Stereographic projection is the one (and only) thatballows that. You can draw any circle (or a straight line) on a stereographic map and it will remain a circle on the globe.
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.
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.
I wonder what size the circle would be if you took in to account the earth’s curvature.
Are there any map projections that allow for accurate projection of circles across arbitrary points?
Stereographic projection is the one (and only) thatballows that. You can draw any circle (or a straight line) on a stereographic map and it will remain a circle on the globe.
https://en.wikipedia.org/wiki/Stereographic_map_projection#Properties
All map projections are arbitrary. The only way to do this is on a globe.
Different projections preserve different properties. From memory there are ones that leave circles circular, so would allow this.
Edit: It’s stereographic projection that maps circles to circles.
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.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.
If you drew in on a globe, it would look deformed in this projection. I think the radius wouldn’t change, but it would look “wider” towards the north
That theorem only applies 2d from my understanding