Can anyone pictorially show me of a tangent plane to a right circular double cone? Since I think it doesn't have a tangent plane through the origin (correct me if I am wrong), I am not able to visualize it.
2026-04-13 13:38:27.1776087507
Pictorial representation of a tangent plane to a right circular cone
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A right circular cone can be constructed by choosing a line to serve as the axis of the cone, choosing a point on the axis to serve as the vertex of the cone, taking a plane perpendicular to the axis, and constructing a circle in that plane whose center is the point where the axis intersects the plane.
From the vertex you can construct a line segment to any point on the circle. If you take the set of all such line segments you get a finite cone. If you extend each segment from the vertex through the circle and continuing indefinitely beyond the circle, you get a single infinite cone. And if you extend each segment indefinitely in both directions, you get a double infinite cone.
In order for a plane to be tangent to the cone (in either the finite, single infinite, or double infinite case), the plane must contain one of the lines through the vertex and the circle, and the tangent plane must intersect the plane of the circle in a line that is tangent to the circle.
Any other plane will cut through the cone forming a conic section, will cut the cone along two intersecting lines, or will pass through the vertex only and no other point of the cone; none of those cases produces a tangent plane.
It follows that the only place the tangent plane touches the cone is along one single line through the vertex and one single point on the given circle. You can choose any of infinitely many such lines, but once you have chosen one, it is the only one that will lie in the tangent plane. It is not possible for a tangent plane to contain two lines that lie on the cone's surface, let alone three.
If a plane is tangent to a finite right circular cone, however, it is also tangent to the infinite single cone extended from that finite cone, because as the tangent line is extended along the surface of the cone it is still in the plane. Likewise the same plane is also tangent to the double cone, because when the line is extended in the opposite direction it again remains in the same plane.