# Poincare Dual

## group dual

The Poincaré 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 dual-coordinates 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