Geometry - Stereographic projection

[Pearce_09]

I know if a cirlce (on S^2) does not contain N (0,0,1) then it is mapped onto the plane H as a circle. Now say the circles on S^2 are lines of latitude. When mapped by the stereographic projection they are cirlces in R^3 on the plane H. Now the only thing I am not sure on is,

my claim:
When lines of latitude are mapped by the stereographic projection the radius of the circle on H, is the same radies as the circle that was projected from S^2.

I know for one thing that if the equator is mapped onto H it is the same on H as on S^2.

(great cirlces containing N are mapped to lines on H)

 

[hypermorphism]

Unless I misunderstood your claim, it cannot be true, since the radii of the projected circles on H increase without bound as you get latitudes closer to the top of the sphere.

 

[Doodle Bob]

Unless I misunderstood your claim, it cannot be true, since the radii of the projected circles on H increase without bound as you get latitudes closer to the top of the sphere.

In addition, given that N, a circle of latitude on the sphere and its stereographic projection, form a cone with N as its apex that is in some sense perpendicular to the planes defined by the two circles, the radii are never equal to each other.