 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2015/3/d/solution

By invariance under projective transformations and part (b), it is enough to calculate on the normalized quadruple

(\infty, 0, 1, λ)

. Direct evaluation of the cross-ratio gives

(0 \infty : λ 1) (0 \infty : 1 λ) = λ, = λ^{- 1}, (1 λ : \infty 0) (\infty 1 : 0 λ) = λ, = 1 - λ .

For instance,

(0 \infty : 1 λ) = (0 - 1) / (0 - λ) = 1/ λ

, and

(\infty 1 : 0 λ) = (1 - λ) / (1 - 0)

. Thus

(P_{0} P_{1} : P_{2} P_{3}) = (P_{1} P_{0} : P_{3} P_{2}) = (P_{2} P_{3} : P_{0} P_{1}),

(P_{0} P_{1} : P_{2} P_{3}) = (P_{1} P_{0} : P_{2} P_{3})^{- 1},

(P_{0} P_{2} : P_{1} P_{3}) = 1 - (P_{0} P_{1} : P_{2} P_{3}) .

All denominators are legitimate because

λ \in / {0, 1}

. The first identity explains why some distinct permutations always give the same cross-ratio.

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