WHAT IS HYPERBOLIC GEOMETRY? - School of Mathematics, TIFR
... Maps such as f are called parabolic translations and those like g are called inversions. The map g is an inversion about a semi-circle of radius R centered at 0. A more geometric description of g is as follows. Take a circle of radius R centered at 0. Then every point in H lies on a unique ray throu ...
... Maps such as f are called parabolic translations and those like g are called inversions. The map g is an inversion about a semi-circle of radius R centered at 0. A more geometric description of g is as follows. Take a circle of radius R centered at 0. Then every point in H lies on a unique ray throu ...
Lost in an Isosceles Triangle
... shape of a convex polygon. Best results can be obtained for shapes with bilateral symmetry since then the best escape path is likely to be have either bilateral symmetry or rotational symmetry and the space of possible paths can be covered in more depth. An isosceles triangle can be specified by the ...
... shape of a convex polygon. Best results can be obtained for shapes with bilateral symmetry since then the best escape path is likely to be have either bilateral symmetry or rotational symmetry and the space of possible paths can be covered in more depth. An isosceles triangle can be specified by the ...
microsoft word document
... 2. Circle – a set of points in a plane equally distant from a fixed point called the center. 3. Geodesics: are the shortest distances between two points on the sphere. They are line segments along a great circle. 4. Great Circle – When a plane passes through the center of a sphere, cutting the spher ...
... 2. Circle – a set of points in a plane equally distant from a fixed point called the center. 3. Geodesics: are the shortest distances between two points on the sphere. They are line segments along a great circle. 4. Great Circle – When a plane passes through the center of a sphere, cutting the spher ...
Geometry – Section 11.1 – Notes and Examples – Lines that
... Geometry – Section 11.2 – Notes and Examples – Arcs and Chords A __________ angle is an angle whose __________ is the _________ of a circle. An _____ is an _____________ part of a circle consisting of _____ points called the ______________ and all the points on the ________ between them. ...
... Geometry – Section 11.2 – Notes and Examples – Arcs and Chords A __________ angle is an angle whose __________ is the _________ of a circle. An _____ is an _____________ part of a circle consisting of _____ points called the ______________ and all the points on the ________ between them. ...
the isoperimetric problem on some singular surfaces
... 2. Existence and regularity We consider piecewise smooth (stratified) n-dimensional closed submanifolds M of R N , with a piecewise smooth, continuous Riemannian metric within a bounded factor of the induced metric, possibly undefined on strata of dimension less than n − 1. We do not allow M to have ...
... 2. Existence and regularity We consider piecewise smooth (stratified) n-dimensional closed submanifolds M of R N , with a piecewise smooth, continuous Riemannian metric within a bounded factor of the induced metric, possibly undefined on strata of dimension less than n − 1. We do not allow M to have ...
Non-Euclidean Geometry
... • A unique straight line can be drawn through any two points A and B • A segment can be extended indefinitely • For any two distinct points A and B, a circle can be drawn with center A and radius AB • All right angles are congruent ...
... • A unique straight line can be drawn through any two points A and B • A segment can be extended indefinitely • For any two distinct points A and B, a circle can be drawn with center A and radius AB • All right angles are congruent ...
Ghost Conical Space - St. Edwards University
... Aside from the geodesics that hit the cone point there are other types of geodesics as well. A good place to start is a cone that has a generating angle of less than 180 degrees. If we start with our flattened model and use a straight edge to draw a line we can observe what happens to the line once ...
... Aside from the geodesics that hit the cone point there are other types of geodesics as well. A good place to start is a cone that has a generating angle of less than 180 degrees. If we start with our flattened model and use a straight edge to draw a line we can observe what happens to the line once ...
Analytical Calculation of Geodesic Lengths and Angle Measures on
... Figure 2 : Tilings of the sphere arising from the Platonic and Archimedean solids. Spherical trigonometry. There are spherical trigonometry identities that are similar to ones in planar geometry. Let a, b, and c be three sides of the triangle and α, β, and γ be three angles as in Figure 3(a). Law of ...
... Figure 2 : Tilings of the sphere arising from the Platonic and Archimedean solids. Spherical trigonometry. There are spherical trigonometry identities that are similar to ones in planar geometry. Let a, b, and c be three sides of the triangle and α, β, and γ be three angles as in Figure 3(a). Law of ...
Lecture 8: Curved Spaces
... was given to spaces where this postulate does not hold. Mathematicians such as, Gauss, Riemann, Lobachevskii formulated the field of non-Euclidean geometry. Let’s begin by examining the subspace R2 (the flat infinite plane) embedded into R3 . Or, to make a more concrete example, consider the flat un ...
... was given to spaces where this postulate does not hold. Mathematicians such as, Gauss, Riemann, Lobachevskii formulated the field of non-Euclidean geometry. Let’s begin by examining the subspace R2 (the flat infinite plane) embedded into R3 . Or, to make a more concrete example, consider the flat un ...
On the equivalence of Alexandrov curvature and
... It is well known that the curvature bounded above (resp. below) in the sense of Alexandrov is stronger than the curvature bounded above (resp. below) in the sense of Busemann (see, e.g., [7, p. 107] or [9, p. 57]). The classical example that shows that the converse statement does not hold is the fin ...
