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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6610 • Analysis of Numerical Methods I

Winter 2004 • Question 4

Question

Math 6610 • Analysis of Numerical Methods I Winter 2004 • Question 4

Bernstein-Bezier Form Let \(T\) be a non-degenerate triangle with vertices \(V_1\), \(V_2\), and \(V_3\). The barycentric coordinates \(b_1\), \(b_2\), and \(b_3\) of a point \(x\in \mathbb {R}^2\) with respect to \(T\) are defined by \[ x=\sum _{i=1}^3 b_i V_i \qquad \text {where}\qquad \sum _{i=1}^3 b_i=1. \tag {6} \] A polynomial \(p\) of degree \(d\) can be written uniquely in its Bernstein-Bezier form as \[ p(x)=\sum _{i+j+k=d}\frac {d!}{i!j!k!}c_{ijk}b_1^i b_2^j b_3^k. \tag {7} \] Assuming you know the coefficients \(c_{ijk}\) of \(p\) as a polynomial of degree \(d\), show how to write \(p\) as a polynomial of degree \(d+1\) in Bernstein-Bezier form. What are the coefficients of \(p\) if \(p(x)=1\)?

Raw LaTeX

{Bernstein-Bezier Form}
Let \(T\) be a non-degenerate triangle with vertices \(V_1\), \(V_2\), and \(V_3\). The
barycentric coordinates \(b_1\), \(b_2\), and \(b_3\) of a point \(x\in\mathbb{R}^2\) with
respect to \(T\) are defined by
\[
  x=\sum_{i=1}^3 b_i V_i
  \qquad\text{where}\qquad
  \sum_{i=1}^3 b_i=1. \tag{6}
\]
A polynomial \(p\) of degree \(d\) can be written uniquely in its Bernstein-Bezier form
as
\[
  p(x)=\sum_{i+j+k=d}\frac{d!}{i!j!k!}c_{ijk}b_1^i b_2^j b_3^k. \tag{7}
\]
Assuming you know the coefficients \(c_{ijk}\) of \(p\) as a polynomial of degree \(d\),
show how to write \(p\) as a polynomial of degree \(d+1\) in Bernstein-Bezier form. What
are the coefficients of \(p\) if \(p(x)=1\)?

Topic Tags

Linear Programming and Optimization Polynomial Interpolation and Error Formulas Variable-Coefficient Diffusion Problems