Plücker coordinates

In geometry, Plücker coordinates, introduced by Julius Plücker in the 19th century, are a way to assign six homogeneous coordinates to each line in projective 3-space, ${\displaystyle \mathbb {P} ^{3}}$. Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in ${\displaystyle \mathbb {P} ^{3}}$ and points on a quadric in ${\displaystyle \mathbb {P} ^{5}}$ (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe k-dimensional linear subspaces, or flats, in an n-dimensional Euclidean space), Plücker coordinates arise naturally in geometric algebra. They have proved useful for computer graphics, and also can be extended to coordinates for the screws and wrenches in the theory of kinematics used for robot control.