( w = 1 \): \( \mathbf{P}(u, 1) \)
#### 3. Diagonals
To find the diagonals, evaluate the parametric surface along lines where \( u = w \) and \( u = 1 - w \).
- For \( u = w \): \( \mathbf{P}(u, u) \)
- For \( u = 1 - w \): \( \mathbf{P}(u, 1 - u) \)
Lastly, visualize these computations to better understand the geometric properties of the bilinear surface patch.
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