There are projections where infinitesimal circles stay circles, e.g. our dear Mercator projection, but that doesn’t hold for finite sized circles, i.e. circles would still be distorted in north-south direction. Tissot indicatrix
That’s a general metric holding for lots of projections. I think the specific projection that works for finite sized circles is stereographic projection.
On a stereographic map you should be able to draw a circle that stays a perfect circle (“small circle”) on a globe.
In addition, in its spherical form, the stereographic projection is the only map projection that renders all small circles as circles.
By small circles they mean circles on a sphere that are not an equator (great circle), not infinitessimally small circles. So basically they just mean circles.
By small circles they mean circles on a sphere that are not an equator (great circle), not infinitessimally small circles. So basically they just mean circles.
This only applies to the circles perpendicular to the axis of projection, i.e. usually the circles of latitude (parallels), though. The Tissot indicatrices still show increasing sizes of the circles from the center of the map to its outside. Thus, any circle that isn’t coaxial with the parallels is distorted on the map.
Stereographic projection preserves circles and angles. That is, the image of a circle on the sphere is a circle in the plane […]. We will outline two proofs of the fact that stereographic projection preserves circles, one algebraic and one geometric.
I also ran across a math stackexchange thread talks about a proof that “Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane”.
Further I notices the mathematical wikipedia page for the projection states it without the weird map terminology simply as “It maps circles on the sphere to circles or lines on the plane”.
I really don’t see any qualifiers anywhere, to the best of my understanding this holds in general for all circles. With the one exception that circles through the point opposite the center turn into lines (infinitely large circles for simplicity).
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.
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.
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.
The circles in the stereographic projection are scaled, they are essentially pulled outwards, when further away from the center. This matches an increasing jacobian.
Exactly. The Jacobi determinant increases in radial direction (longitudinal on the globe).
But they stay circular, the stretching happens in the right way for that to hold true.
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.
Wichtig: Die Kreistreue gilt generell nur für die Kreislinie. Der Mittelpunkt des Objektkreises wird z. B. nicht als Mittelpunkt des Bildkreises abgebildet. Hiervon sind nur die Fälle, bei denen der Kreiskegel gerade ist, ausgenommen.
Important: The circle fidelity generally only applies to the circle line. The centre of the object circle, for example, is not depicted as the centre of the image circle. The only exceptions to this are cases where the cone of the circle is straight.
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.
There are projections where infinitesimal circles stay circles, e.g. our dear Mercator projection, but that doesn’t hold for finite sized circles, i.e. circles would still be distorted in north-south direction.
Tissot indicatrix
That’s a general metric holding for lots of projections. I think the specific projection that works for finite sized circles is stereographic projection.
On a stereographic map you should be able to draw a circle that stays a perfect circle (“small circle”) on a globe.
By small circles they mean circles on a sphere that are not an equator (great circle), not infinitessimally small circles. So basically they just mean circles.
This only applies to the circles perpendicular to the axis of projection, i.e. usually the circles of latitude (parallels), though. The Tissot indicatrices still show increasing sizes of the circles from the center of the map to its outside. Thus, any circle that isn’t coaxial with the parallels is distorted on the map.I couldn’t find the video I was thinking about, which is a bummer.
I did find one written argument (uiuc - Stereographic Projection)
And one video proof youtube - Stereographic Projection Circle to Circle Proof.
I also ran across a math stackexchange thread talks about a proof that “Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane”.
Further I notices the mathematical wikipedia page for the projection states it without the weird map terminology simply as “It maps circles on the sphere to circles or lines on the plane”.
I really don’t see any qualifiers anywhere, to the best of my understanding this holds in general for all circles. With the one exception that circles through the point opposite the center turn into lines (infinitely large circles for simplicity).
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.