I have a globe image that is centered at lat,lon. At this lat,lon, by definition x=0, y=0, z=1.
A line of latitude drawn on this map can either be completely visible (in which case it would be drawn as an oval like the 60°S line in the image below), or it can be completely hidden like the 60°N latitude in the picture (all points on the line are z < 0).
The third possibility is that a line of constant latitude has a start point at some longitude and an end point at some other longitude where z=0. How can I find these two longitudes?
For example if the center lat,lon is -50, -130 (between New Zealand and S. America), the line of latitude near 40°S starts near a longitude near Western Australia and ends in The Atlantic between S. America and Africa. But at what longitude does this line get to z=0 on both ends?
So in the picture below, the two red dots are on either side of the 40°S line. What are the two points of longitude? They will be at z=0, lat=-40 (in degrees) and long=???
In the picture below, gridlines are shown every 10 degrees with the center of the projection at Lat/Lon (-30,0). In this view, all latitude lines above 60N are invisible because they fall on the backside of the earth where z<0. All latitude lines below 60S are completely visible because the whole line in in an area where z>0.
The area between 60S and 60N has some of each latitude line visible. They end at the following longitudes (calculated by hand with trial and error and interpolation using excel):
60N 0.000 50N 46.523 40N 61.023 30N 70.529 20N 77.869 10N 84.157 Equ 90.000 10S 95.843 20S 102.130 30S 109.471 40S 118.977 50S 133.477 60S 180.000
I need a formula to calculate these values.
Given: A latitude for the center of the projection (-30 in this case).
Given: A line of latitude (20 for example)
Find: The number of degrees E/W that the line remains visible (z>=0): 77.869 in this example.