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great circle
If your question is " What is the shape of a sphere's tangent?" , then the answer is : a plane. A plane basically touches the sphere at only one point and hence , it should be a plane. It cannot be a straight line , as a straight line is a tangent for 2D objects,( ie.) a circle.
A Circle.
A circle, which could degenerate to a point.
We're having a hard time making out the dark grey area from here. But the intersection of a plane and a sphere is always a circle.