I have $a^2 = (4x-8)^2 + 4y^2$ and $b^2=(4x+8)^2 + 4y^2$ which I switch between every $t=\frac{n\pi}2$
How do I draw this touching the origin, and moving outwards, noting that
$$4x+8=b\cos(t+\beta)$$ $$2y=-b\sin(t+\beta)$$
These are ellipses I can see, and it would seem I am going to get the top right quarter of each repeating. But I can't figure out how to graph this. Any ideas how to graph, would be greatly appreciated.
$$b^2=16(x+2)^2+4y^2\\ 1=\frac{(x+2)^2}{\left(\frac b4\right)^2}+\frac{y}{\left(\frac b2\right)^2}$$
This is an ellipse with semi-major axis $b/2$ (along the $y$-axis), semi-minor axis $b/4$ (along the $x$-axis) and centre $(-2,0)$.
You should be able to draw this easily.