Can $x^4 + y^4 = 1$ be parameterized in terms of $\theta $?

Question

In general, can equations of the form $(\frac {x}{a})^n + (\frac {y}{a})^n= 1 $ be parameterized in some polar form with $ r $ and $\theta $? (Where $ n $ is an even integer $> 2 $)

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My assertion is that it cannot be parameterized.

Considering the equation in the title, if we simply put $ y= mx $ (I.e line through origin making slope $ m $)(and, $ m= \tan {\theta} $) we can solve for $ x $ and $ y $ in terms of $\theta $. However you get a polynomial of degree 4. This means that $x$ (at some angle $\theta $) will have 4 values (two of which are complex). We are not able plot this in 2d.

Contrast this with a circle ($ x^2 + y^2=1 $). Varying $\theta $ always gives x and y values which are real. Because of which I can plot say only the y ordinate with respect to $\theta $ (which in other words is the sine function). Here you can get a function which we are able to plot in 2d.

Answer 1

The same parametrization of ellipse can be adapted with its exponent manipulated:

$$ x = \cos ^ { 2/n} \theta \quad y = \sin ^ { 2/n} \theta $$

Answer 2

$x^4 + y^4 = 1$ can be parameterized as

$\left\{\begin{array}{rcl} r_{12}(t)&=&(\pm\sqrt{\cos t},\; \sqrt{\sin(t)}),\; t \in [0,\pi/2]\\[0.2cm] r_{34}(t)&=&(\sqrt{\cos t},\; \pm\sqrt{\sin(t)}),\; t \in [0,\pi/2]\\[0.2cm] \end{array}\right.$