Sphere (3D)
Click to project point
→
Plane (2D Projection)
Click to find pre-image
Projection Settings
1.0
1.0
Display Options
Grid Density
8
Rotation
0.005
Animation
About Stereographic Projection
What is it?
Stereographic projection maps points from a sphere to a plane by projecting from a fixed point (the "north pole"). It's a fundamental concept in complex analysis and geometry.
Key Properties
- Conformal: Preserves angles between curves
- Circle-preserving: Circles on sphere map to circles on plane (or lines)
- Bijective: One-to-one correspondence (except projection point)
- Smooth: Infinitely differentiable
Projection Formula
For a point (x, y, z) on the unit sphere, the stereographic projection to the plane is:
X = x / (1 - z)
Y = y / (1 - z)
Y = y / (1 - z)
The inverse projection (plane to sphere) is:
x = 2X / (1 + X² + Y²)
y = 2Y / (1 + X² + Y²)
z = (X² + Y² - 1) / (1 + X² + Y²)
y = 2Y / (1 + X² + Y²)
z = (X² + Y² - 1) / (1 + X² + Y²)
Current Point
Click on either canvas to see projection