Sentence examples for cross ratio from inspiring English sources

Cross ratio, in projective geometry, ratio that is of fundamental importance in characterizing projections.

The latter formulation reveals the cross ratio as a ratio of ratios of distances.

There is one more important invariant under projective mappings, known as the cross ratio (see the figure).

In 1847 the German mathematician Karl G.C. von Staudt showed how to effect this separation by defining the cross ratio without reference to length.

Although the cross ratio was used extensively by early 19th-century projective geometers in formulating theorems, it was felt to be a somewhat unsatisfactory concept because its definition depended upon the Euclidean concept of length, a concept from which projective geometers wanted to free the subject altogether.

In 1873 the German mathematician Felix Klein showed how the basic concepts in Euclidean geometry of length and angle magnitude could be defined solely in terms of von Staudt's abstract cross ratio, bringing the two geometries together again, this time with projective geometry occupying the more basic position.

There are many interesting geometric characterisations of these sets via criteria involving cross ratios and bounded turning initially discovered by Ahlfors, see [9] for a comprehensive survey.

It starts by showing that ratios of lengths can in some cases be invariants, though ratios of ratios of lengths called "cross ratios"—are required in the case of full perspective projection.

Therefore, high zero-crossing ratio (HZCRR) is used in our experiments, and it is defined as follows: (5).

Then the cross-ratio inequality with respect to f holds.

In these works the existence of conjugation was shown by estimating cross-ratio distortions (i.e., proving cross-ratio inequality).