Parametric equation of a curve

Question

What is the curve that has a parametric equation as, $(a\cos^{4}t, a\sin^{4}t)$? Does it have any special name?

Answer 1

This parametric equation is a special case of a family of curves called superellipses, which have equations:

$$x^{n} + y^{n} = a^{n}$$

Or, parametrically:

$$x = a\cos^{\frac{2}{n}}(t)$$

$$y = a\sin^{\frac{2}{n}}(t)$$

Thus, the parametric equation you are inspecting is a superellipse with $n = \frac{1}{2}$. It has equation:

$$\sqrt{x} + \sqrt{y} = \sqrt{a}$$

$$\sqrt{y} = \sqrt{a}-\sqrt{x}$$

$$y = a + x - 2a\sqrt{x}$$

With $0\leq x,y\leq a$. It happens that this equation produces a segment of a parabola with its axis of symmetry as the line $y=x$.

Many curves are superellipses (for example, a circle is $n=2$, a astroid is $n=\frac{2}{3}$, and a line segment is $n=1$.)