Unit 7: Lines of Curvature, Asymptotic Curves & The Dupin Indicatrix

1. Definition of Lines of Curvature

Definition 7.1 (Line of Curvature): A regular curve $C$ on a surface $S$ is called a line of curvature (or curvature line) if its tangent vector at every point points along a principal direction of the surface.

2. Rodrigues' Formula

Theorem 7.1 (Rodrigues' Formula, 1815): A regular curve $\mathbf{r}(t)$ on a surface $S$ is a line of curvature if and only if there exists a scalar function $\kappa(t)$ (the principal curvature) such that:

Proof: By definition, $\mathbf{r}'(t)$ is an eigenvector of the Shape Operator $S_p$ if and only if:

Recall that the Shape Operator is defined by $S_p(\mathbf{w}) = -d\mathbf{n}_p(\mathbf{w})$. Applying this to $\mathbf{w} = \mathbf{r}'(t)$:

This establishes Rodrigues' formula directly. $\blacksquare$

3. Differential Equation of Lines of Curvature

Let $d\mathbf{r} = \mathbf{r}_u \, du + \mathbf{r}_v \, dv$ and $d\mathbf{n} = \mathbf{n}_u \, du + \mathbf{n}_v \, dv$. By Rodrigues' formula, $d\mathbf{n}$ is collinear with $d\mathbf{r}$. Hence, their cross product must vanish in $\mathbb{R}^3$:

Since both $d\mathbf{n}$ and $d\mathbf{r}$ lie in the tangent plane $T_p S$, the vector $d\mathbf{n} \times d\mathbf{r}$ is parallel to the normal vector $\mathbf{n}$. Therefore:

Using the Weingarten equations to express $d\mathbf{n}$ in terms of $E, F, G$ and $L, M, N$, this determinant expands into the classical Monge-Darby determinant:

Theorem 7.2 (Differential Equation of Lines of Curvature): The lines of curvature on a surface patch satisfy the second-order quadratic ODE:

Expanding this $3 \times 3$ determinant:

Corollary 7.1 (Coordinate Curves as Lines of Curvature): The coordinate curves $u = \text{const}$ and $v = \text{const}$ form a family of lines of curvature if and only if:

In this case, the principal curvatures are simply $\kappa_1 = L/E$ and $\kappa_2 = N/G$.

1. Statement and Proof of Euler's Formula

Let $p \in S$ be a non-umbilical point ($\kappa_1 \ne \kappa_2$). Choose an orthonormal basis $\{\mathbf{e}_1, \mathbf{e}_2\}$ in $T_p S$ along the principal directions:

Any unit tangent vector $\mathbf{u} \in T_p S$ can be represented as:

where $\theta \in [0, 2\pi)$ is the angle made by $\mathbf{u}$ with the first principal direction $\mathbf{e}_1$.

Theorem 7.3 (Euler's Theorem, 1760): The normal curvature $\kappa_n(\theta)$ along the direction making an angle $\theta$ with the first principal direction is given by:

Proof: Recall from Section 5.2 that the normal curvature along a unit tangent vector $\mathbf{u}$ is:

Substituting $\mathbf{u} = \cos\theta \, \mathbf{e}_1 + \sin\theta \, \mathbf{e}_2$:

Taking the inner product with $\mathbf{u}$:

Since $\mathbf{e}_1 \cdot \mathbf{e}_1 = 1$, $\mathbf{e}_2 \cdot \mathbf{e}_2 = 1$, and $\mathbf{e}_1 \cdot \mathbf{e}_2 = 0$:

2. Orthogonal Pairs and the Invariance of Mean Curvature

Consider two mutually perpendicular tangent directions: $\theta$ and $\theta + \pi/2$. The normal curvature in the perpendicular direction is:

Adding the two normal curvatures together:

Corollary 7.2 (Invariance of Sum of Orthogonal Curvatures): For any pair of orthogonal unit tangent vectors on a surface, the sum of their normal curvatures is constant and equals twice the Mean Curvature:

1. Construction of the Dupin Indicatrix

The Dupin indicatrix is a geometric construction in the tangent plane $T_p S$ that visually characterizes the local second-order shape of the surface. Let Cartesian coordinates $(\xi, \eta)$ be chosen in $T_p S$ along the principal directions $\mathbf{e}_1, \mathbf{e}_2$. For each direction defined by angle $\theta$ ($\xi = r \cos\theta, \eta = r \sin\theta$), plot a segment of length:

Squaring and multiplying by $|\kappa_n(\theta)|$:

Substituting $\xi = r \cos\theta$ and $\eta = r \sin\theta$:

Definition 7.2 (The Dupin Indicatrix): The Dupin indicatrix at a point $p \in S$ is the quadratic curve in $T_p S$ given by:

2. Geometric Shape Depending on Point Type

1. At an Elliptic Point ($K > 0$):

$\kappa_1$ and $\kappa_2$ have the same sign. The equation is:

This is an ellipse with semi-axes $a = 1/\sqrt{\kappa_1}$ and $b = 1/\sqrt{\kappa_2}$. If the point is an umbilic ($\kappa_1 = \kappa_2$), the ellipse becomes a circle.

