02 Spherical Geometry Basics
... other point is not the south pole, the shortest distance along the sphere is obvsiouly to go due south. We are, from now on, going to rule out pairs of antipodal points such as the north and south poles, because there are infinitely many geodesics between them. Otherwise, just as in geometry for the ...
... other point is not the south pole, the shortest distance along the sphere is obvsiouly to go due south. We are, from now on, going to rule out pairs of antipodal points such as the north and south poles, because there are infinitely many geodesics between them. Otherwise, just as in geometry for the ...
lengths of geodesics on riemann surfaces with boundary
... known bound is Lg ≤ 21(g − 1) ([3, Remark 5.2.5, p. 129]). A surface of genus g will be called maximal for Bers’ partition constant if it is a surface on which a minimal partition P satisfies l(P) = Lg . A Y -piece can be separated into two symmetric isometric right angled hyperbolic hexagons. To do ...
... known bound is Lg ≤ 21(g − 1) ([3, Remark 5.2.5, p. 129]). A surface of genus g will be called maximal for Bers’ partition constant if it is a surface on which a minimal partition P satisfies l(P) = Lg . A Y -piece can be separated into two symmetric isometric right angled hyperbolic hexagons. To do ...
Math 11 Adv Distance on a sphere – another application of
... latitude are not. Thus, traveling along a line of longitude is traveling on a great circle, while traveling along a line of latitude between two points may seem like the most direct path, but it is not. Our geographic coordinate system is great for figuring distances that are parallel to lines of la ...
... latitude are not. Thus, traveling along a line of longitude is traveling on a great circle, while traveling along a line of latitude between two points may seem like the most direct path, but it is not. Our geographic coordinate system is great for figuring distances that are parallel to lines of la ...
Einstein memorial lecture.
... the curve C that you get in the plane is a (piece of) a straight line. For the sphere, the curves c which roll out to straight lines in the plane are exactly the great circles. But we can make this definition for any curve on any surface. It is then a mathematical theorem that this definition of geo ...
... the curve C that you get in the plane is a (piece of) a straight line. For the sphere, the curves c which roll out to straight lines in the plane are exactly the great circles. But we can make this definition for any curve on any surface. It is then a mathematical theorem that this definition of geo ...
class summary - Cornell Math
... plane (for proof, draw three segments, and draw circles about all three points with sides a, b, and c as radii). But is it true on the sphere? Think of possible counterexamples (along the same lines of SAS and ASA, “big” usually leads to bad). Is it true when restricting to “small” triangles? Now th ...
... plane (for proof, draw three segments, and draw circles about all three points with sides a, b, and c as radii). But is it true on the sphere? Think of possible counterexamples (along the same lines of SAS and ASA, “big” usually leads to bad). Is it true when restricting to “small” triangles? Now th ...
Statistical Analysis of Shapes of Curves and Surfaces
... metrics. I will show two methods — shooting and path-straightening — to compute geodesics on these spaces. Using geodesics, one can define and compute sample statistics, impose probability distributions, and study hypothesis testing involving shapes. As an example, I will present examples of generati ...
... metrics. I will show two methods — shooting and path-straightening — to compute geodesics on these spaces. Using geodesics, one can define and compute sample statistics, impose probability distributions, and study hypothesis testing involving shapes. As an example, I will present examples of generati ...
Holographic dual of a time machine
... timelike intervals Geodesics of the second type reach the Poincare horizon r=0 at λ=0. We can treat this point in a similar way as the complex infinity in the theory of complex functions: Two disconnected geodesics possessing the same invariants E and J but emerging from two different boundary point ...
... timelike intervals Geodesics of the second type reach the Poincare horizon r=0 at λ=0. We can treat this point in a similar way as the complex infinity in the theory of complex functions: Two disconnected geodesics possessing the same invariants E and J but emerging from two different boundary point ...
Internal geometry of surfaces
... A fellow took a morning stroll. He first walked 2 km South, then 2 km West, and then 2 km North. It so happened that he found himself back at his house door. How can this be? ...
... A fellow took a morning stroll. He first walked 2 km South, then 2 km West, and then 2 km North. It so happened that he found himself back at his house door. How can this be? ...
THE GEOMETRY OF SURFACES AND 3
... Or a sphere – the normal vector to the sphere always points toward/away from the center, and a path on the sphere which only accelerates toward the center is a great circle. So that’s one definition of a “line” – a curve that goes “straight” (or at least as straight as possible). It is a remarkable ...
