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I am working in a project of calibration of a laser set using a geometry with know dimensions, I am using a cone to do my calibration, the cone in 2D section gives an ellipse , my problem right now I searching for a method that help me to find the coordinates of a point on the ellipse with respect the cone coordinates.

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Intelligenti pauca
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abdulbagi abdulaziz
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  • You should explain better what you mean by "to find the coordinates of a point on the ellipse". I suppose you have in mind the coordinates with respect to a pair of axes on the plane itself, but that's not clear. See if you can find something useful here: https://math.stackexchange.com/questions/3102248/is-the-right-intersection-of-an-oblique-circular-cone-an-ellipse/3103027#3103027 – Intelligenti pauca Mar 14 '20 at 15:27
  • thanks for your answer, to explain more if I take the center of the circle at the bottom of the cone as (0,0,0) coordinate in 3D, I want to know the coordinates of the major and minor axis in 2D with respect (0,0,0) – abdulbagi abdulaziz Mar 14 '20 at 23:35
  • "the coordinates of the major and minor axis in 2D with respect (0,0,0)" - what does that mean? The equations of lines $AB$ and $CD$ in figure below? Or else? I explained in my answer how those points are defined, but of course you must know some 3-d analytic geometry to find them. – Intelligenti pauca Mar 14 '20 at 23:55

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The plane perpendicular to the plane of the ellipse and passing through vertex $V$, intersects the cone along two rays $VI$ and $VJ$, cutting the ellipse at $A$ and $B$. Segment $AB$ is then the major axis of the ellipse and its midpoint $O$ is the center of the ellipse.

To find minor axis $CD$, just intersect the cone with the line through $O$ perpendicular to plane $VIJ$. You can easily find all 3-D coordinates of these points, if cone and cutting plane are given.

You can then set up a local coordinate system on the plane of the ellipse, with origin $O$ and axis directions given by unit vectors $$ \hat x={A-O\over \overline{AO}},\quad \hat y={C-O\over \overline{CO}}. $$ With respect to this coordinate system the equation of the ellipse is $$ {x^2\over a^2}+{y^2\over b^2}=1, $$ where $a=\overline{AO}$ and $b=\overline{CO}$.

Any point $P$ on the ellipse, having local coordinates $(x_P,y_P)$ has then a position in space given by $$ P=x_P\,\hat x + y_P\,\hat y. $$

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Intelligenti pauca
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  • Thanks for your answer, to explain more the laser will highlight the cone geometry giving an ellipse in 2D, all I have is the dimensions of the cone, if I take the center of the circle at the bottom of the cone as (0,0,0) coordinate in 3D, I want to know the coordinates of the major and minor axis in 2D with respect to (0,0,0), the coordinates of the intersection between the cone and the ellipse – abdulbagi abdulaziz Mar 14 '20 at 23:58
  • @abdulbagiabdulaziz If you want help on finding the coordinates of points $ABCD$ then you should provide a specific example of cone and plane. – Intelligenti pauca Mar 16 '20 at 11:12