1.4. Minimal surfaces 17 Figure 1.10. Helicoid Jean-Baptiste-Marie-Charles Meusnier (1785) soon gave equation (1.47) a geometric interpretation. At each point P of a smooth sur- face, choose a vector normal to the surface, cut the surface with nor- mal planes (that contain the normal vector but that differ in orien- tation), and obtain a series of plane curves. For each plane curve, determine the curvature at P . Find the minimum and maximum curvatures (from amongst all the plane curves passing through P ). These are your principal curvatures . Meusnier showed that the minimal surface equation implies that the mean curvature (the average of the principal curvatures) is zero at every point of the minimizing surface. As a result, any surface with zero mean curvature is typically referred to as a minimal surface, even if it does not provide an absolute or relative minimum for surface area. Meusnier also discovered that the catenoid and the helicoid , the surface formed by line segments perpendicular to the axis of a circular helix as they go through the helix (see Figure 1.10), satisfy Lagrange’s
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