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this post was submitted on 29 Sep 2025
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There is no qualifier on wikipedia and I do remember seeing some neat geometry tricks you can do with the property long ago.
The Tissot thing to me looks like a visualization for the jacobian, so the factor by which the area at that point is scaled, plus the gradient.
The circles in the stereographic projection are scaled, they are essentially pulled outwards, when further away from the center. This matches an increasing jacobian. But they stay circular, the stretching happens in the right way for that to hold true.
If you wait a bit I'll see if I can find some further things relying on this property, or at least stating it more unambiguously.
Essentially, the tissot indicatrices are a visualization of the eigenvalues and eigenvectors of the projection in any point. So, in 2d, the areas of these ellipses correspond to the Jacobi determinant, the product of the two eigenvalues of the Jacobian at that point.
Exactly. The Jacobi determinant increases in radial direction (longitudinal on the globe).
If you draw a circle on a globe, that is not coaxial to the parallels and apply the projection, the radius of said circle becomes elongated in outward direction in the same way the circles of the Tissot indicatrices increase in size.
Or in other words, any slice oncrement of the circle along a fixed degree of latitude changes in size depending on the value of the Jacobi determinant at that degree of latitude.
~~Thus, the circle on the globe becomes somehow like a rounded triangle on the map.~~
Edit: That shifts only the center of the mapped circle towards the outside of the original, but the circle remains a circle.
I sent you plenty of proofs that the circles are magnified but stay circular in the other message. Take the video and go to 10:10 for example. Sadly it's not animated, which the video I remember was. But it does show an arbitrary off-axis circle that still is mapped to a (much larger and further out) circle.
I get it now. The German Wikipedia article contains an explanation of my missconception:
Great that's cleared up. This is saying the center-points differ between the globe and the projection? That would match my expectations too.
Exactly, unless it's the cone of the circle is straight, i.e. the circle is around the (north) pole, the centers of the circles on the globe and on the map aren't identical.