Conformality & the Jacobian

Lesson 2 · why holomorphic functions preserve angles — and where they don't

f(z) =

z-plane (input)

w-plane w = f(z)

base point z₀

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f(z₀)

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input angle ∠(v₁,v₂)

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output angle ∠(f′v₁,f′v₂)

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rotation arg(f′(z₀))

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scale |f′(z₀)|

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Drag the white dot z₀ and the two vector tips (coral / cyan) on the left. The right panel shows each vector multiplied by the derivative f′(z₀) as a complex number — a rotation by arg(f′) and a scaling by |f′|. Because the same rotation hits both vectors, the angle between them is unchanged: that's conformality. For z̄ the local map is a reflection (Jacobian [[1,0],[0,−1]]), which flips orientation and reverses the angle. Where f′(z₀)=0 (origin for z², z³) the map collapses and angles are destroyed.

Conformality & the Jacobian

Interactive: drag a base point and two tangent vectors in the z-plane; see their images under f and whether the angle between them is preserved.

z-plane (input)

w-plane w = f(z)