Orthogonal affine coordinate systems are rectangular, and others are referred to as oblique.

Basis vectors that are the same at all points are global bases, and can be associated only with linear or affine coordinate systems.

A collection of n affinely independent affine functions is an affine coordinate system on A.

For defining a polynomial function over the affine space, one has to choose an affine coordinate system.

Hence, the abovementioned formulas for Cartesian coordinates are applicable in any affine coordinate system.

A model of natural texture based on a structural component which uses affine coordinate transformations and a stochastic residual component is presented.

There is a unique affine structure on this maximal spectrum that is compatible with the filtration on the affine coordinate ring.

An affine coordinate system on A sets up a bijection of A with the complex coordinate space , whose elements are n-tuples of complex numbers.

In each affine coordinate domain the coordinate vector fields form a parallelization of that domain, so there is an associated connection on each domain.

Conversely, it is possible to recover the points of the affine space as the set of algebra homomorphisms from the affine coordinate ring into the complex numbers.

In mathematics, an affine coordinate system is a coordinate system on an affine space where each coordinate is an affine map to the number line.

The algebra of polynomials in the affine functions on A defines a ring of functions, called the affine coordinate ring in algebraic geometry.

This technically defines the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.

The theorem can be refined to include a chain of ideals of R (equivalently, closed subsets of X) that are finite over the affine coordinate subspaces of the appropriate dimensions.

For the Riemann sphere, above, the notion of a polynomial is not defined globally, but simply with respect to an affine coordinate chart, namely that consisting of the complex plane (all but the north pole of the sphere).

More precisely, the choice of an affine coordinate system (or, in the real case, a Cartesian coordinate system) for an affine plane P over a field F induces an isomorphism of affine planes between P and F.

It was realised in the nineteenth century that the ring of integers of a number field has analogies with the affine coordinate ring of an algebraic curve or compact Riemann surface, with a point or more removed corresponding to the 'infinite places' of a number field.