ARML talk poster Gauss Map
Using Alfred Gray’s Mathematica programming we constructed animations to support Differential Geometry courses. Animations elucidate simple geometric ideas even when the symbolic mathematical representations are complex.
Here one studies the ‘catenoid’ surface on the left by transferring information from it to the geometrically simpler sphere on the right. The catenoid red curve is transferred to the sphere yellow curve by the “Gauss map” method. The blue vector on the catenoid is moved so its base point is the sphere center. The yellow tip ‘paints’ the yellow curve on the sphere, carrying geometric information from the catenoid, in the simpler setting.
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The above graphic is one frame of an animated view of the Gauss map of a catenoid. On the left side, the catenoid, the surface of revolution with profile curve the catenary {v, cosh[v]}, is shown with a red spiral curve on it. This curve obtained by mapping a spiral centered at the origin of the u-v plane to the catenoid by the parameterization catenoid[u,v] = {cos[u]*cosh[v],sin[u]cosh[v],v}. The blue 'outward' unit normal with a yellow tip is shown at a point of this curve. On the right side of the graphic the blue unit normal with yellow tip is shown translated to the origin, with its yellow tip on a gray sphere (shown in wireframe view).The yellow curve on the sphere is the image of the red curve under the Gauss map of the catenoid. We can think of the unit normal as 'painting' that portion of the sphere corresponding to the image of the red spiral under the Gauss map. If we imagine the outermost part of the spiral on the catenoid as enclosing a two-dimensional region D, then it appears from the graphic that the image under the Gauss map of D will be a two-dimensional region on the sphere. The area of this image is controlled by the Gauss curvature of the catenoid. If p is any point on the catenoid, and D any two-dimensional region with p in D, then the ratio Area(G(D))/Area(D), where G is the Gauss map, is a certain number. The limiting value of such numbers (with an appropriate sign) as D 'shrinks down to p' is the Gauss curvature of the surface at p, K(p). This is how Gauss introduced the idea of what we now call Gauss curvature (Gauss called it the "measure of curvature"). For the catenoid, the Gauss curvature is strictly negative at every point, taking a minimum value of -1 at the "center" of the red spiral (the negativity of the curvature is reflected in the opposing motions of the normals on the catenoid as they move along the red spiral and the sense in which the image of the spiral is traced out on the sphere). That K = -1 at the center of the spiral can be interpretted to mean that near the center of the red spiral,the area of a small disc will have approximately the same area as its image under the Gauss map, and have an opposite orientation. Contrast this animation with the next one, the Gauss map for a cylinder. There the Gauss curvature is identically 0 and any such disc will be mapped to a curve. Click on the picture to see the animated view.
Normal Curvature:
Using Alfred Gray’s Mathematica programming we constructed animations to support Differential Geometry courses. Animations elucidate simple geometric ideas even when the symbolic mathematical representations are complex.
Here one studies the red ‘saddle’ surface by studying black ‘slice’ curves on it. These curves arise from slicing the saddle with rotating yellow planes built geometrically from the saddle.
As the plane rotates, the blue curve on the right measures how sharply bent up (towards the green vector) or down the black slice curves are, in terms of the plane’s rotation angle. Fundamental geometric surface information (‘normal curvature’) comes from these curve attributes.
The surface z = y^2 - x^2 is shown in red. The green 'upward' normal is located at (0,0,0). The normal curvature is computed in the direction of the blue tangent vector at (0,0,0). The blue and green vectors span the yellow slicing plane. The normal curvature is (up to sign) the ordinary space curve curvature of the curve given by the intersection of the slicing plane and the surface. The normal curvature is positive when the slice curve 'bends up towards' the chosen normal, and negative when it 'bends down away'. The intersection curves are shaded black. It is an exercise in the course to prove that these curves are the images of straight lines through {0,0} in the u-v parameter plane under the parameterization { u , v , -u^2 + v^2 }. In the right-hand frame the normal curvature is graphed as a function of the 'polar angle' of the blue tangent vector. The 'polar angle' is defined using the parameterization for a basis of the tangent space, and applying the Gram-Schmidt process to that basis. Explaining this was also part of an exercise. Click on the picture to see the animated view.
Using Alfred Gray’s Mathematica programming we constructed animations to support Differential Geometry courses. Animations elucidate simple geometric ideas even when the symbolic mathematical representations are complex.
This animation illustrates a premier result in Differential Geometry, the Theorema Egregium of Gauss. The green-yellow hemisphere is deformed so that the distance between points measured along the surface does not change. Three measures of curvedness of the deforming hemisphere are also shown. While the red and blue ‘principal curvatures’ change, the purple Gauss curvature (the product of the principal curvatures) never changes. That is what the Theorema Egregium establishes in symbolic mathematical form.