0
votes
2answers
52 views

Is there a general way to parameterize all implicit functions?

We all know some curves can be described by $y=f(x)$ and some surfaces can be described by $z=f(x,y)$ However, there exists curves and surfaces which cannot be described by those, such as a circle and ...
10
votes
1answer
123 views

Why can't elliptic curves be parameterized with rational functions?

Background: For our abstract algebra class, we were asked to prove that $\mathbb{Q}(t, \sqrt{t^3 - t})$ is not purely transcendental. It clearly has transcendence degree $1$, so if it is purely ...
2
votes
2answers
124 views

Implicit form of a parametric surface

Let $\Sigma$ be the surface in $\mathbb{R}^3$ parametrized by $$ (u,v) \mapsto \Big(\;p_X(u,v),\; p_Y(u,v),\; p_Z(u,v)\;\Big), $$ where $p_X, p_Y, p_Z$ are polynomials. Is there a standard way to ...
0
votes
0answers
74 views

Torus equation in terms of tangent

So if I have an equation for a torus in $F(a,b) = (X, Y, Z)$ where $X = (R + r\cos a)\cos b$ and $0 < r < R$, how would I go about rewriting this equation for $X$ in terms of $\tan(a/2)$ and ...
0
votes
1answer
1k views

Converting parametric equation to implicit form

So I have the equation defined in homogeneous coordinates $[w; x, y]$ as $[1+t^2; 1-t^2, 2t]$ $$w = 1+t^2$$ $$x = 1-t^2$$ $$y = 2t$$ If I do $w+x-y$ I get $-2t+2$, so $t = -(w+x-y-2)/2$. I was then ...
0
votes
0answers
875 views

Converting standard equation for a paraboloid to a parametric one

I have the equation for a hyperbolic paraboloid in $x$, $y$, and $z$: $$\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}$$ I also have the parametric equations for the same parabaloid: $$x = a u ...