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Consider 2 oddly shaped surfaces and pick a point on each of the surfaces. At the point considered we can always draw a normal to the surface. Now if the two surfaces are in contact at this same point will the normal of one surface always be coincident to the normal of the other surface?.

I am asking this because, when studying about gears, there is always talk of the common normal and I am trying to understand if the presence of common normal is due to the design of the gear tooth or there is always a common normal between 2 surfaces in contact.

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If two surfaces are tangentially in contact meaning they have the same partial derivatives at that point; then they do have a common normal.

And it is the cross product of the two tangent vectors,

$ \frac{\partial f}{\partial x} \space and\space \frac{\partial f}{ \partial y}.$

With f being the surface function.

However two surfaces can be in contact intersecting each other or touching at an edge, where they won't have a common normal!

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