geodesy geometry and gauss’ work onpeople.math.umass.edu/~tevelev/475_2018/geodesy.pdf ·...
TRANSCRIPT
Gauss’ Work on Geometry and GeodesyAlex Sellers and Sydney Hauver
Overview
● Who was Gauss● How to Measure the World● Surface Integrals and Divergence Theorem● Differential geometry● Class Activity/Map Projections● Gaussian Curvature● Gauss-Bonnet Theorem● Geodesy● Magnetism● Summary● Questions
Who Was Gauss (1777-1855)
● Contributions to many fields:○ number theory, algebra, statistics, analysis, differential geometry,
geodesy, geophysics, mechanics, electrostatics, magnetic fields astronomy, matrix theory, and optics.
● Nick-Names:○ Princeps mathematicorum( "the foremost of mathematicians")○ "the greatest mathematician since antiquity",
● Bottom Line:● Gauss had an exceptional influence in many fields of mathematics and
science, and is ranked among history's most influential mathematicians● Most Well Known for:● From Brunswick, summed 1-101 as 5050● It was only Carl Gauss who gave proofs of the fundamental theorem that are
still considered valid, by making use of the geometrical interpretation of complex numbers that was unknown to Euler
Measuring the World
● Gauss and his journeys with French explorer Aimé Bonpland○ Their many groundbreaking
ways of taking the world's measure
● Combination of a triangle and a pentangle give 15-gon
● Zimmerman hired land surveying
Surface Integrals and the Divergence Theorem
In 1813 Gauss used divergence theorem in considering the gravitational attraction of an elliptical spheroid
But Gauss went further than Lagrange in showing how to calculate an integral with respect to dS in the case where the surface S is given parametrically by three functions x = x(p, q), y = y(p, q),z = z(p, q). Using a geometrical argument, he demonstrated that:
How to calculate curvature
● Curvature is a local property on a surface S. ● To be defined, it is clear that the curvature may vary from
point to point
Differential Geometry
● Gauss was finally able by 1827 to put on paper the results of his thoughts of over a quarter century on the subject of curved surfaces. Gauss noted in the abstract of his work Disquisitiones generales circa superficies curvas (General Investigations of Curved Surfaces)○ He realized how curvature could be calculated in terms of an analytic
description of the surface in question○ A sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat
plane and a cylinder have Gaussian curvature 0 everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
Gauss-Chern-Bonnet theorem
▪ Gauss, he also argued that the precise nature of physical space could not be determined but only by “experience”
▪ Paths with zero geodesic curvature
Gaussian Curvature and How to Measure It
● Geometry in which Euclid’s parallel postulate did not hold
● Gauss established a relationship between curvature and the sum of the angles of a triangle on the surface
Curvature and the Theorema Egregium
Bottom line: plane can be developed onto a cylinder, the curvature of the cylinder equals that of the plane, namely, 0.
How to Visualize Gaussian Curvature Class Activity
Goode homolosine projection (or interrupted Goode homolosine projection) is a pseudocylindrical, equal-area, composite map projection used for world maps
Geodesy
Accurately measuring and understanding three of Earth’s fundamental properties:
▪ Geometric shape▪ Orientation in space▪ Gravitational field
History
In 1801 Gauss became famous for determining the next appearance of the Ceres from behind the sun with only three observations. Others tried but he was by far the most successful.
Method of Least Squares
Given more sets of equations than there are unknowns, try to make the best guess. Approximation
Method of Least Squares
Approximate the unknowns by reducing the residual error from each equation. For linear least squares it becomes a simple matrix transpose:
Practice problem
Uses of Least Squares
Legendre published his method in 1805 and used it on existing data to calculate the shape of the Earth. Geodesists at the time were stunned and eager to use it.
Gauss published his more rigorous and complete method in 1809.
Triangulation and surveying
The survey of Hannover was a long endeavor that took 14 years but led to the development of the Heliotrope
"All the measurements in the world are not worth one
theorem by which the science of eternal truth is genuinely
advanced" - Gauss
Gauss and Wilhelm Weber
After surveying, Gauss became close friend with Wilhelm Weber, a physics professor at the University of Göttingen. The two were interested in electricity and magnetism
First Electromagnetic Telegraph
Gauss and Weber developed the first electromagnetic telegraph and, at the time, was the longest telegraph in history
Electricity and Magnetism
Gauss’s law for magnetism:
A magnetic field with zero divergence does not change in magnitude or direction, essentially stating there is no existence of a monopole
Summary
● Who was Gauss● How to Measure the World● Differential Geometry● Class Activity/Map Projections● Gaussian Curvature● Gauss-Bonnet Theorem● Geodesy● Magnetism
Questions
Works Cited
▪ The Gauss-Bonnet-Chern Theorem on Riemannian Manifolds Yin Li 28 Nov 2011▪ Snyder, John P. (1993). Flattening the earth: two thousand years of map projections. University of Chicago
Press. p. 1. ISBN 0-226-76746-9.▪ Thomas Banchoff; Terence Gaffney; Clint McCrory; Daniel Dreibelbis (1982). Cusps of Gauss Mappings.
Research Notes in Mathematics. 55. London: Pitman Publisher Ltd. ISBN 0-273-08536-0. Retrieved 10 April 2018.
▪ Lecture on Measuring Curvature, Dr. Robert Kusner, 3 April 2018.▪ http://bolvan.ph.utexas.edu/~vadim/classes/17f/divrot.pdf http://www2.sjs.org/raulston/mvc.10/topic.6.lab.1.htm▪ https://www.magcraft.com/johann-carl-friedrich-gauss▪ https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/
pioneers/carl-friedrich-gauss https://thatsmaths.com/2014/07/10/gausss-great-triangle-and-the-shape-of-space/▪ https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/
museum/gauss-weber-telegraph▪ http://wlym.com/archive/pedagogicals/geodesy.html
https://www.encyclopediaofmath.org/index.php/Gauss,_Carl_Friedrich