Equations for infinite double cones in the Cartesian, cylindrical and spherical coordinate systems.
We start with the equation for a circle centered at the origin, but instead of treating
the radius as a constant, we make it a variable in the third dimension.
$$x^2 + y^2 = z^2$$
All points \((x, y, z)\) that satisfy this equation are on a double cone.
We can picture the double cone as an infinite number of circles parallel to the x-y plane.
For each radius \(\sqrt{x^2 + y^2}\) there are two circles, one at \(z = \sqrt{x^2 + y^2}\) and one at \(z =
-\sqrt{x^2 + y^2}\).
If we make a cut at \(y = 0\), we get the following shape.
Another way of conceptualizing the double cone is as a line \(z = mx\) with slope \(m = \tan \alpha\) rotated about the z-axis. Our above equation uses the radius for z without any scaling, so \(m = 1\). But in general, we have $$ (mx)^2 + (my)^2 = z^2. $$
To get the cylindrical coordinates from Cartesian coordinates, we simply replace the x- and y-coordinates with polar coordinates and keep z. Each point then is described with a triple \((r, \theta, z)\), where \(r^2 = x^2 + y^2\) and \(\tan \theta = y/x\). That's very convenient for our double cone because we only need to describe how r relates to z. $$ z = mr $$ As long as this equation holds, points \((r, \theta, z)\) with any angle \(\theta\) are on the double cone.