1. Polynomial curves

By a polynomial curve in    R$^n$ let us mean simply a parametric curve given by a function $P (t)$ for which each coordinate function is a polynomial. Examples , , and are polynomial curves. So is this curve:

Example 1.1 . $P(t) = (1-t)^3 (4,1) + 3(1-t)^2 t (0,-3) + 3(1-t)t^2 (0,3) + t^3 (4,-1)$. (See Figure .)

Figure: Example 5

book/07dir/example_5.eps

The degree of a polynomial curve is the maximum of the degrees of the coordination functions. If a polynomial curve $P (t)$ is described as a time-varying linear combination of points, you can see that the degree of $P (t)$ is no larger than the largest degree of the functions $f_i (t)$. (The degree could be less, if powers of $t$ cancel when adding polynomials.)