... It is well known that the curvature bounded above (resp. below) in the sense of Alexandrov is stronger than the curvature bounded above (resp. below) in the sense of Busemann (see, e.g., [7, p. 107] or [9, p. 57]). The classical example that shows that the converse statement does not hold is the fin ...
con escuadra y catabon
... property is its construction, due to the fact that is impossible to move the point E once is drawn on paper. Is that for, we will use GeoGebra, a dynamic geometry software, that allow us to move the point E, and then verify compliance with this property for any isosceles triangle and any position of ...
... property is its construction, due to the fact that is impossible to move the point E once is drawn on paper. Is that for, we will use GeoGebra, a dynamic geometry software, that allow us to move the point E, and then verify compliance with this property for any isosceles triangle and any position of ...
Some Geometry You Never Met 1 Triangle area formulas
... As on the sphere, hyperbolic lines are curved but, if you examine them closely, you will see that each of them is perpendicular to the edge of the region containing them. Think of that region made infinitely large so that the circle represents the “edge” of space. Then you can think of those lines a ...
... As on the sphere, hyperbolic lines are curved but, if you examine them closely, you will see that each of them is perpendicular to the edge of the region containing them. Think of that region made infinitely large so that the circle represents the “edge” of space. Then you can think of those lines a ...
DIFFERENTIAL GEOMETRY HW 3 32. Determine the dihedral
... Let A, B and C be the sides of the triangle formed by the Ni as pictured above. Since the Ni are normal vectors to the Si , the angle between Ni and Nj is equal to π − φ, where φ is the dihedral angle between Si and Sj (which is, of course, the same for all i 6= j); hence, since they lie on the unit ...
... Let A, B and C be the sides of the triangle formed by the Ni as pictured above. Since the Ni are normal vectors to the Si , the angle between Ni and Nj is equal to π − φ, where φ is the dihedral angle between Si and Sj (which is, of course, the same for all i 6= j); hence, since they lie on the unit ...
5 The hyperbolic plane
... 2. Every straight line contains a least two points. 3. There are at least three points not lying on the same straight line. ...
... 2. Every straight line contains a least two points. 3. There are at least three points not lying on the same straight line. ...
GeoPCA: a new tool for multivariate analysis of dihedral angles
... pi = (ai1, ai2, . . ., aim) can be treated as a point on the m-dimensional unit sphere, representing the ith conformation. For our test data set (see below), the nucleotides all have the same C30 -endo sugar pucker conformation. Hence, the input data consist of the seven conventional torsion angles ...
... pi = (ai1, ai2, . . ., aim) can be treated as a point on the m-dimensional unit sphere, representing the ith conformation. For our test data set (see below), the nucleotides all have the same C30 -endo sugar pucker conformation. Hence, the input data consist of the seven conventional torsion angles ...
Right Angle Trig Apps
... Key Steps in Trigonometry Applications 1) Draw a picture of the problem, if a drawing is not included in the original problem. 2) Find right-triangle(s) inherent and useful to the problem. If you can only find non-right triangles perhaps there are similar triangles to work with. Perhaps you can cut ...
... Key Steps in Trigonometry Applications 1) Draw a picture of the problem, if a drawing is not included in the original problem. 2) Find right-triangle(s) inherent and useful to the problem. If you can only find non-right triangles perhaps there are similar triangles to work with. Perhaps you can cut ...
Overview - Connecticut Core Standards
... investigation we want them to focus on the surface. One way to get them to see this is to view the sphere as a solid of rotation. This idea was first introduced in question 10 of Activity 6.4.2 where students rotated a semicircle about its diameter. We can think of a sphere as formed by a double rot ...
... investigation we want them to focus on the surface. One way to get them to see this is to view the sphere as a solid of rotation. This idea was first introduced in question 10 of Activity 6.4.2 where students rotated a semicircle about its diameter. We can think of a sphere as formed by a double rot ...
274 Curves on Surfaces, Lecture 5
... or the cross-ratio of a, b, c, d. This can be used to describe the moduli space of ideal quadrilaterals. In general, we expect the moduli space of ideal n-gons to have dimension n − 3. We can think of the n as the number of parameters describing vertices and the 3 as the dimension of PSL2 (R). There ...
... or the cross-ratio of a, b, c, d. This can be used to describe the moduli space of ideal quadrilaterals. In general, we expect the moduli space of ideal n-gons to have dimension n − 3. We can think of the n as the number of parameters describing vertices and the 3 as the dimension of PSL2 (R). There ...
examples of non-polygonal limit shapes in iid first
... and hence with positive probability (which can be made arbitrarily close to 1 by decreasing ε), for each i there is a positive density of n such that BLi +nvi (xj , xi ) > 0 for all j 6= i. For such an n, take yi,n ∈ Li + nvi to be the closest point (in the sense of passage times) to xi ; assuming u ...
... and hence with positive probability (which can be made arbitrarily close to 1 by decreasing ε), for each i there is a positive density of n such that BLi +nvi (xj , xi ) > 0 for all j 6= i. For such an n, take yi,n ