Stereographic projection of circles on Riemann sphere
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This example demonstrates the stereographic projection of circles on the Riemann sphere. It involves core mathematical concepts such as spherical geometry, stereographic projection, trigonometry, and coordinate geometry. The construction uses GeoGebra commands ImplicitCurve and objects a, b, c. It is suitable for high school and above students to understand the geometric properties and calculations of spherical projections.
GeoGebra Commands & Parameters
ImplicitCurve(x^(2) + (y - cos(a) / (b / 10 - sin(a)))^(2) - (1 - b^(2) / 100) / (b / 10 - sin(a))^(2)) → c
AI Drawing Prompt Reference
Draw a Riemann sphere with a circle on its surface, and show its stereographic projection onto the equatorial plane. The sphere is centered at the origin with radius 1, and the projection point is the north pole. Choose a circle on the sphere determined by parameters a and b, where a controls the tilt of the circle's plane and b controls its size. Project the circle from the north pole onto the equatorial plane, resulting in a planar circle. This projected circle has its center on the y-axis, with position and radius computed from a and b. The figure should include the sphere, the circle on the sphere, and the projected circle in the plane, with relevant angles and distances labeled.
The prompt above was compiled from the original drawing commands and is for reference only.