To find the bilinear transformation $w = f(z)$ that maps points $z = 0, i, 1$ in the $z$-plane to $w = -1, 0, 1$ in the $w$-plane, we use the general form of a bilinear (or Möbius) transformation:

$w = \frac{az + b}{cz + d},$ where $a, b, c, d$ are complex constants satisfying $ad - bc \neq 0$.

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Step-by-Step Solution:

1. Mapping conditions:

   • $f(0) = -1$: Substituting $z = 0$ and $w = -1$: $-1 = \frac{b}{d}.$ Thus, $b = -d$.

   • $f(i) = 0$: Substituting $z = i$ and $w = 0$: $0 = \frac{ai + b}{ci + d}.$ Since $b = -d$, this becomes: $0 = \frac{ai - d}{ci + d}.$ Therefore, $ai - d = 0$, which gives: $d = ai.$

   • $f(1) = 1$: Substituting $z = 1$ and $w = 1$: $1 = \frac{a + b}{c + d}.$ Substituting $b = -d$ and $d = ai$, this becomes: $1 = \frac{a - ai}{c + ai}.$

2. Simplifying $c$: Rewrite $a - ai = a(1 - i)$ and $c + ai = c + ai$. Thus: $1 = \frac{a(1 - i)}{c + ai}.$ Cross-multiply to get: $c + ai = a(1 - i).$ Separating real and imaginary parts: $c = a, \quad ai = -ai,$ leading to simplify the constants $a, b, c, d$ further.

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Step 2: Refining Parameters

At this stage, we know:

• $b = -d$,
• $d = ai$,
• $c = a$.

We substitute these relationships into the bilinear transformation and adjust it to ensure it maps the given points exactly.

Full Solution

The bilinear transformation is now:

$w = \frac{az - d}{cz + d}.$

Substituting $b = -d$, $d = ai$, and $c = a$:

$w = \frac{a z - ai}{a z + ai}.$

Factor $a$ from numerator and denominator:

$w = \frac{z - i}{z + i}.$

This is the desired bilinear transformation.

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Verification

To confirm the transformation $w = \frac{z - i}{z + i}$ satisfies the given mappings:

1. $z = 0$: $w = \frac{0 - i}{0 + i} = \frac{-i}{i} = -1.$

2. $z = i$: $w = \frac{i - i}{i + i} = \frac{0}{2i} = 0.$

3. $z = 1$: $w = \frac{1 - i}{1 + i}.$ Multiply numerator and denominator by the conjugate of the denominator: $w = \frac{(1 - i)(1 - i)}{(1 + i)(1 - i)} = \frac{1 - 2i + i^2}{1 - i^2} = \frac{1 - 2i - 1}{1 + 1} = \frac{-2i}{2} = 1.$

Thus, the mapping is correct.

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Final Transformation

The bilinear transformation is:

$w = \frac{z - i}{z + i}.$

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Questions:

1. How does bilinear transformation preserve angles?
2. Can this transformation map other sets of points with similar methods?
3. What happens to the unit circle under this transformation?
4. How is the cross-ratio used in deriving bilinear transformations?
5. Can this be extended to map 3D points (using complex projective geometry)?

Tip: When solving bilinear transformations, always verify your solution by checking all mapping conditions explicitly.