Post by Admin on Nov 21, 2016 7:24:05 GMT
Gary
Here follows my attempt at this suggested exercise:
H-lines
Since stereographic projection preserves circles as well as angles, then a circle whose arc in the Poicaré disc is an h-line orthogonal to the unit circle will be a circle on the Riemann sphere orthogonal to the unit circle at the same two points, and will therefore be a semicircular vertical section of the hemisphere.
Equidistant curves
Equidistant curves of an h-line in the disc are arcs of circles which pass through the end points of the h-line and make the same angle at each end with the unit circle. So if the vertical plane in the last paragraph is rotated about the line joining the end points of the h-line, then the arcs of the circles of intersection of the plane with the hemisphere will be the images of equidistant curves on the disc.
Horocycles
Horocycles in the disc are circles which touch the unit disc. So they are like equidistant curves for a zero length h-line, and their images on the hemisphere will be similar to the equidistant curves except that they will all touch the unit circle rather than intersect it.
Vasco
Here follows my attempt at this suggested exercise:
H-lines
Since stereographic projection preserves circles as well as angles, then a circle whose arc in the Poicaré disc is an h-line orthogonal to the unit circle will be a circle on the Riemann sphere orthogonal to the unit circle at the same two points, and will therefore be a semicircular vertical section of the hemisphere.
Equidistant curves
Equidistant curves of an h-line in the disc are arcs of circles which pass through the end points of the h-line and make the same angle at each end with the unit circle. So if the vertical plane in the last paragraph is rotated about the line joining the end points of the h-line, then the arcs of the circles of intersection of the plane with the hemisphere will be the images of equidistant curves on the disc.
Horocycles
Horocycles in the disc are circles which touch the unit disc. So they are like equidistant curves for a zero length h-line, and their images on the hemisphere will be similar to the equidistant curves except that they will all touch the unit circle rather than intersect it.
Vasco
