CQ2

Sep. 17

Given a cubic Bezier curve with Bernstein basis function polynomials,

$\vec f(u)=\vec b_0(1-u)^3+\vec b_13u(1-u)^2+\vec b_13u^2(1-u)+\vec b_3u^3$

describe in 1-2 sentences what you would do to find the coefficients of an equivalent monomial basis function representation.

$f(u)=a_0+a_1u+a_2u^2+a_3u^3$

This shows that the monomial basis function representation produces the same shape curve as the Bezier curve.

What are the advantages of the Bezier curve (Bernstein polynomial representation) over the monomial form?

Q1

describe in 1-2 sentences what you would do to find the coefficients of an equivalent monomial basis function representation.

Firstly, expand the Bernstein basis polynomials into monomial form and then collect and equate coefficients for each power of (u).

Q2

What are the advantages of the Bezier curve (Bernstein polynomial representation) over the monomial form?

The advantages of the Bezier curve representation over the monomial form include its intuitive control via geometric points, which facilitates curve design and modification, and its properties such as the curve always lying within the convex hull of its control points, which provides predictable and stable behavior.