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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
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.
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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