Geometric Invariance Math Example 4

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Example 4

hard
The cross-ratio of four collinear points is (AC⋅BD)/(BC⋅AD)(AC \cdot BD)/(BC \cdot AD), an invariant under projective transformations. If A,B,C,DA, B, C, D are at positions 0,1,3,60, 1, 3, 6 on a number line, compute the cross-ratio.

Solution

  1. 1
    Step 1: AC=∣3−0∣=3AC = |3-0| = 3, BD=∣6−1∣=5BD = |6-1| = 5, BC=∣3−1∣=2BC = |3-1| = 2, AD=∣6−0∣=6AD = |6-0| = 6.
  2. 2
    Step 2: Cross-ratio =3×52×6=1512=54= \dfrac{3 \times 5}{2 \times 6} = \dfrac{15}{12} = \dfrac{5}{4}.

Answer

Cross-ratio =54= \dfrac{5}{4}.
The cross-ratio is a classical projective invariant that remains unchanged under any projective transformation, including perspectivities. It is fundamental in projective geometry and has applications in computer vision and the study of conics.

About Geometric Invariance

A property or measurement of a geometric figure that remains unchanged when a particular transformation is applied.

Learn more about Geometric Invariance →

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