I'm looking for a geometric construction which would allow me to draw an ellipse, which is supposed to be an orthographic projection of a great circle of a sphere, given two points on it.
The following picture shows what I need:
I have some circle drawn, which is a projection of that sphere.
Then, I have two points drawn (red), which are projections of two points on that sphere.
Now I need to find a way to draw an ellipse (blue), which is a projection of the great circle passing through these two points. That is, I need to find the orientation of its axes, and the ratio of major to minor axis, geometrically.
What I know is that the major axis is the same as the diameter of the circle & sphere. Those two points are coplanar: their radii (red lines) lie on the same plane, and in the picture these are projections of the radius of the sphere & circle.
I can find constructively the minor axis of the ellipse representing a great circle with arbitrary tilt to the view plane, knowing the point which is the end point of its axis on the sphere (its pole). What I'm missing is how to find that pole's position on my drawing from these two red points I know, through some geometric construction using these two points.
Edit 1: Here's my construction I use to get from the pole's position to the ellipse:
First, I draw a perpendicular to my axis (the axis is green). Next, I draw a circle (blue) to find the same distance as my axis on the perpendicular. Then I project this point perpendicularly to meet the circle, and again perpendicularly to meet my axis. This point designates the minor radius of my ellipse, so I can draw it. The second picture (on the right) explains why does this work: I can view my ellipse from the side, as some diagonal line (blue-green) laying at some angle to the level. I can find the sine of this angle by just projecting the point, where the blue-green line meets the circle, straight down (orange). The cosine is then the red line, and coversine is blue. Since the axis (green) is always perpendicular to the blue-green plane of the great circle, It is at the same angle to the vertical direction, and has the same sine, cosine and coversine, but rotated 90 degrees. So when I know the distance of the pole from the center on my drawing, I can reverse this process to find these sines and cosines and then the minor radius of my ellipse I'm looking for.
So my problem reduces now to finding the location of this pole point from my known red points.
Any ideas how to find it?
To be clear, I know how to do it analytically, through vector cross products, matrices etc., and how to calculate it, but that's not what I'm looking for. I need to do it just by geometrical constructions. And I suppose it is possible, since all those cartographers in the past somehow managed to draw all those maps, and astronomers all those sky domes, right? ;-J
Edit 2: OK, I have some idea. Not much pretty, but it works :-> I can use my two red radii as axes for two other great circles -- this is what I can do already: draw an ellipse being a projection of a great circle for a given axis of that circle, so why not to exploit it here too? ;-) Then, I will get two such great circles crossing each other at two points, since they lie on two planes intersecting each other in a line. And this line has to be perpendicular to both red radii! :-> That's the green line I'm looking for. Having the green radius, I would be able then to draw the blue ellipse I'm looking for, on which both these points lie.
Now I need to find out some simplification of this procedure, which wouldn't require me drawing two additional ellipses, since it's quite expensive construction. I only need their two points of intersection.