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I'm trying to plot the conic intersection of a plane and a cone in python. I have a plane that is defined as x, y, z=0 and I have a unit vector and an angle at some height which I want to generate a cone from that angle extending out of the unit vector. Then where the extended cone and plane intersect I am interested in the projection of the intersection between the cone and plane. I would expect with angles at around 45 degrees that the projection would be a parabola. I'm trying to plot this in python. I think for the plane I would create a meshgrid for x,y within some bounds, and leave z=0. But I'm not sure how to create an array of points for the cone. If I had that my goal would then to look at the points of intersection between the x_plane, y_plane, and x_cone,y_cone to extract a 2D array of x's and y's that are the data points of the conic intersection, which should be able to be plotted with plt.plot() to get a parabola?

Graham Chapman
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  • Your question is not clear: Do you need the equation of the cone? Or something else? – Intelligenti pauca Mar 29 '23 at 13:14
  • See if this can be of interest: https://math.stackexchange.com/questions/4079720/on-the-equation-of-the-ellipse-formed-by-intersecting-a-plane-and-cone – Intelligenti pauca Mar 29 '23 at 13:16
  • Note that, to have a parabola, one of the generatrices of the cone must be parallel to the plane. – Intelligenti pauca Mar 29 '23 at 13:17
  • See also https://math.stackexchange.com/questions/2368523/find-two-points-and-one-angle-so-that-cone-intersects-the-xy-plane-in-the-para/ – Intelligenti pauca Mar 29 '23 at 13:19
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    Sorry, to clear things up, I am looking for how to generate an equation for a cone given a 3d unit vector, a height, and an angle. I believe I can generate the equation for the plane, it is just x,y grid values at z=0. I think if I have the equation for a cone I can find the intersection between the arrays of x,ys of the plane and the cone. Then that could be plotted as x,y values of their own in whatever shape needed. – Graham Chapman Mar 29 '23 at 15:29
  • See here for the equation of the cone: https://math.stackexchange.com/questions/3233966/help-finding-the-equation-of-a-cone-in-space – Intelligenti pauca Mar 29 '23 at 17:01

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