How do I find the intersection of three cones?

[Loubrainz]

Homework Statement

tl;dr: looking for a way to find the intersection of three cones.

I'm currently working with a team to build a Compton camera and I've taken up the deadly task of image reconstruction.

Background Theory:
https://en.wikipedia.org/wiki/Compton_scattering
For a single Compton scattering experiment, photons from a source deflect off of the surface electrons of some object, imparting some of its energy, and then enters a detector. The scattering angle are energy are related by the second formula found in that web page. So for a source with a known photopeak, the angle of scatter can be deduced by the amount the photopeak has shifted.

Problem:
We intend to use three detectors in tandem to pinpoint the location of a source. The scattering angle could be horizontal or vertical. Therefore, we want to find the point of intersection of three cones (given the position of their apexes and opening angles).
To begin with, the apexes of these cones will lie on a single x-y plane and all have the same axes (like in the picture), but then I want to generalise it to cones of any position and direction (such as in an equilateral triangle, pointing inwards).

Homework Equations

http://mathworld.wolfram.com/Cone.html
The equation of a cone with apex position (x0,y0,z0) and axis parallel to the z-axis is: (x - x0)^2 + (y - y0)^2 = c^2 (z - z0)^2, where c is the cone gradient: c = r/h, r being the radius of the base circle and h being the perpendicular height of the cone. The opening angle is theta = 2*arctan(c). I have no idea how to include axis vector/angle into that, or how to get purely in terms of the most relevant coordinate system and input data.

The Attempt at a Solution

Clearly this involves a) solving three simultaneous cone equations in terms of the most relevant coordinate system and input data or b) modelling the cones in Matlab and determining the intersect computationally. Both of these I get stuck with and lack confidence to continue. The additionally degree of freedom granted by the arbitrary cone axes makes it more confusing.

What do you think?

 

[Ray Vickson]

Homework Statement

tl;dr: looking for a way to find the intersection of three cones.

I'm currently working with a team to build a Compton camera and I've taken up the deadly task of image reconstruction.

Background Theory:
https://en.wikipedia.org/wiki/Compton_scattering
For a single Compton scattering experiment, photons from a source deflect off of the surface electrons of some object, imparting some of its energy, and then enters a detector. The scattering angle are energy are related by the second formula found in that web page. So for a source with a known photopeak, the angle of scatter can be deduced by the amount the photopeak has shifted.

Problem:
We intend to use three detectors in tandem to pinpoint the location of a source. The scattering angle could be horizontal or vertical. Therefore, we want to find the point of intersection of three cones (given the position of their apexes and opening angles).
To begin with, the apexes of these cones will lie on a single x-y plane and all have the same axes (like in the picture), but then I want to generalise it to cones of any position and direction (such as in an equilateral triangle, pointing inwards).

Homework Equations

http://mathworld.wolfram.com/Cone.html
The equation of a cone with apex position (x0,y0,z0) and axis parallel to the z-axis is: (x - x0)^2 + (y - y0)^2 = c^2 (z - z0)^2, where c is the cone gradient: c = r/h, r being the radius of the base circle and h being the perpendicular height of the cone. The opening angle is theta = 2*arctan(c). I have no idea how to include axis vector/angle into that, or how to get purely in terms of the most relevant coordinate system and input data.

The Attempt at a Solution

Clearly this involves a) solving three simultaneous cone equations in terms of the most relevant coordinate system and input data or b) modelling the cones in Matlab and determining the intersect computationally. Both of these I get stuck with and lack confidence to continue. The additionally degree of freedom granted by the arbitrary cone axes makes it more confusing.

What do you think?

Google is your friend. See, eg.,
https://mathoverflow.net/questions/120126/intersection-of-cones-in-three-space
or
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.487.5298&rep=rep1&type=pdf

 

[Loubrainz]

Google is your friend. See, eg.,
https://mathoverflow.net/questions/120126/intersection-of-cones-in-three-space
or
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.487.5298&rep=rep1&type=pdf

I found these in my initial googling. I didn't find them helpful. Maybe if you echoed the information in your way it would help me understand.