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I have an axis made up of 2 points in 3D space. This is the axis of a cone which extends outwards from one of these two points into space. The apex angle of this cone is known, theta.

I have a 2D plane at some distance from this cone and I am hoping to figure out the projection of this cone on the 2D slice. This question is fairly confusing so here is a picture of the scenario:

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The points which make the axis can only be found in the cubic areas they sit it and the cone always extends in the forward direction. My maths is a bit rusty and I have tried to solve this myself to no avail. I am hoping to get a mathematical solution to determine the shape of the ellipses on the 2D plane.

Any help which can be given would be greatly appreciated.

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    Who said it'd be an ellipse? The projection would be a conic section. – 2017-02-25
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    Hi frederick99, you are of course right. Depending on the axis and angle of the cone this shall be a conic section or an ellipse. I wrote this question on the train in a bit of a rush. Thank you for highlighting this. – 2017-02-25
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    I am not great at maths as well. But if the angle made by the axis and normal to the plane and the semi-vertical angle of the cone are complementary, you'd have a parabola. And if the angle is zero, obviously the conic is a circle. An angle b/w zero and the magical angle at which we get a parabola, the conic formed is an ellipse and for every angle beyond, you have a hyperbola. – 2017-02-25
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    The circle equation should be quite straight forward and I think the parabola would not be very difficult. When it comes to ellipses and hyperbolas, I start to lose my confidence :P – 2017-02-25
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    I think mathematicians here would understand this question better if you replaced the word "projection" by "cross section". This is standard geometry that goes back to the Greeks. There's lots of information on the wikipedia page https://en.wikipedia.org/wiki/Conic_section . – 2017-02-25

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