Stereographic Projection IV Opposite of a Pole

[@insunghan7120]

Dear Professor, is it normally required to represent P' in case that P' is outside the primitive circle? Just out of curiosity...

[@rajeshprasadlectures]

Well, if we wish to represent P', the projection of a pole 180 degrees away from P, using the same projection point, i.e., the South pole, then we will have to go out of the primitive as described here. If you do not wish to go out of primitive, then you can change the projection point to the North Pole. Then the opposite pole will come inside the primitive. But you have to carefully label this to indicate that you are using a different projection point. In the example of this video @3:00 instead of joining M to C and finding the intersection P' with the diameter AB, we will join M to D and mark the intersection point P" (not shown) with AB. You can see that P" will lie inside the primitive. We can mark this point with an open circle to indicate that the North, instead of the South pole has been used as the projection point. In fact, in this case, the construction is much simplified. You can show geometrically, that OP=OP". Thus you really don't have to construct the perpendicular diameter CD at all. Just mark the point P" on AB such that OP"=OP. I now feel that I should have considered this case as well. Thanks for asking the interesting question. However, In the next video of the series, kzbin.info/www/bejne/j57MhYWYjt2qbMU, you will note that we do need the opposite pole outside the primitive to be able to draw the great circle through two given poles.

[@insunghan7120]

@@rajeshprasadlectures Thank you very much for your explanation! Very helpful lecture to build a stereographic projection background.