Principal curvatures

The maximum and minimum normal curvatures at a point $p$ on a regular surface are called the principal curvatures at $p$ , usually denoted $k_1$ and $k_2$ , respectively. The corresponding directions, $e_1$ and $e_2$ , are called the principal directions at $p$ .

The principal curvatures and directions have an inuitive interpretation in terms of the second fundamental form (sec:SecondFundamentalForm) -- the principal directions are its eigenvectors^2.5. The second fundamental form (and thus the normal curvatures) takes its minimum and maximum values at the eigenvectors, and those values are the negative of the eigenvalues. (The negative sign comes from the definition of the second fundamental form: $II_p(v) = -\ensuremath{\langle dN_p(v), v \rangle}$ .)

The principal directions form a convienient vector basis for determining curvatures of any direction in the tangent plane. If $v$ is an arbitrary unit vector in the tangent plane, then clearly $v = e_1 \cos \theta + e_2 \sin \theta$ , where $\theta$ is the angle between $v$ and $e_1$ . Working from the definition of the second fundamental form, we have

$$k_n = II_p(v)$$ (2.36)

$$= -\ensuremath{\langle dN_p(v), v \rangle}$$ (2.37)

$$= -\ensuremath{\langle dN_p(e_1 \cos \theta + e_2 \sin \theta), e_1 \cos \theta + e_2 \sin \theta \rangle}$$ (2.38)

$$= \ensuremath{\langle e_1 k_1 \cos \theta + e_2 k_2 \sin \theta, e_1 \cos \theta + e_2 \sin \theta \rangle}$$ (2.39)

$$= k_1 \cos^2 \theta + k_2 \sin^2 \theta.$$ (2.40)

The last expression is the Euler formula, and is the expression of the second fundamental form in the basis $\{e_1, e_2\}$ .

Footnotes

... eigenvectors^2.5

Recall that if some operator $A$ maps vectors to vectors of the same dimension, then $v$ is called an eigenvector if $Av = \lambda v$ , and $\lambda$ is called its eigenvalue of $dN_p$ . Note that this definition applies for any mapping of vectors to vectors, not just matrix operators.