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EXERCISE 2.2 Derive the expression of the Bezier curve for the four points Po = ( = (0, 0, 0), P₁ = (1, 0, 0), P₂ = (2, 1, 0), and P3 = (3, 0, 1). Derive the normal vector K(t) expression

EXERCISE 2.2 Derive the expression of the Bezier curve for the four points Po = ( = (0, 0, 0), P₁ = (1, 0, 0), P₂ = (2, 1, 0), and P3 = (3, 0, 1). Derive the normal vector K(t) expression

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 2.2** **Objective:** Derive the expression of the Bézier curve for the four points \( P_0 = (0, 0, 0) \), \( P_1 = (1, 0, 0) \), \( P_2 = (2, 1, 0) \), and \( P_3 = (3, 0, 1) \). **Tasks:** 1. Derive the normal vector \( \mathbf{K}(t) \) expression of the parametric curve. 2. Calculate \( \mathbf{K}(0) \), \( \mathbf{K}(0.5) \), and \( \mathbf{K}(1) \). 3. Establish the osculating plane of the curve and find its equations for \( t = 0 \), \( t = 0.5 \), and \( t = 1 \). ### Steps: 1. **Derive the Expression of the Bézier Curve:** - Use the given control points to formulate the Bézier curve equation. 2. **Derive the Normal Vector \( \mathbf{K}(t) \):** - Differentiate the parametric equation of the Bézier curve to find the tangent vector. - Find the normal vector \( \mathbf{K}(t) \). 3. **Calculate Specific Normal Vectors:** - Evaluate \( \mathbf{K}(t) \) at \( t = 0 \), \( t = 0.5 \), and \( t = 1 \). 4. **Establish the Osculating Plane:** - Determine the osculating plane at the specified \( t \) values. - Formulate the equations for these planes. ### Detailed Walkthrough: Given Points: - \( P_0 = (0, 0, 0) \) - \( P_1 = (1, 0, 0) \) - \( P_2 = (2, 1, 0) \) - \( P_3 = (3, 0, 1) \) Formulate the Bézier curve \( \mathbf{P}(t) \) for \( 0 \leq t \leq 1 \): \[ \mathbf{P}(t) = (1-t)^3P_0 + 3(1-t)^2tP
Transcribed Image Text:**Exercise 2.2** **Objective:** Derive the expression of the Bézier curve for the four points \( P_0 = (0, 0, 0) \), \( P_1 = (1, 0, 0) \), \( P_2 = (2, 1, 0) \), and \( P_3 = (3, 0, 1) \). **Tasks:** 1. Derive the normal vector \( \mathbf{K}(t) \) expression of the parametric curve. 2. Calculate \( \mathbf{K}(0) \), \( \mathbf{K}(0.5) \), and \( \mathbf{K}(1) \). 3. Establish the osculating plane of the curve and find its equations for \( t = 0 \), \( t = 0.5 \), and \( t = 1 \). ### Steps: 1. **Derive the Expression of the Bézier Curve:** - Use the given control points to formulate the Bézier curve equation. 2. **Derive the Normal Vector \( \mathbf{K}(t) \):** - Differentiate the parametric equation of the Bézier curve to find the tangent vector. - Find the normal vector \( \mathbf{K}(t) \). 3. **Calculate Specific Normal Vectors:** - Evaluate \( \mathbf{K}(t) \) at \( t = 0 \), \( t = 0.5 \), and \( t = 1 \). 4. **Establish the Osculating Plane:** - Determine the osculating plane at the specified \( t \) values. - Formulate the equations for these planes. ### Detailed Walkthrough: Given Points: - \( P_0 = (0, 0, 0) \) - \( P_1 = (1, 0, 0) \) - \( P_2 = (2, 1, 0) \) - \( P_3 = (3, 0, 1) \) Formulate the Bézier curve \( \mathbf{P}(t) \) for \( 0 \leq t \leq 1 \): \[ \mathbf{P}(t) = (1-t)^3P_0 + 3(1-t)^2tP
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