The proper distance between two points with coordinate separations dx in the x direction and dy in the y direction is $$ ds = \frac{1}{y} \sqrt{dx^2 + dy^2} $$

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1. Robert McNees (mcnees@mastodon.social)'s status on Thursday, 01-Dec-2022 15:53:20 UTC Robert McNees

   in reply to

   We can express the actual / proper / physical distance between two nearby points in terms of their separation on the sort of grid we would usually use to describe locations and distances on a plane.

   In a little local patch where the space looks approximately flat, the coordinates of two nearby points differ by dx along the x-axis and dy along the y-axis. The proper distance between them, which we call ds, is: