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In geometry, collinearity of a set of points is the property of their lying on a single line. [1] A set of points with this property is said to be collinear (sometimes spelled as colinear [2]). In greater generality, the term has been used for aligned objects, that is, things being "in a line" or "in a row".
The collinearity equations are a set of two equations, used in photogrammetry and computer stereo vision, to relate coordinates in a sensor plane (in two dimensions) to object coordinates (in three dimensions).
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated ...
5 Definition in homogeneous coordinates. 6 Role in non-Euclidean geometry. ... The cross ratio of the four collinear points A, B, C, and D can be written as
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 ...
The 1st Brocard cubic is the locus of X for which the points X A, X B, X C are collinear. The 1st Brocard cubic passes through the centroid, symmedian point, Steiner point, other triangle centers, and the vertices of the 1st and 3rd Brocard triangles.
In photogrammetry and computer stereo vision, bundle adjustment is simultaneous refining of the 3D coordinates describing the scene geometry, the parameters of the relative motion, and the optical characteristics of the camera(s) employed to acquire the images, given a set of images depicting a number of 3D points from different viewpoints.
The usual lifting transform that relates the Delaunay triangulation to the bottom half of a convex hull (i.e., giving each point p an extra coordinate equal to |p| 2) shows the connection to the planar view: Four points lie on a circle or three of them are collinear exactly when their lifted counterparts are not in general linear position.