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 Question from Tom: Is there a way to derive an equation that describes the perimeter of the intersection of a plane and a cone regardless of the angle of the plane to the cone. Assume that the plane does not cut through the base of the cone, the x, y, z location of the vertex is known, the distance from the vertex to the plane through the axis is know., and that the angle of the cone is known.

It is not clear to me what your goal is. It should first be understood that a single “equation” involving three variables (x, y, and z) will necessarily represent a surface, not a curve. To describe a curve one needs a single independent variable, say x. A curve will then be described by equations for y, and z as functions of x. One way to obtain such equations for your conic would be to obtain the equations of the cone and the plane. It would make sense to put the cone in standard position:

$z^2 = x^2 + y^2,$

and let the plane be $z = ax + by + c.$

Plug $z = ax + by + c$ into the equation of the cone to get a quadratic equation that you can solve for $y;$ then plug the resulting expression for $y(x)$ into the second equation to obtain $z(x).$

It sounds as if you want to use the general equation of a cone, in which case the resulting expressions for $y(x)$ and $z(x)$ will be even more complicated.

Chris

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