Finding the Equation of a Rotated Cone: A Step-by-Step Guide

[eddo]

I have two questions. How do you find the equation of a cone given data points? I've found lots of info on the equation of a cone, but can't find anything on one that is rotated and not centered at the origin. What is the equation for a rotated translated cone?
Second, given the equation of a cone, how do you figure out the equation of the axis?

Side question: Does anyone know of a good way to find an equation of a cone given more data points than necesarry, which won't fit perfectly to the cone? Does anyone know of a good way to approximate the cone equation which takes into account more points? Thanks for any help.

 

[fresh_42]

There is no single equation for a cone: it has a circle as a bottom, is of finite height and has a mantle. At best you have a rotational right triangle.

I would write down the equation(s) of a standard cone. Then I would apply possible transformations, like stretching along a certain plane or whatever you allow as transformation. This gives you a couple of equations with the transformation coefficients as parameters. Finally you can try to solve for these parameters given your actual data.

If we have more data than necessary, than I would try to use a linear regression algorithm first and operate with its result instead of the raw data.

I'm not aware of a ready made answer to what you might have in mind. My solution requires some work to do.