Geometry of the Sky: Stereographic Projection Explained

If the South Pole is directly opposite the North Pole on the z-axis, and if the plane is at the equator, then a line from the North Pole through the South Pole passes through the exact center of the plane; therefore, the coordinates are (0, 0).

If the equator is on the same plane as the map, and if the sphere has a radius of 1, then the points on the equator are already on the plane; therefore, they form a circle with a radius of 1.

If the projection is centered on the pole, it provides a clear view of the surrounding area; if it is conformal, it preserves shapes, and its geometric property of mapping circles to circles makes it easy to construct.

If we use geometry to solve for the intersection of the ray with the equatorial plane, then the distance from the center follows the half-angle tangent of the co-latitude.

If the Riemann Sphere is a way to visualize the set of complex numbers plus infinity, and if infinity is represented by the North Pole, then the plane is being mapped to a sphere.

If the North Pole is the source of the rays, then a ray cannot be drawn from the North Pole through itself to the plane. If the ray is undefined at that point, then the projection point itself has no coordinate on the 2D plane.

If crystals have symmetrical faces in 3D that need 2D plots, or if maps and wide-angle lenses need to project 3D light onto 2D sensors or paper, then these fields utilize stereographic principles.

If the ray originates at the North Pole and hits a point very near it, the angle of the ray will be very shallow; if the ray is nearly horizontal, then it will travel a very long distance before hitting the equatorial plane.

If the standard projection goes from 3D to 2D, then the "inverse" must reverse the process; if it starts on the flat map, then it starts on the 2D plane.

If straight lines on the map represent circles on the sphere that pass through the projection point, and if there are no infinitely long straight lines on a finite sphere, then the statement is false.

If we want to represent the curved surface of a sphere on a flat sheet of paper, then we must use a projection method. If that method involves drawing lines from a specific point on the sphere through to a plane, then we are performing a stereographic projection.

If the projection rays must travel from the North Pole to the South Pole and hit a plane at the equator, they hit near the center. If they must hit points further up the sphere, the rays become nearly horizontal and hit the plane very far away, creating massive stretching.

If an astrolabe is a flat instrument used to track the 3D sky, then it needs a way to put 3D coordinates on a 2D surface. If stereographic projection maps the celestial sphere to a disk, then it is the perfect mathematical system for the astrolabe.

If we imagine a flat sheet cutting through the middle of the sphere horizontally, then that sheet sits on the plane of the equator.

If a projection is conformal (preserves angles), then it typically cannot be equal-area. If the regions near the South Pole are stretched more than regions near the equator in this projection, then the area is not preserved.

If a circle on the sphere contains the North Pole, and if we project every point on that circle onto a plane, then those points will extend to infinity in two directions; if this happens, the result on the flat plane is a straight line.

If the projection point itself cannot be mapped to a finite spot on the plane, then it is sent to infinity. If the mapping preserves angles (conformal), then local shapes are kept, but area is distorted because regions further from the center are stretched.

If a circle is drawn on the sphere's surface, and if it does not touch the focal point (North Pole), then the geometric transformation of that circle onto the plane results in another perfect circle.

If we need a starting point for our projection rays and we choose the highest point on the vertical axis of the sphere, then that point is traditionally referred to as the North Pole.

If a mapping is conformal, then the angles measured on the 3D surface remain the same when measured on the 2D map. If stereographic projection follows this mathematical property, then it is a conformal mapping.