Cylinders over Curves

Given a curve X(t) and a unit vector V , we may form the cylinder over X(t) in the direction of V by defining Z_V(t,u) =X(t) +uV , where t runs through the domain of the curve and u takes on all real values. The cylinder is another example of a ruled surface .

For certain curves, it is possible to find a direction so that the cylinder over the curve in that direction is embedded, i.e. so 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 cylinder along any direction not parallel to that plane will necessarily be embedded, while the cylinder along any direction lying in the plane will be embedded only if no tangent vector to the curve is parallel to the direction.

Demonstration 2: Cylinders over Curves

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For the space cardioid, which directions lead to embedded cylinders?

For an ellipse in the plane, which directions lead to embedded cylinders?

What can you say about the intersection of a plane and the cylinder over a curve along a particular direction?