Polynomial Curves

Consider an $n$ th degree polynomial curve $\boldsymbol{R}$ parametrized by $t$ :

\boldsymbol{R}(t) = \sum_{i=0}^n \boldsymbol{A_i} t^i,

with the initial conditions:

\boldsymbol{R}(t_j) = \boldsymbol{P_j},

where $t_i \in [0, 1]$ .

The initial conditions imply:

\sum_{i=0}^n \boldsymbol{A_i} t_j^i = \boldsymbol{P_j},

this may be represented as matrix multiplication:

\begin{bmatrix}t_j^0 & \dots & t_j^n\end{bmatrix} \begin{bmatrix} \boldsymbol{A_0} \\ \vdots \\ \boldsymbol{A_n} \end{bmatrix} = \boldsymbol{P_j},

or:

\begin{bmatrix} t_0^0 & \dots & t_0^n \\ \vdots & \ddots & \vdots \\ t_n^0 & \dots & t_n^n \end{bmatrix} \begin{bmatrix} \boldsymbol{A_0} \\ \vdots \\ \boldsymbol{A_n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{P_0} \\ \vdots \\ \boldsymbol{P_n} \end{bmatrix}.

Solving for $A_i$ :

\begin{bmatrix} \boldsymbol{A_0} \\ \vdots \\ \boldsymbol{A_n} \end{bmatrix} = \begin{bmatrix} t_0^0 & \dots & t_0^n \\ \vdots & \ddots & \vdots \\ t_n^0 & \dots & t_n^n \end{bmatrix}^{-1} \begin{bmatrix} \boldsymbol{P_0} \\ \vdots \\ \boldsymbol{P_n} \end{bmatrix}.

x_0

y_0

t_0

x_1

y_1

t_1

:Show axes:

\begin{align*} x(t) &= t, \\ y(t) &= t. \end{align*}