Parametric equation of a cone
I usually use the following parametric equation to find the surface area of a regular cone $z=\sqrt{x^2+y^2}$: $$x=r\cos\theta$$ $$y=r\sin\theta$$ $$z=r$$
And make $0\leq r \leq 2\pi$, $0 \leq \theta \leq 2\pi$.
I've now have a cone $z=\sqrt{2x^2+2y^2}$ and I think the parametric equation I normally use won't work anymore. Which would be a more suitable one in this case? Is there any generic parametric equation for cones, because one of the form $z=\sqrt{4x^2+y^2}$ would also have a different one.
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$\begingroup$Let $z=\sqrt{a^2 x^2 + b^2 y^2}$ where $a>0$ and $b >0$
Then let $z=r$, $x=\frac{r}{a} cos(\theta)$ and $y= \frac{r}{b} sin(\theta)$
$\endgroup$ $\begingroup$I like doing it with $\sin(r)$ or $\cos(r)$ as the scalar quantity rather than $r$. When I make algebraic spheres and cones it works out better. The points on a sphere and cone look the same in algebraic chaos. The color function also makes more sense when done this way. Also rational triangles don't divide evenly between $0$ and $1$.
You probably have no idea what I mean. I suppose I can elaborate a little further. This is just an example that makes it easier to describe what I'm talking about. In actuality I'm making spheres and cones with a different algorithm. I don't consider this an algebraic method but it's closer than the other way.
cone
\begin{align} x &= \sin(x \arctan(y/x))\sin(y \arctan(y/x))\\ y &= \sin(x \arctan(y/x))\cos(y \arctan(y/x))\\ z &= \sin(x \arctan(y/x)) \end{align}
sphere \begin{align} x &= \sin(x \arctan(y/x)) \sin(y \arctan(y/x))\\ y &= \sin(x \arctan(y/x)) \cos(y \arctan(y/x))\\ z &= \cos(x \arctan(y/x)) \end{align}
with $x\in [-49, 49]$, $y\in [-49, 49]$.
The color function is just the product of all three. You can do an area formula on the sphere of each point and it just ends up being the product. Well sort of. The min/max in Maple is -max,..max
Everybody seems to use $r$ as the scalar. But nobody seems to say why. Is it just because it's convenient? Or is there a particular reason that r is better?
$\endgroup$ $\begingroup$Correct formula (from: ) in Mathematica Language is:
ParametricPlot3D[{r*(h - u)/h Cos[t], r*(h - u)/h Sin[t], u}, {u, 0, h}, {t, 0, 2 Pi}]where:$h$ is height of cone, $r$ is radius of cone (constants);$t, u$ are parameters.
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