I have an equation x² + y² + z³ - z² = 0
This is an alpha-loop rotated around the z-axis.
Can someone please help me to convert this to three parametric equations for x, y, and z (varying for u and v)?
I ask because I am an artist trying to make models of equations (not a new idea but great fun).
I am almost there, but there is an interesting problem with my surface. The tear drop is upside down... I assume it should be rounded at the top and come to a point (a singularity???) where it meets a similar point on the top of the mountain.
The reason for my assumption is the curve (before rotation) has the shape I am struggling to describe. See: Image of curve from Wolfram
I have used parametric equations as suggested by Ted and Compacto [thanks to both].
Any idea why my curve is inside out or upside down? It should be the same as the curve but rotated around an axis.
Image of surface from equation
Here is the successful result: Success!
If one knows beforehand it's a surface of revolution, then it will be generated by a curve
$$x(t) = f(t), z(t) = g(t)$$
Which is the intersection of the surface with the plane $y = 0$. Rotating this curve around the $z$ axis gives the parametric equations for the surface:
$$(x(u,v), y(u,v), z(u,v)) = (f(v)\cos (u), f(v)\sin (u), g(v))$$
So you just need to obtain the functions $f, g$ which parametrize the curve. As I said before (and was hinted by Ted in the comments), the generating curve is obtained by making $y = 0$ in the implicit equation, giving
$$x^2 = z^2-z^3 = z^2(1-z)$$
And so, $x = |z|\sqrt{1-z}$ ($x$ is assumed to be positive so it lies "on the right" of the axis and forms the part of the curve that will be rotated) so we can put $x(t) =f(t) = |t|\sqrt{1-t}, z(t) = g(t) = t$, and so the parametric equation for the surface is
$$(x(u,v), y(u,v), z(u,v)) = (|v|\sqrt{1-v}\cos (u), |v|\sqrt{1-v}\sin (u), v)$$