8.6: Meusnier's Theorem

Meusnier's theorem concerns the osculating circles of curves tangent to the same fixed direction at a given point. The osculating circle of the normal section X_n(t) has its center at X_n(t₀) + (1/κ)P_n(t₀) and a radius of (1/κ). (Remember that for normal sections, κ_N(t₀) = κ(t₀)). Meusnier's theorem states that at a point on a surface, the osculating circles of all curves tangent to a fixed direction lie on a sphere with center at X(t_0,) + (1/κ)P_n(t₀) and radius (1/κ) , where κ is still equal the curvature of the normal section.

Proof of Meusnier's Theorem

Recall that we defined normal curvature as the normal component of κP, so that κ_N = κcos(α), where α is the angle between the normal vector N and the principal normal vector P. From this, we can verify that in the case of the normal section, P(t₀) = N(t₀), and therefore α = 0 and κ_N(t₀) = κ(t₀). In Meusnier's Theorem, we are given a point on a surface and a fixed direction, and we look at the osculating circle of any curve on the surface that is tangent to that point. For such a curve X(t), the center of its osculating circle is located at X(t₀) + (1/κ)P(t₀). We can write this expression in terms of the normal curvature:

X(t₀) + (cos(α)/κ_N)P(t₀)

The radius of an osculating circle is equal to cos(α)/κ_N. But since κ_N is fixed for any given point and direction, the radius depends only on the angle α:

R(α) = (1/κ_N)cos(α)

Demonstration

In this demonstration, we provide an illustration of Meusnier's Theorem. In the domain window, we choose a point using the white dot and then specify a direction using the yellow vector. In a separate window we draw the surface as well as the normal plane containing both the point and the direction vector. The intersection of the normal plane and the surface is the normal section which is colored red. Remember that for a normal section, P(t₀) = N(t₀) and κ_N(t₀) = κ(t₀). The osculating circle for the normal section is drawn in magenta. We also draw the sphere from Meusnier's Theorem with center at X_n(t₀) + (1/κ)P_n(t₀) and radius 1/κ_N. This sphere is known as the Meusnier Sphere. In order to verify Meusnier's theorem, we need to look at the osculating circles of curves for which P N (or α 0). To do this we draw a second rotatable plane containing the white point and the yellow direction vector. The angle between this plane and the normal plane is determined by the variable α in the control panel. The intersection of this plane and the surface produces a second planar curve such that the angle between P and N is α. We then draw the osculating circle for this curve in magenta. Observe that the magenta osculating circle lies on the Meusnier Sphere for any value of α

Some curves on the surface will not have an osculating circle at t = t₀. In particular, this will happen whenever the yellow vector points in an asymptotic direction. Hence, for example, if α = π/4 on our default surface, the hyperbolic paraboloid, one obtains an asymptotic direction and thus no osculating circle.

Exercises

1. Which curves on the surface of a hyperbolic paraboloid go through both pole of the Meusnier's Sphere?, How about curves on the surface of a torus? Why? Can these hypotheses be generalized?