Math 2.0

Differential geometry Definition

Principal curvature

Builds on Shape operatorRegular surfaceSecond fundamental formEigenvalueSelf-adjoint operator

Definition

Let S⊆ℝ3S \subseteq \R^3 be a regular surface with a smooth unit normal field NN near a point p∈Sp \in S. Let SpS_p be the shape operator at pp. Since SpS_p is self-adjoint, it has two real eigenvalues k1≥k2k_1 \ge k_2, counted with multiplicity. These are the principal curvatures of SS at pp. A unit eigenvector of SpS_p is a principal direction at pp.

Remarks

The principal directions can be chosen orthonormal, by the spectral theorem. Equivalently, k1k_1 and k2k_2 are the maximum and the minimum of the second fundamental form I ⁣Ip(w)I\!I_p(w) over unit vectors w∈TpSw \in T_p S. Replacing NN by −N-N changes k1,k2k_1, k_2 into −k2,−k1-k_2, -k_1. Example: for the saddle z=xyz = xy at the origin, with N=(0,0,1)N = (0, 0, 1), one has I ⁣I0(a,b)=2abI\!I_0(a, b) = 2ab on T0S=ℝ2×{0}T_0 S = \R^2 \times \set{0}. So k1=1k_1 = 1 and k2=−1k_2 = -1, with principal directions (1,1,0)/2(1, 1, 0)/\sqrt{2} and (1,−1,0)/2(1, -1, 0)/\sqrt{2}. Non-example: on the same saddle, the normal curvature in the direction (1,0,0)(1, 0, 0) is I ⁣I0(1,0)=0I\!I_0(1, 0) = 0, which is neither k1k_1 nor k2k_2, so (1,0,0)(1, 0, 0) is not a principal direction.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Regular surface, concept, no evidence levelRegular surfaceconceptSecond fundamental form, concept, no evidence levelSecond fundamental fo…conceptShape operator, concept, no evidence levelShape operatorconceptVector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionInner product space, definition, no evidence levelInner product spacedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionPrincipal curvature, definition, no evidence levelPrincipal curvaturedefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Regular surface concept
  2. Second fundamental form concept
  3. Shape operator concept
  4. Vector space definition
  5. Inner product definition
  6. Linear map definition
  7. Inner product space definition
  8. Endomorphism definition
  9. Adjoint of a linear map definition
  10. Eigenvalue definition
  11. Self-adjoint operator definition
  12. Principal curvature definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Principal_curvature
About this page
Tag8QBWWC354F
URIm2://8QBWWC354F
Kinddefinition
WritingW0 stub
MSC53A05
DefinesPrincipal curvature
UsesShape operator, Regular surface, Second fundamental form, Eigenvalue, Self-adjoint operator
Changed2 November 2026

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