1. A Bzier curve is defined by the following equation with four control vertices as shown...
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1. A Bézier curve is defined by the following equation with four control vertices as shown in the figure P(1) =B₁0 (1)V₁ + B₁, (1)V₁ + B₁2 (1)V₂ + B₁3 (1)V3. The four blending functions of the control vertices are defined by Bernstein polynomials B₁(t)=(1-1)=31(1-1)², = (3) ₁a-1)² * = (-1)², =(√)ra-0²¹ = (3) 2² (1-1)³² ² = 31 ›={3}*a-1)³³ = r²³ B₁₂ (1)=²(1-1)³² = 31² (1-1), B₁(t)= B₁3(t) = (a) What is the degree of the Bézier curve? (b) Prove that a Bézier curve goes through the first and the last control vertices. (c) Also prove that the tangent vector of the Bézier curve at the start point is parallel to the first segment of its control polygon, and that the tangent vector of the Bézier curve at the end point is parallel to the last segment of its control polygon. (d) Given Vo-[2, 3], V-[3, 6], V2-[8, 8], V-[10, 4] being vertices of the Bézier polygon, determine six points on the Bézier curve at parameter 1 = 0.0, 0.3, 0.4, 0.6, 0.7, 1.0 (the figure below with five marked points is for illustration only). Bezier curve Hints: For (b) and (c), can be done by simple verification. For (d), please also refer to laboratory document for points evaluation on a Bézier curve. 1. A Bézier curve is defined by the following equation with four control vertices as shown in the figure P(1) =B₁0 (1)V₁ + B₁, (1)V₁ + B₁2 (1)V₂ + B₁3 (1)V3. The four blending functions of the control vertices are defined by Bernstein polynomials B₁(t)=(1-1)=31(1-1)², = (3) ₁a-1)² * = (-1)², =(√)ra-0²¹ = (3) 2² (1-1)³² ² = 31 ›={3}*a-1)³³ = r²³ B₁₂ (1)=²(1-1)³² = 31² (1-1), B₁(t)= B₁3(t) = (a) What is the degree of the Bézier curve? (b) Prove that a Bézier curve goes through the first and the last control vertices. (c) Also prove that the tangent vector of the Bézier curve at the start point is parallel to the first segment of its control polygon, and that the tangent vector of the Bézier curve at the end point is parallel to the last segment of its control polygon. (d) Given Vo-[2, 3], V-[3, 6], V2-[8, 8], V-[10, 4] being vertices of the Bézier polygon, determine six points on the Bézier curve at parameter 1 = 0.0, 0.3, 0.4, 0.6, 0.7, 1.0 (the figure below with five marked points is for illustration only). Bezier curve Hints: For (b) and (c), can be done by simple verification. For (d), please also refer to laboratory document for points evaluation on a Bézier curve.
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Related Book For
Materials Science and Engineering An Introduction
ISBN: 978-0470419977
8th edition
Authors: William D. Callister Jr., David G. Rethwisch
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