... Or a sphere – the normal vector to the sphere always points toward/away from the center, and a path on the sphere which only accelerates toward the center is a great circle. So that’s one definition of a “line” – a curve that goes “straight” (or at least as straight as possible). It is a remarkable ...
Basics of Hyperbolic Geometry
... point 0. This map is a hyperbolic isometry, and also fixes every point on the vertical geodesic. As another example, the map Let R(z) = 1/z is a hyperbolic isometry that fixes every point on the geodesic connecting −1 to 1. Such maps are called hyperbolic reflections. That is, a hyperbolic reflectio ...
... point 0. This map is a hyperbolic isometry, and also fixes every point on the vertical geodesic. As another example, the map Let R(z) = 1/z is a hyperbolic isometry that fixes every point on the geodesic connecting −1 to 1. Such maps are called hyperbolic reflections. That is, a hyperbolic reflectio ...
Euclid`s Postulates We have been playing with Non
... have visited: taxi cab, spheres, cylinders, cones, annular hyperbolic planes (the stuff we crocheted), and Poincaré Discs. Make sure to include diagrams in your notes as you investigate. 1. A geodesic segment can be drawn joining any two points. 2. A geodesic segment can be extended indefinitely. 3. ...
... have visited: taxi cab, spheres, cylinders, cones, annular hyperbolic planes (the stuff we crocheted), and Poincaré Discs. Make sure to include diagrams in your notes as you investigate. 1. A geodesic segment can be drawn joining any two points. 2. A geodesic segment can be extended indefinitely. 3. ...
Geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesyspecifically with the solution of triangulation networks. Thefigure of the Earth is well approximated by anoblate ellipsoid, a slightly flattened sphere. A geodesicis the shortest path between two points on a curved surface, i.e., the analogueof a straight line on a plane surface. The solution of a triangulationnetwork on an ellipsoid is therefore a set of exercises in spheroidaltrigonometry (Euler 1755).If the Earth is treated as a sphere, the geodesics aregreat circles (all of which are closed) and the problems reduce toones in spherical trigonometry. However, Newton (1687)showed that the effect of the rotation of the Earth results in itsresembling a slightly oblate ellipsoid and, in this case, theequator and the meridians are the onlyclosed geodesics. Furthermore, the shortest path between two points onthe equator does not necessarily run along the equator. Finally, if theellipsoid is further perturbed to become a triaxial ellipsoid (withthree distinct semi-axes), then only three geodesics are closed and oneof these is unstable.The problems in geodesy are usually reduced to two main cases: thedirect problem, given a starting point and an initial heading, findthe position after traveling a certain distance along the geodesic; andthe inverse problem, given two points on the ellipsoid find theconnecting geodesic and hence the shortest distance between them.Because the flattening of the Earth is small, the geodesic distancebetween two points on the Earth is well approximated by the great-circledistance using themean Earth radius—the relative error isless than 1%. However, the course of the geodesic can differdramatically from that of the great circle. As an extreme example,consider two points on the equator with a longitude difference of179°59′; while the connecting great circle follows theequator, the shortest geodesics pass within180 km of either pole (theflattening makes two symmetric paths passing close to the poles shorterthan the route along the equator).Aside from their use in geodesy and related fields such as navigation,terrestrial geodesics arise in the study of the propagation of signalswhich are confined (approximately) to the surface of the Earth, forexample, sound waves in the ocean (Munk & Forbes 1989) and theradio signals from lightning (Casper & Bent 1991). Geodesics areused to define some maritime boundaries, which in turn determine theallocation of valuable resources as suchoil and mineral rights. Ellipsoidal geodesics alsoarise in other applications; for example, the propagation of radio wavesalong the fuselage of an aircraft, which can be roughly modeled as aprolate (elongated) ellipsoid(Kim & Burnside 1986).Geodesics are an important intrinsic characteristic of curved surfaces.The sequence of progressively more complex surfaces, the sphere, anellipsoid of revolution, and a triaxial ellipsoid, provide a usefulfamily of surfaces for investigating the general theory of surfaces.Indeed, Gauss's work on thesurvey of Hanover, which involvedgeodesics on an oblate ellipsoid, was a key motivation for hisstudy of surfaces(Gauss 1828). Similarly, the existence of three closed geodesicson a triaxial ellipsoid turns out to be a general property ofclosed, simply connected surfaces; this wasconjectured by Poincaré (1905) and proved byLyusternik & Schnirelmann (1929)(Klingenberg 1982, §3.7).