Schwarz-Christoffel Preview

Complex Analysis · Axiom Academy

How do you wrap a line around a corner? A Möbius transformation can slide the upper half-plane onto a disk, but it can never produce a corner — its images are always smooth. So how could a single map fold a perfectly straight line into the sharp-cornered boundary of a triangle, a square, or any polygon? Watch one do it. A point runs along the real axis on the left. On the right, its image traces the boundary of a square: it walks a straight edge, and every time the runner crosses one of the marked points — the prevertices — the image snaps through a turn, planting a corner. Four crossings, four corners, one square. That turn at each prevertex is the whole secret — and the running total lands on exactly , one full lap around the polygon. The turns always add up to one full circle Slide to choose how many sides the polygon has. The prevertices spread out along the real axis, and at each one the boundary turns by the exterior angle . The interior corner is whatever is left over, — which the Schwarz–Christoffel formula writes as . No matter how many sides, the turns always sum to a full . Each corner uses up a slice of turning; the slices always close the loop. That fixed total is why the angles in the S-C formula must satisfy . Why the corners land where they should

This is the written version of the interactive lesson above. See the full Complex Analysis course.