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![**EXERCISE 4.3** The four corner points of a bilinear surface are \( P_{00} = (0, 0, 1) \), \( P_{10} = (1, 0, 0) \), \( P_{01} = (1, 1, 1) \), and \( P_{11} = (0, 1, 0) \). Derive the parametric expression of the bilinear surface.
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### Explanation for Educational Website:
**Exercise 4.3** presents a problem involving the derivation of the parametric expression of a bilinear surface. The given corner points define the bilinear surface in a 3-dimensional space.
**Given Points:**
- \( P_{00} = (0, 0, 1) \)
- \( P_{10} = (1, 0, 0) \)
- \( P_{01} = (1, 1, 1) \)
- \( P_{11} = (0, 1, 0) \)
**Objective:**
To derive the parametric expression for the bilinear surface defined by these corner points.
A bilinear surface can be represented parametrically using two parameters, \( u \) and \( v \), where \( u, v \in [0, 1] \). The general form for the parametric equation of a bilinear surface is:
\[ \mathbf{P}(u, v) = (1-u)(1-v)\mathbf{P}_{00} + u(1-v)\mathbf{P}_{10} + (1-u)v\mathbf{P}_{01} + uv\mathbf{P}_{11} \]
Here, each term represents a weighted combination of the given corner points, with the weights determined by \( u \) and \( v \). The weights ensure that all combinations of \( u \) and \( v \) within the unit square result in a point on the surface.
By substituting the coordinates of the given points into the parametric equation:
\[ \mathbf{P}(u, v) = (1-u)(1-v)(0, 0, 1) + u(1-v)(1, 0, 0) + (1-u)v(1, 1, 1) + uv(0, 1, 0) \]
Expanding this](https://content.bartleby.com/qna-images/question/780f9839-f737-4aba-91a2-6210989911b1/842ba39c-72fd-4be7-8666-35dd4ede14ab/wq57szc_processed.png)
Transcribed Image Text:**EXERCISE 4.3** The four corner points of a bilinear surface are \( P_{00} = (0, 0, 1) \), \( P_{10} = (1, 0, 0) \), \( P_{01} = (1, 1, 1) \), and \( P_{11} = (0, 1, 0) \). Derive the parametric expression of the bilinear surface.
---
### Explanation for Educational Website:
**Exercise 4.3** presents a problem involving the derivation of the parametric expression of a bilinear surface. The given corner points define the bilinear surface in a 3-dimensional space.
**Given Points:**
- \( P_{00} = (0, 0, 1) \)
- \( P_{10} = (1, 0, 0) \)
- \( P_{01} = (1, 1, 1) \)
- \( P_{11} = (0, 1, 0) \)
**Objective:**
To derive the parametric expression for the bilinear surface defined by these corner points.
A bilinear surface can be represented parametrically using two parameters, \( u \) and \( v \), where \( u, v \in [0, 1] \). The general form for the parametric equation of a bilinear surface is:
\[ \mathbf{P}(u, v) = (1-u)(1-v)\mathbf{P}_{00} + u(1-v)\mathbf{P}_{10} + (1-u)v\mathbf{P}_{01} + uv\mathbf{P}_{11} \]
Here, each term represents a weighted combination of the given corner points, with the weights determined by \( u \) and \( v \). The weights ensure that all combinations of \( u \) and \( v \) within the unit square result in a point on the surface.
By substituting the coordinates of the given points into the parametric equation:
\[ \mathbf{P}(u, v) = (1-u)(1-v)(0, 0, 1) + u(1-v)(1, 0, 0) + (1-u)v(1, 1, 1) + uv(0, 1, 0) \]
Expanding this
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