Querverhältnisse

Querverhältnisse, commonly translated as cross ratios, are an invariant quantity in projective geometry that describe the relationship between four collinear points or four concurrent lines. Given four collinear points A, B, C, D, the cross ratio [A,B;C,D] is defined as (AC/BC)/(AD/BD), where the distances are measured along the line. The value is invariant under projective transformations, making it a fundamental tool for studying properties conserved by perspective projections.

In the dual setting, for four concurrent lines l₁, l₂, l₃, l₄ intersecting at a point, the cross ratio is defined via the sines of the angles between adjacent lines: [l₁,l₂;l₃,l₄] = (sin∠l₁l₃ / sin∠l₂l₃) / (sin∠l₁l₄ / sin∠l₂l₄). This dual definition yields the same invariant value as for points, and again remains unchanged under projective transformations.

Querverhältnisse play a key role in classical projective theorems such as Pascal’s and Pappus’s theorems, where the equality of certain cross ratios leads to conclusions about collinearity or concurrency. In analytic geometry, the cross ratio can be expressed using homogeneous coordinates, and it appears in formulae for the harmonic conjugate of a point with respect to a pair of others.

Applications extend beyond pure mathematics. In computer vision, cross ratios are used for 3D reconstruction from 2D images because they offer a way to relate points in different views without requiring explicit knowledge of the camera’s internal parameters. Similarly, in architectural design and the analysis of perspective drawings, the invariant property of cross ratios ensures consistency of proportions across different viewpoints.