8.1: Normal Curvature of Curves on Surfaces

In lab 3 , we resolved the acceleration vector of a space curve X(t) into a linear combination of the the unit tangent vector T(t) and the principal normal vector P(t). If we now consider a curve on a surface X(u,v), the most natural pair of vectors to look at are the same unit tangent vector T(t) and the unit normal N(t) to the surface at the point, X(t). Since this vector N(t), is perpendicular to T(t), we may express it as a linear combination of P(t) and B(t). And since it is a unit vector, for some angle α(t), we have

N(t) = cos(α(t))P(t)+sin(α(t))B(t).

The normal component of the acceleration vector is defined to be κ_N(t)s'(t)², or

X''(t) · N(t) = κ_N(t)s'(t)²,     where
κ_N(t)=κP(t) · N(t)    

We may select a point on the domain and a direction θ (in the control panel a is used to indicate the angle). We thus determine a line through the chosen point C with slope tan(θ) given by

(u(t),v(t))=(C₁ + tcos(θ),C₂ + tsin(θ))

In the control panel, the angle can be changed from 0 tp 2π. Observe that orange normal vector N(t) always lies in the plane spanned by the vectors P(t) and B(t). We may observe how the angle between P(t) and N(t) changes as the red line is rotated.

In this demonstration, we choose a line in the parameter domain of a paraboloid (the default), and we observe how the normal to the surface relates to the Frenet frame of the space curve.

Exercises

1. Observe in particular that the angle θ = π/4 gives a version of the space cardioid on the torus. What happens as goes from zero to 2π?

Normal curvature and the Weingarten Map

κ_N(t)s'(t)² = X''(t) · N(t)

(d/dt)(X'(t) · N(t)) = X''(t) · N(t) + X'(t) · N'(t)

X''(t) · N(t) = - X'(t) · N'(t)

κ_N(t)s'(t)² = -X'(t) · L(X'(t))

κ_N(t) = -T(t) · L(T(t))

The following demonstration provides a geometric representation of the relationship between the Wiengarten Map and the normal curvature.

In one window we see a surface (monkey saddle by default) and a green curve (u₁(t), v₁(t)) that lies on the surface. At the point (u₁(0), v₁(0)) on the curve, two vectors are drawn. The white vector is the velocity vector X'(t) and the cyan vector is N'(t). Both vectors lie in the tangent plane to the surface at that point. In a second window, we see the tangent space which contains the X'(t) and N'(t) vectors. In order to simplify things, the magnitude of the velocity vector is set equal to 1 so that s'(t) = 1 and X'(t) = T(t). From our previous calculations we know that L(X'(t)) = N'(t), so the cyan vector is just the image of the white vector under the L-transformation.

To use this demo, start by picking a point on the surface using the hotspot in the "Domain" window, then choose a direction by rotating the white tangent vector in the "Tangent Space" window. Note that the white vector is the tangent vector of the green curve on the surface. So as the white vector rotates, the green curve changes as well.

The window labeled "Normal Curvature" shows how the the Weingarten Map is connected with normal curvature. Given a point on a surface and a direction T(t), we can calculate the normal curvature. In fact, according to our last result, κ_N(t) = -T(t) · L(T(t)), so the normal curvature is just the negative of the dot product of the white vector and the cyan vector. Here we show a graph of the normal curvature at (u₁(0), v₁(0)) as a function of angle. The yellow bar indicates the current angle of the tangent vector. Observe that the graph is sinusoidal.

Principal and Asymptotic Directions

Back in lab 6, we defined the principal curvatures κ₁ and κ₂ in terms of the Gaussian and mean curvatures Κ and H. Since then, there have been two geometric representations of the principal curvatures. We first looked at parallel surfaces and defined the radii of curvature as the distances at which a point on the parallel surfaces becomes singular. The inverses of the radii of curvature are what we called the principal curvatures. After that we described a linear transformation known as the Weingarten (or L_i^j) Map that acts on vectors in the tangent plane. Writing this linear transformation as a matrix, we looked its characteristic equation and found that the eigenvalues are the principal curvatures κ₁ and κ₂. Using the demo in this lab, we arrive at a third geometric representation of the principal curvatures.


We can define the prinicpal curvatures (κ₁ and κ₂) at a point on a surface as the maximum and minimum values of the normal curvature κ_N. The directions along which the normal curvature is principal are called principal directions. The directions along which the normal curvature is equal to zero are called asymptotic directions. For the monkey saddle surface in the demonstration, we observed that the graph of normal curvature as a function of angle is sinusoidal. There will, therefore, always be minimum and maximum values for κ_N. Points on a surface for which the normal curvature is constant in all directions are called umbilic points.

Exercises

1. Using the demo, what can we say about the orientation of T(t) and L(T(t)) when the T(t) points along a principal direction. When T(t) points along an asymptotic direction?

2. Prove or disprove the claim that asymptotic directions do not exist in regions of positive Gaussian curvature. Use the demo to verify your answer.