Affine coordinate system

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A rectilinear coordinate system in an affine space. An affine coordinate system on a plane is defined by an ordered pair of non-collinear vectors and (an affine basis) and a point (the coordinate origin). The straight lines passing through the point and parallel to the basis vectors are known as the coordinate axes. The vectors and define the positive direction on the coordinate axes. The axis parallel to the vector is called the abscissa (axis), while that parallel to the vector is called the ordinate (axis). The affine coordinates of a point are given by an ordered pair of numbers which are the coefficients of the decomposition of the vector by the basis vectors:

The first number is called the abscissa, while the second number is called the ordinate of .

An affine coordinate system in three-dimensional space is defined as an ordered triplet of linearly-independent vectors and a point . As in the case of the plane, one defines the coordinate axes — abscissa, ordinate and applicate — and the coordinates of a point. Planes passing through pairs of coordinate axes are known as coordinate planes.

How to Cite This Entry:
Affine coordinate system. A.S. Parkhomenko (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098