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In an n-dimensional space where n ≥ 3, a set of k points {, , …, } are coplanar if and only if the matrix of their relative differences, that is, the matrix whose columns (or rows) are the vectors , , …, is of rank 2 or less.
The vectors ¯, ¯ are parallel to the vectors ¯, ¯ and therefore the coplanarity constraint holds if we substitute these vectors. If we call y , y ′ {\displaystyle y,y'} the coordinates of the projections of P {\displaystyle P} onto the left and right image planes, then the coplanarity constraint may be written as
In computer vision, the fundamental matrix is a 3×3 matrix which relates corresponding points in stereo images.In epipolar geometry, with homogeneous image coordinates, x and x′, of corresponding points in a stereo image pair, Fx describes a line (an epipolar line) on which the corresponding point x′ on the other image must lie.
The general formulation of covariance and contravariance refers to how the components of a coordinate vector transform under a change of basis (passive transformation). Thus let V be a vector space of dimension n over a field of scalars S, and let each of f = (X 1, ..., X n) and f′ = (Y 1, ..., Y n) be a basis of V.
Most of these methods operate in O(n) time (linear time), where n is the number of edges (or vertices) in the graph, which is asymptotically optimal. Rather than just being a single Boolean value, the output of a planarity testing algorithm may be a planar graph embedding , if the graph is planar, or an obstacle to planarity such as a ...
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
When the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points (,,, …) and the interpolation problem consists of yielding values at arbitrary points (,,, …).
Image rectification is used in computer stereo vision to simplify the problem of finding matching points between images (i.e. the correspondence problem), and in geographic information systems (GIS) to merge images taken from multiple perspectives into a common map coordinate system.