We now state the algorithm in projective coordinates.

2.

By Hilbert's basis theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian.

3.

Similarly, it may be shown that this topology is defined intrinsically by sets of elements of the projective coordinate ring, by the same formula as above.

4.

Here some efficient algorithms of the addition and doubling law are given; they can be important in cryptographic computations, and the projective coordinates are used to this purpose.

5.

Such a rational parameterization may be considered in the projective space by equating the first projective coordinates to the numerators of the parameterization and the last one to the common denominator.

6.

Mathematically, " x " and " y " are projective coordinates and the colors of the chromaticity diagram occupy a region of the real projective plane.

7.

To know more about the speeds of addition and doubling in projective coordinates on Edwards curves, standard coordinates on twisted Edwards curves, inverted coordinates on Edwards curves and inverted coordinates on twisted Edwards curves refer to the table in:

8.

Historically, homographies ( and projective spaces ) have been introduced to study field ( the above definition is based on this version ); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies.

9.

In other words, a projective variety is a projective algebraic set, whose homogeneous coordinate ring is an integral domain, the " projective coordinates ring " being defined as the quotient of the graded ring or the polynomials in " n " + 1 variables by the homogeneous ( reduced ) ideal defining the variety.

10.

A point P = ( x, y ) on the elliptic curve in the Montgomery form By ^ 2 = x ^ 3 + Ax ^ 2 + x can be represented in Montgomery coordinates P = ( X : Z ), where P = ( X : Z ) are projective coordinates and x = X / Z for Z \ ne 0.