Principal Curvatures & Directions

The extremal curvatures and their directions

Principal Curvatures

Turn around at a point on a surface and the bend changes with direction. The steepest and gentlest bends, the principal curvatures, always sit at right angles.

Principal Direction Field

Each cross marks the two principal directions, colored red or blue by the sign of the curvature. Switch surfaces and look for yellow circles: umbilical points, where every direction bends equally.

k₁ direction (positive)
k₁ direction (negative)
Umbilical point (k₁ = k₂)
Go deeper:Principal Curvatures

At each point on a surface, the principal curvatures k1 and k2 are the maximum and minimum values of normal curvature over all tangent directions.

The corresponding directions are called principal directions, and they are always perpendicular (except at umbilical points where k1 = k2).

Go deeper:Eigenvalues of the Shape Operator

The principal curvatures are the eigenvalues of the shape operator S, and the principal directions are the corresponding eigenvectors.

S(e1) = k1 e1
e1 is the direction of maximum curvature
S(e2) = k2 e2
e2 is the direction of minimum curvature

Curvature Lines

These curves always follow a principal direction. Toggle the k₁ and k₂ lines to see them cross at right angles, and notice how they swirl around umbilical points.

k₁ curvature lines
k₂ curvature lines

Curvature lines are curves on the surface that always point in principal directions. They form an orthogonal network (except at umbilical points) and reveal the intrinsic geometry of the surface.

Go deeper:Curvature Lines

Curvature lines (or lines of curvature) are curves on the surface that always point in a principal direction. They form an orthogonal network that reveals the intrinsic geometry of the surface.

Euler's Formula

Sweep the direction slider θ and watch the normal curvature trace a smooth wave between k2 and k1.

k₁ (max)
-0.6497
k₂ (min)
-2.4998
κ(θ)
-0.6497
κ(θ) = k₁·cos²θ + k₂·sin²θ

Euler's formula: normal curvature in any direction is a weighted average of the principal curvatures

Go deeper:Euler's Formula

Euler's formula tells us the normal curvature in any direction: it's a weighted average of the principal curvatures, with weights determined by the angle to the principal directions.

Go deeper:Classification of Surface Points
Elliptic
k1k2 > 0

Both curvatures same sign (bowl-like). K > 0.

Hyperbolic
k1k2 < 0

Opposite signs (saddle-like). K < 0.

Parabolic
k1k2 = 0

One curvature is zero (cylinder-like). K = 0.

Umbilical
k1 = k2

Equal curvatures (sphere-like). All directions are principal.

Go deeper:Relating to Gaussian and Mean Curvature

The principal curvatures are the fundamental building blocks. All other curvature quantities can be expressed in terms of k1 and k2:

K = k1 · k2
Gaussian curvature (intrinsic)
H = (k1 + k2) / 2
Mean curvature (extrinsic)

Key Takeaways

  • k1 and k2 are the largest and smallest normal curvatures at a point.
  • Principal directions are perpendicular eigenvectors of the shape operator.
  • Euler's formula: κ(θ) = k1cos²θ + k2sin²θ.
  • At umbilical points k1 = k2, so every direction is principal.

Spot a mistake or have an idea?

Search