Search results
Results from the WOW.Com Content Network
If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example ...
Alternatively, a line can be described as the intersection of two planes. Let L be a line contained in distinct planes a and b with homogeneous coefficients (a 0 : a 1 : a 2 : a 3) and (b 0 : b 1 : b 2 : b 3), respectively. (The first plane equation is =, for example.) The dual Plücker coordinate p ij is
Switching to homogeneous coordinates using the embedding (a, b) ↦ (a, b, 1), the extension to the real projective plane is obtained by permitting the last coordinate to be 0. Recalling that point coordinates are written as column vectors and line coordinates as row vectors, we may express this polarity by:
The coordinates in the higher-dimensional space are an example of homogeneous coordinates. If the original space is Euclidean , the higher dimensional space is a real projective space . The advantage of using homogeneous coordinates is that one can combine any number of affine transformations into one by multiplying the respective matrices.
Since the homogeneous coordinates given by the minors are 6 in number, this means that the latter are not independent of each other. In fact, a single quadratic relation holds between the six minors, as was known to nineteenth-century geometers. This example was the first known example of a Grassmannian, other than a projective space. There are ...
Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C ...
For example, Plücker coordinates are used to determine the position of a line in space. [11] When there is a need, the type of figure being described is used to distinguish the type of coordinate system, for example the term line coordinates is used for any coordinate system that specifies the position of a line.
Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preserving commutativity and other properties), it becomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (a ...