It sounds like you're describing a plane intersecting a cone in a specific way. When a plane is not parallel to the base, doesn't pass through the base, and makes an angle with the horizontal that is less than the angle made by the slant height of the cone, it intersects the cone in an ellipse. This intersection forms an elliptical shape on the cone's surface. The exact shape and size of the ellipse depend on the angle of intersection and the dimensions of the cone.
