Editing Plücker coordinates (section)

{{short description|Method of assigning coordinates to every line in projective 3-space}}
{{About|the classic case of lines in projective 3-space|general Plücker coordinates|Plücker embedding}}

{{No footnotes|date=February 2011}}

In [[geometry]], '''Plücker coordinates''', introduced by [[Julius Plücker]] in the 19th century, are a way to assign six [[homogeneous coordinates]] to each [[line (mathematics)|line]] in [[projective space|projective 3-space]], {{tmath|\mathbb P^3}}. Because they satisfy a quadratic constraint, they establish a [[one-to-one correspondence]] between the 4-dimensional space of lines in {{tmath|\mathbb P^3}} and points on a [[quadric]] in {{tmath|\mathbb P^5}} (projective 5-space). A predecessor and special case of [[Grassmann coordinates]] (which describe {{mvar|k}}-dimensional linear subspaces, or [[Flat (geometry)|''flats'']], in an {{mvar|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 [[screw theory|screws and wrenches]] in the theory of [[kinematics]] used for [[robot control]].