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Stereographic projection.
Note that any point can serve as the source; there is no distinguished point.
This is an example of a conformal map, a map which preserves angles. But it does not preserve distances (so it distorts the metric). Circles map to circles. In particular perpendicular circles map to perpendicular circles. Contiguity relations are preserved.
How do we measure the angle between two arbitrary curves that intersect? Approximate each curve at the point of intersection P by lines (of best fit) through P. Measure the angle between the lines.
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