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huyichen
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How to show that a transverse intersection is clean, but not conversely?
A transverse intersection is a type of intersection between two objects, where the objects intersect at a right angle or perpendicular to each other. In other words, the intersection occurs in a "clean" or non-tangled manner without any overlapping or tangential points.
A clean intersection in the context of transverse intersections refers to an intersection where the tangent spaces of the two objects at the intersection point are distinct and span the entire space. This means that the two objects are not "sticking" or overlapping at the intersection point, and there are no tangential directions between them.
To show that a transverse intersection is clean, you can use the definition of a clean intersection and check if the tangent spaces of the two objects at the intersection point are distinct and span the entire space. This can also be demonstrated by examining the Jacobian matrix of the intersection equation, which should have full rank at the intersection point.
Yes, it is possible for a transverse intersection to be clean but not conversely. This means that the intersection is clean, but the converse may not hold true. In other words, the objects may intersect cleanly, but their tangent spaces may not be distinct and span the entire space. This can happen when the objects have a tangential direction at the intersection point, but it is not significant enough to cause tangling or overlapping.
Understanding clean transverse intersections is important in various fields of science and mathematics, such as differential geometry, topology, and physics. It is particularly relevant in studying critical points and singularities, as well as in applications like robotics, computer graphics, and computer vision.