Suppose we have a cone passing through the origin of $xyz$ coordinate system. Now, the question is that whether we can find an invertible transformation on this coordinate system that turns the cone into a line or plane, let say on $x'y'z'$ coordinate system? Is the simple argument that a cone consists of infinitely many lines passing through the origin and thus they cannot be packed together and form a line by an invertible transformation true?
2026-04-04 14:12:26.1775311946
Cone under similarity transformation
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For a circular cone, all generating lines have the same angle with the axis of the cone. For an elliptic cone, that angle lies in a range which is maximized for lines lying in one symmetry plane and minimized for the other. Similarity transformations are characterized by preserving angle. A line would be a degenerate cone where all generating lines have angle zero with the axis of symmetry. Conversely, a plane would be a degenerate cone where the minimal angle is $0°$ and the maximal angle is $90°$, in order to stay flat and still cover the whole plane. Neither of these properties can be achieved from a non-degenerate cone while preserving angles.