Reflections

Let A,B ∈X_N with A≤B and let f,f_r∈S(N). Then f_r is the reflection of f from A to B if:

1. i<A⇒f_r(i)=f(i) and
2. i>B⇒f_r(i)=f(i) and
3. A≤i≤B⇒f_r(i)=f(A+B-i)

For example, let f=(0.37, 0.26, 0.16, 0.10, 0.07, 0.05). Then the reflection of f from 3 to 5 is (0.37, 0.26, 0.07, 0.10, 0.16, 0.05)

• For each A, the reflection of f from A to A has no effect, and so is an identity mapping.
• Each reflection is its own inverse.
  • If r is any reflection and c is any contraction, then rcr (that is, r^-1cr) reflects the centre of c
• To interchange f(A) and f(B):-
  1. If B≤A+2, reflect f from A to B;
  2. If B>A+2 reflect f from A to B and then reflect the result from A+1 to B-1