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at coordinates (x, y) and the fixed pivot point (-1, 0); similarly, t is the opposite of the slope of the line through the points at coordinates (x, y) and (+1, 0). The inverse mapping from s to (x, y) is given by

It can be confirmed that x² + y² = 1 for all values of s and t. These two charts provide a second atlas for the circle, with the transition map

(that is, one has this relation between s and t for every point where s and t are both nonzero).

Each

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