2. At a Hyperbolic Point ($K < 0$):

$\kappa_1$ and $\kappa_2$ have opposite signs (say $\kappa_1 > 0, \kappa_2 < 0$). The Dupin indicatrix consists of a pair of conjugate hyperbolas:

The asymptotes of these hyperbolas are given by $\kappa_1 \xi^2 + \kappa_2 \eta^2 = 0$, which correspond to the asymptotic directions of the surface!

3. At a Parabolic Point ($K = 0, \kappa_1 \ne 0, \kappa_2 = 0$):

The equation reduces to:

This is a pair of parallel straight lines parallel to the direction of zero curvature.

1. Asymptotic Directions and Curves

Definition 7.3 (Asymptotic Direction and Curve):

1. A tangent direction $(du, dv)$ at $p \in S$ is called an asymptotic direction if the normal curvature along that direction is zero:

1. A regular curve on $S$ whose tangent vector at every point points along an asymptotic direction is called an asymptotic curve (or asymptotic line).

From Euler's theorem, $\kappa_n(\theta) = \kappa_1 \cos^2\theta + \kappa_2 \sin^2\theta = 0$:

• At an elliptic point ($K > 0$): $\kappa_1 / \kappa_2 > 0$, so $\tan^2\theta < 0$, which has no real solutions. There are no asymptotic directions at elliptic points!
• At a parabolic point ($K = 0$): there is one unique asymptotic direction (along the direction of $\kappa_2 = 0$).
• At a hyperbolic point ($K < 0$): $\tan\theta = \pm \sqrt{-\kappa_1 / \kappa_2}$, yielding two distinct real asymptotic directions symmetric about the principal directions!

2. The Beltrami-Enneper Theorem

Along an asymptotic curve, $\kappa_n = \kappa \cos \theta = 0$. Assuming the space curvature $\kappa > 0$, Meusnier's theorem dictates that $\cos \theta = 0 \implies \mathbf{N} \cdot \mathbf{n} = 0$. Thus, the principal normal to the curve $\mathbf{N}$ is perpendicular to the surface normal $\mathbf{n}$! Consequently, the binormal vector $\mathbf{B} = \mathbf{T} \times \mathbf{N}$ is parallel to $\mathbf{n}$:

The osculating plane of an asymptotic curve coincides with the tangent plane to the surface!

Theorem 7.4 (Beltrami-Enneper Theorem, 1870): Let $C$ be an asymptotic curve with non-vanishing space curvature $\kappa > 0$ on a surface with negative Gaussian curvature $K < 0$. Then the torsion $\tau$ of the asymptotic curve satisfies:

Proof: Along the asymptotic curve, the binormal satisfies $\mathbf{B} = \mathbf{n}$ (up to a sign $\pm 1$). By the Serret-Frenet formulas:

Taking the norm squared:

Since $\mathbf{B} = \mathbf{n}$, $\mathbf{B}'(s) = d\mathbf{n}(\mathbf{T}) = -S_p(\mathbf{T})$. Thus:

By the fundamental form identity (Theorem 5.4):

Since the curve is asymptotic, $II(\mathbf{T}) = 0$. Since $\mathbf{T}$ is a unit vector, $I(\mathbf{T}) = 1$. Substituting these values:

Since $K < 0$, $-K > 0$, taking the square root gives:

1. Conjugate Directions

Definition 7.4 (Conjugate Directions): Two tangent directions $\mathbf{w}_1 = (du, dv)$ and $\mathbf{w}_2 = (\delta u, \delta v)$ at $p \in S$ are called conjugate directions if:

In coordinates, the conjugacy condition is:

Theorem 7.5 (Geometric Properties of Conjugate Directions):

1. The principal directions are the only mutually orthogonal conjugate directions.
2. An asymptotic direction is self-conjugate ($II(du, dv) = 0$).
3. Two directions $\theta_1, \theta_2$ relative to the principal frame are conjugate if and only if:

2. Triply Orthogonal Systems & Dupin's Theorem

Definition 7.5 (Triply Orthogonal System): A system of three families of surfaces in $\mathbb{R}^3$ is called a triply orthogonal system if through each point there passes exactly one surface from each family, and the three surfaces intersect each other pairwise orthogonally.

Theorem 7.6 (Dupin's Theorem, 1813): The intersection curves of the surfaces of any triply orthogonal system are lines of curvature on all three intersecting surfaces!

Example: Confocal quadrics (ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets sharing the same focal conics) form a triply orthogonal system. By Dupin's Theorem, their curves of intersection are automatically the lines of curvature on each quadric surface!