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Stereographic Projection — Definition, Formula & Examples

Stereographic projection is a mapping that projects points from a sphere onto a plane by drawing a line from a fixed pole (typically the north pole) through each point on the sphere and finding where that line intersects the plane.

Given a unit sphere S2S^2 centered at the origin and a projection point N=(0,0,1)N = (0, 0, 1) (the north pole), stereographic projection is the map σ:S2∖{N}→R2\sigma: S^2 \setminus \{N\} \to \mathbb{R}^2 that sends each point (x,y,z)∈S2(x, y, z) \in S^2 to the point in the equatorial plane where the line through NN and (x,y,z)(x, y, z) intersects that plane.

Key Formula

(u,v)=(x1−z,  y1−z)(u, v) = \left(\frac{x}{1 - z},\; \frac{y}{1 - z}\right)
Where:
  • (x,y,z)(x, y, z) = Coordinates of a point on the unit sphere $x^2 + y^2 + z^2 = 1$, with $z \neq 1$
  • (u,v)(u, v) = Coordinates of the projected point in the plane $z = 0$

How It Works

Imagine a sphere sitting on a flat plane. Pick the topmost point (north pole) as your projection center. For any other point on the sphere, draw a straight line from the north pole through that point and continue until it hits the plane. The spot where it hits is the projected image. Points near the north pole get flung far from the origin, while points near the south pole land close to the origin. Every point on the sphere except the north pole maps to exactly one point on the plane, and the map is conformal — it preserves angles between curves.

Worked Example

Problem: Find the stereographic projection of the point (35,45,0)(\frac{3}{5}, \frac{4}{5}, 0) on the unit sphere, projecting from the north pole onto the z=0z = 0 plane.
Verify the point is on the unit sphere: Check that x2+y2+z2=1x^2 + y^2 + z^2 = 1.
(35)2+(45)2+02=925+1625=1  ✓\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 + 0^2 = \frac{9}{25} + \frac{16}{25} = 1 \;\checkmark
Apply the projection formula: Use (u,v)=(x1−z,  y1−z)(u,v) = \left(\frac{x}{1-z},\; \frac{y}{1-z}\right) with z=0z = 0.
u=3/51−0=35,v=4/51−0=45u = \frac{3/5}{1 - 0} = \frac{3}{5}, \quad v = \frac{4/5}{1 - 0} = \frac{4}{5}
Answer: The projected point is (35, 45)(\frac{3}{5},\, \frac{4}{5}). Because this point lies on the equator (z=0z = 0), it maps to itself in the plane — a characteristic feature of stereographic projection from the north pole onto the equatorial plane.

Why It Matters

Stereographic projection is central to complex analysis, where the Riemann sphere uses it to extend the complex plane by a point at infinity. Cartographers use it to create map projections that preserve angles (conformal maps), which is critical for navigation charts. In topology, it provides the standard example of a homeomorphism between a punctured sphere and Euclidean space.

Common Mistakes

Mistake: Using 1+z1 + z instead of 1−z1 - z in the denominator.
Correction: The denominator 1−z1 - z comes from projecting from the north pole (0,0,1)(0,0,1). If you project from the south pole (0,0,−1)(0,0,-1) instead, then you use 1+z1 + z. Always match the denominator to your choice of projection point.