10.1 Computation of Parametric Functions

Since the parametric representation significantly simplifies the creation of curves, we now intend to define functions that, from the parametric description of a curve, produce an array of points that belong to that curve. To generate these points we will specify the limits of the interval of the parameter $t$ and the progressive increment of its value $\Delta_t$, producing $t_0, t_0+\Delta_t, t_0+2\Delta_t,\ldots,t_1$. For example, to generate $n$ uniformly spaced points that form a circle of radius $r$ we need to consider the interval $[0,2\pi[$ and a angular separation of $\Delta_t=\frac{2\pi}{n}$. Note that the interval is open at $t=2\pi$ because the corresponding point coincides with the one at $t=0$.

To generate values of $t$ uniformly spaced by $\Delta_t$ we can use the function enumerate, defined in section Enumerations, which serves precisely this purpose, so that we can map the parametric function:

map(t -> pol(r, t), enumerate(0, 2*pi, dt))

As usual, we can encapsulate the previous expression with a parametrized function, which we will generalize to allow the specification of the circle center "p":

circle_points(p, r, dt) =

map(t -> p + vpol(r, t), enumerate(0, 2*pi, dt))

Figref{fig:pontosCirc} illustrates the points generated by the circle_points function for different $\Delta_t$ values.

Points along a circle. From left to right, $\Delta_t=\frac{2\pi}{10}$, $\Delta_t=\frac{2\pi}{20}$, $\Delta_t=\frac{2\pi}{30}$, e $\Delta_t=\frac{2\pi}{40}$.

To see a more practical example, let us consider the Archimedean spiral: a curve described by a point which moves with constant velocity $v$ along a radius that rotates around a center point with constant angular velocity $\omega$. In polar coordinates, the equation of the Archimedean spiral has the form

$$\rho=\frac{v}{\omega}\phi$$

or, with $\alpha=\frac{v}{\omega}$,

$$\rho=\alpha\phi$$

which, as can be seen, is a parametric description in terms of $\phi$.

To convert the polar representation of the Archimedean spiral to the parametric form, we only have to replace $\phi$ with $t$ and add an equation in terms of $\phi$, that is

$$\left\{ \begin{aligned} \rho(t)&=\alpha t\\ \phi(t)&= t \end{aligned}\right.$$

The computation of the coordinates of the Archimedean spiral can now be accomplished by mapping the parametric equation onto the array of values of $t$:

archimedean_spiral(alpha, t0, t1, dt) =

map(t -> pol(alpha*t, t),

enumerate(t0, t1, dt))

In the previous definition, the Archimedean spiral is positioned at the origin. If we want to make the center of a spiral a parameter $p$, we only need to operate a simple translation, i.e.:

archimedean_spiral(p, alpha, t0, t1, dt) =

map(t -> p+vpol(alpha*t, t),

enumerate(t0, t1, dt))

This figure shows three Archimedean spirals drawn from the following invocations:

> spline(archimedean_spiral(xy(0, 0), 2.0, 0, 4*pi, 0.1))

Spline(...)

> spline(archimedean_spiral(xy(50, 0), 1.0, 0, 6*pi, 0.1))

Spline(...)

> spline(archimedean_spiral(xy(100, 0), 0.2, 0, 36*pi, 0.1))

Spline(...)

Archimedean spirals. From left to right, the parameters are $\alpha=2, t \in [0,4\pi]$, $\alpha=1, t \in [0,6\pi]$, and $\alpha=0.2, t \in [0,36\pi]$.