Curvature and the Shape Operator

Total Curvature of a Surface

This interactive demo shows a plot of the surface z = sin(x)cos(y) with x and y from -3 to 3.

Unlike the last demo, this also shows a unit sphere below it. In red we see the path that the normal takes on the surface according to the radius on the slider. The red shape on the sphere represents the path traced out if the normal vector to the surface were anchored at the origin while it traced out the circle on the surface. The area of this shape on the sphere is called the total curvature of the red surface.

Notice that when the local curvature is 0 (i.e. at the beginning of the demo), the shape on the surface of the sphere is just an arc and therefore has an area of zero.

Additionally, as the radius shrinks to zero, the ratio between the area of the shape on the sphere and of the red path on the surface approaches the local curvature that we've computed in the last two demos!

Use the sliders to manipulate the position of the normal vector and see the current curvature at a point. Note that this demo displays the upward- pointing normal vector because it is easier to see. Be sure to keep track of when the curvature is positive, negative, and zero.

In addition, the following controls are available to manipulate the image:


Curvature at (, , ):

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