Poincare Dual
group dual
The PoincarĂ© Dual of an element is the "subspace complement" of the argument with respect to the pseudoscalar in the exterior algebra. In practice, it is a relabeling of the coordinates to their dualcoordinates and is used most often to implement a "join" operation in terms of the exterior product of the duals of each operand.
Ex: The dual of the point \(\mathbf{e}_{123} + 3\mathbf{e}_{013}  2\mathbf{e}_{021}\) (the point at \((0, 3, 2)\)) is the plane \(\mathbf{e}_0 + 3\mathbf{e}_2  2\mathbf{e}_3\).
Summary
Members  Descriptions 

public KLN_INLINE point KLN_VEC_CALL operator! (plane in) noexcept 

public KLN_INLINE plane KLN_VEC_CALL operator! (point in) noexcept 

public KLN_INLINE line KLN_VEC_CALL operator! (line in) noexcept 

public KLN_INLINE branch KLN_VEC_CALL operator! (ideal_line in) noexcept 

public KLN_INLINE ideal_line KLN_VEC_CALL operator! (branch in) noexcept 

public KLN_INLINE dual KLN_VEC_CALL operator! (dual in) noexcept 