Polar correspondence

A correspondence between two surfaces such that at corresponding points the radius vector of one of them is parallel to the normal of the other, and vice versa. For every smooth surface $ F $ in $ E ^ {3} $ with radius vector $ x $ there exists (under certain conditions) a surface $ F ^ { * } $ polar with it and with radius vector $ x ^ {*} = - n / ( x , n ) $, where $ n $ is the normal and $ ( x , n ) $ is the support function to $ F $, so that

$$ ( x ^ {*} , x ) = 1 ,\ ( x _ {i} , x ) = \ ( x _ {i} , x ^ {*} ) = 0 . $$

Sometimes these conditions are also included in the definition of a polar correspondence.

The concept of polar correspondence shows itself particularly clearly (in the sense of a complete duality) in centro-affine geometry.