Cones over Curves

Given a curve X(t) and a point P , we may form the cone over X(t) from P by defining C_P(t,u) =uX(t) +(1-u)P , where t runs through the domain of the curve and u runs from 0 to 1. The point P is called the apex. The cone is an example of a ruled surface since every point of the surface lies in at least one straight line lying in the surface.

For certain curves, it is possible to find a position for the apex so that the cone over the curve is embedded, i.e., so that it does not intersect itself. In other cases, this is not possible: when the curve forms a knot, for example. For a plane curve that does not intersect itself, the cone from any point not in its plane will necessarily be embedded, and the cone from a point in the plane will be embedded if and only if the curve is star-shaped with respect to the point.

Demonstration 1: Cones over Curves

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For the space cardioid, whose equation is

X(t) =((1+cos(t))cos(t), (1+cos(t)) sin(t),csin(t))

which positions of the cone point lead to embedded cones?

For an ellipse in the plane, which positions of the cone point lead to embedded cones?

What can you say about the intersection of a plane and the cone over a curve from a point?