Eight or infinity sign trajectory

Question

When I want to move object around point I do:

    point.x *= cosf(timer.timeElapsed);
    point.y *= sinf(timer.timeElapsed);

How to make point move on eight or infinity sign trajectory?

Answer 1

Some possibilities:

Lemniscate of Bernoulli

Lemniscate of Gerono

Lemniscate of Booth

Watt's curve

Answer 2

As Marton notes, there are several "figure of eight" curves that might fit your needs. Perhaps the simplest is the lemniscate of Gerono, which has the parametrization:

x = cos(t);
y = sin(2*t) / 2;

and looks like this:

However, the lemniscate of Bernoulli may be visually more pleasing; it has a parametrization very similar to the lemniscate of Gerono, except that both axes are scaled by a factor of 1/(sin(t)^2 + 1) = 2/(3 - cos(2*t)):

scale = 2 / (3 - cos(2*t));
x = scale * cos(t);
y = scale * sin(2*t) / 2;

It looks like this:

(Animations made with Maple 13, compressed with GIFsicle.)