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, . Because they satisfy a quadratic constraint, they establish a one-to-one correspondence between the 4-dimensional space of lines in and points on a quadric in (projective 5-space). A predecessor and special case of Grassmann coordinates (which describe -dimensional linear subspaces, or flats, in an -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.
Geometric intuition
A line in 3-dimensional Euclidean space is determined by two distinct points that it contains, or by two distinct planes that contain it. Consider the first case, with points and The vector displacement from to is nonzero because the points are distinct, and represents the direction of the line. That is, every displacement between points on is a scalar multiple of . If a physical particle of unit mass were to move from to , it would have a moment about the origin. The geometric equivalent to this moment, is a vector whose direction is perpendicular to the plane containing and the origin, and whose length equals twice the area of the triangle formed by the displacement and the origin. Treating the points as displacements from the origin, the moment is , where "×" denotes the vector cross product. For a fixed line, , the area of the triangle is proportional to the length of the segment between and , considered as the base of the triangle; it is not changed by sliding the base along the line, parallel to itself. By definition the moment vector is perpendicular to every displacement along the line, so , where "⋅" denotes the vector dot product.
Although neither nor alone is sufficient to determine , together the pair does so uniquely, up to a common (nonzero) scalar multiple which depends on the distance between and . That is, the coordinates
may be considered homogeneous coordinates for , in the sense that all pairs , for , can be produced by points on and only , and any such pair determines a unique line so long as is not zero and . Furthermore, this approach extends to include points, lines, and a plane "at infinity", in the sense of projective geometry. In addition a point lies on the line if and only if .
Example. Let and . Then .
Alternatively, let the equations for points of two distinct planes containing be
Then their respective planes are perpendicular to vectors and , and the direction of must be perpendicular to both. Hence we may set , which is nonzero because are neither zero nor parallel (the planes being distinct and intersecting). If point satisfies both plane equations, then it also satisfies the linear combination
That is,
is a vector perpendicular to displacements to points on from the origin; it is, in fact, a moment consistent with the previously defined from and .
Proof 1: Need to show that
what is "r"?
Without loss of generality, let
Point is the origin. Line passes through point and is orthogonal to the plane of the picture. The two planes pass through and and are both orthogonal to the plane of the picture. Points and are the closest points on those planes to the origin , therefore angles and are right angles and so the points lie on a circle (due to a corollary of Thales's theorem). is the diameter of that circle.
Angle is a right angle due to the following argument. Let . Since (by side-angle-side congruence), then . Since , let . By the inscribed angle theorem, , so . ; , ; therefore, . Then must be a right angle as well.
Angles are right angles, so the four points lie on a circle, and (by the intersecting secants theorem)
that is,
Proof 2:
Let
This implies that
According to the vector triple product formula,
Then
When the line passes the origin with direction . If the line has direction ; the plane that includes the origin and the line has normal vector ; the line is tangent to a circle on that plane (normal to and perpendicular to the plane of the picture) centered at the origin and with radius
Example. Let , and , . Then .
Although the usual algebraic definition tends to obscure the relationship, are the Plücker coordinates of .
Algebraic definition
Primal coordinates
In a 3-dimensional projective space , let be a line through distinct points and with homogenous coordinates and .
The Plücker coordinates are defined as follows: