0. Parametric curves in general

Let's work with curves in    R$^2$, although curves in    R$^3$ are treated the same. Recall that a curve given parametrically is the same thing as a vector-valued function $P (t)$, i.e., a function $P :$   R$\rightarrow$   R$^2$ or a function on just part of    R$$ to    R$^2$. We can either write $P (t)$ directly or write the function using coordinates: $P(t) = (x(t), y(t))$.

The curve in    R$^2$ is really the image of the function, or in other words, the path swept out by the moving point described by $P (t)$, if you think of $t$ as time. Usually, though, we just say ``the parametric curve'' when we mean the function $P (t)$.

In this course, such functions are usually given one of two ways:

1. By giving the coordinate functions themselves:

   Example 0.1 . $P(t) = (\cos t, \sin t)$

   Example 0.2 . $P(t) = (t^3, t^2)$ (Figure )

   Figure: Examples 2 and 4

   book/07dir/example_2.eps book/07dir/example_4.eps

2. By expressing the curve as a linear combination of given points, where the coefficients are functions of $t$. The curve might go through some of the points or it might not.

   Example 0.3 . $P(t) = (1-t) P_0 + t P_1$ (a line).

   Example 0.4 . $P(t) = (t^2 - 2t + 1) P_0 + (2 t - 2t^2) P_1 + t^2 P_2$ (Figure ).

An expression such as the one of Example may look unfamiliar at first, but notice that for each $t$, each coefficient is some particular number, so this kind of computation is really nothing new. For instance, in Example 4, $P( {\frac 1 2})$ $=$ ${\frac 1 4 } P_0 + {\frac 1 2} P_1 + {\frac 1 4} P_2$.

We could call an expression of the second kind a time-varying linear combination of points. In graphics the coefficient functions of $t$ are called blending functions. For a curve given this way, it is easy to find the coefficient functions:

Observation. For a curve $P(t) = f_0 (t) P_0 + \dots + f_n (t) P_n$, write $P_i$ $=$ $(x_i, y_i)$ and $P (t)$ $=$ $(x(t),y(t))$. Then

$x(t) = f_0 (t) x_0 + \dots + f_n (t) x_n$,

$y(t) = f_0 (t) y_0 + \dots + f_n (t) y_n$.

Interestingly, $x(t)$ and $y(t)$ can also be regarded as linear combinations of the functions $f_i$ with the numbers $x_i$ or $y_i$ as coefficients, instead of as linear combinations of numbers with functions as coefficients.

In all applications in this course, the coefficient functions will add up to $1$ at all times $t$ used . This property is needed to ensure that if the points $P_i$ are translated by some vector    b$$, then the curve is also translated by    b$$. (See the Exercises.)