Solution Found!
Let be a cubic Bézier curve determined by points p0, p1,
Chapter 8, Problem 3E(choose chapter or problem)
Let x(t) be a cubic Bézier curve determined by points \(\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2} \text {, and } \mathbf{p}_{3}\)
a. Compute the tangent vector x’(t). Determine how x’(0) and x’(1) are related to the control points, and give geometric descriptions of the directions of these tangent vectors. Is it possible to have x0 .1/ D 0?
b. Compute the second derivative x’’(t) and determine how x’’(0) and x’’(1) are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of x’’(0). [Hint: Use p1 as the origin of the coordinate system.]
Questions & Answers
QUESTION:
Let x(t) be a cubic Bézier curve determined by points \(\mathbf{p}_{0}, \mathbf{p}_{1}, \mathbf{p}_{2} \text {, and } \mathbf{p}_{3}\)
a. Compute the tangent vector x’(t). Determine how x’(0) and x’(1) are related to the control points, and give geometric descriptions of the directions of these tangent vectors. Is it possible to have x0 .1/ D 0?
b. Compute the second derivative x’’(t) and determine how x’’(0) and x’’(1) are related to the control points. Draw a figure based on Figure 10, and construct a line segment that points in the direction of x’’(0). [Hint: Use p1 as the origin of the coordinate system.]
ANSWER:Solution 3E1. when 2. Plug in , then Plug in , then Geometrically, is on the li