Recipe

Find a Mobius transformation sending three points to three points

Use when: you're given three distinct points z_1,z_2,z_3 and three distinct images w_1,w_2,w_3, and asked for the one Mobius transformation T with T(z_k)=w_k

Goal: the explicit transformation T(z)=\dfrac{az+b}{cz+d} satisfying all three point conditions at once

Taught in Möbius transformations

1. 1

   Write (z,z_1,z_2,z_3)=(w,w_1,w_2,w_3). Use the cross-ratio (z,z_1,z_2,z_3)=\dfrac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} on each side.

   Example

   Send z_1=0,z_2=1,z_3=i to w_1=1,w_2=-1,w_3=i. Then \dfrac{z(1-i)}{z-i}=\dfrac{(w-1)(-1-i)}{(w-i)(-1-1)}.

   Why: every Mobius transformation leaves the cross-ratio unchanged, so equating the two says "T sends these three points to those three."

2. 2

   Plug in the three z_k and the three w_k. Keep z and w as the only unknowns.

   ⚡ Shortcut

   If one of the six points is \infty, cross out the two factors that contain it. Do not substitute \infty directly. The ratio of those two factors tends to 1.

   Why: putting \infty into a fraction is not ordinary arithmetic, but the cross-ratio's limit makes dropping those factors safe.

3. 3

   Clear both denominators. Rearrange the equation into the form w=\dfrac{az+b}{cz+d}.

   Example

   Cross-multiplying and solving the equation from Step 1 gives T(z)=\dfrac{-z-i}{(1+2i)z-i}.

   Why: this one algebra step turns the cross-ratio identity into the explicit formula for a Mobius transformation.

4. 4

   Check T(z_1)=w_1, T(z_2)=w_2, and T(z_3)=w_3 against the formula from Step 3.

   Example

   T(0)=\dfrac{-i}{-i}=1, T(1)=\dfrac{-1-i}{1+2i-i}=\dfrac{-1-i}{1+i}=-1, and T(i)=\dfrac{-2i}{(1+2i)i-i}=\dfrac{-2i}{-2}=i. All three match.

   Why: three conditions pin T down completely, so checking only one or two could miss an arithmetic slip.