Questions: Möbius Transformations

Question 1

You need a conformal map sending the upper half-plane to the unit disk, mapping i ↦ 0, the point 0 ↦ -1, and ∞ ↦ 1. How many Möbius transformations satisfy all three conditions?

• A. None — the upper half-plane and unit disk are not conformally equivalent
• B. Infinitely many — there are many ways to construct a Möbius transformation between these domains
• C. Exactly one — the three-point determination property uniquely fixes the map
• D. Exactly two — the map and its inverse both satisfy the conditions

Answer: (C)

The three-point rule states that given any two triples of distinct points in ℂ ∪ {∞}, there is exactly one Möbius transformation sending the first triple to the second. Specifying where three boundary points go pins down the map completely — no free parameters remain. This is precisely what makes Möbius transformations so powerful for conformal mapping: you can construct the exact map you need by selecting three convenient boundary correspondences, then verifying the result sends the entire domain correctly.

Question 2

Under a Möbius transformation, a circle C in the complex plane maps to a line rather than to another circle. Which of the following must be true?

• A. The map violates conformality at the image, since circles and lines have different curvature
• B. The circle C passes through the pole of the transformation — the point mapped to ∞
• C. The transformation is degenerate, meaning ad - bc = 0
• D. C must be the unit circle |z| = 1, since only it has the symmetry required to map to a line

Answer: (B)

In the extended complex plane, 'lines' are circles through ∞. A circle maps to a line (instead of another circle) precisely when it passes through the pole of the transformation — the point z = -d/c that gets sent to ∞. When the pole lies on C, its image must pass through ∞, which in the standard plane is a line. This is a beautiful illustration of why Möbius transformations are best understood on the Riemann sphere: the distinction between 'circle' and 'line' disappears, and the theorem becomes simply 'circles map to circles.'

Question 3

The composition of two Möbius transformations is always another Möbius transformation — they form a group under composition.

• T. True
• F. False

Answer: True

This group property is what makes Möbius transformations so useful in conformal mapping applications. You can chain them: map a disk to a centered disk, then map the centered disk to the upper half-plane, and the composition is automatically another Möbius transformation — exact and conformal, with no approximation. The group structure also means there is always an inverse (another Möbius transformation), and composition is associative. The group of all Möbius transformations is isomorphic to PSL(2,ℂ), the projective special linear group.

Question 4

The condition ad - bc ≠ 0 in a Möbius transformation f(z) = (az+b)/(cz+d) ensures that f is defined for most complex numbers z, including z = -d/c.

• T. True
• F. False

Answer: False

The condition ad - bc ≠ 0 does NOT make f defined at z = -d/c — it maps that point to ∞. The pole z = -d/c is exactly the point where the denominator vanishes, so f is undefined there in the ordinary complex plane; on the Riemann sphere, this point is mapped to ∞. The condition ad - bc ≠ 0 ensures the transformation is non-degenerate (not collapsing to a constant function) — if ad = bc, the 'map' would send every point to a/c. The distinction is important: the condition prevents degeneracy, not the existence of a pole.

Question 5

State the three-point rule for Möbius transformations and explain why it is useful when constructing conformal maps between specific domains.

Model answer: The three-point rule: given any two triples of distinct points (z₁, z₂, z₃) and (w₁, w₂, w₃) in ℂ ∪ {∞}, there exists exactly one Möbius transformation sending zᵢ ↦ wᵢ. This is useful because it reduces the problem of finding a conformal map between domains to the geometric problem of matching boundary points. To map a disk to the upper half-plane, choose three boundary points of the disk and specify where they go on the real axis; the unique Möbius transformation doing this is the desired map. No solving systems of equations is needed beyond the three-point matching.

This rule underpins virtually all explicit conformal mapping constructions involving Möbius transformations. In practice, you identify three convenient points (often using 0, 1, ∞, or boundary intersection points), specify their images, and read off the map. The uniqueness is as important as the existence: it tells you there is no ambiguity once three correspondences are fixed, and it means you can verify a proposed map by checking just three points.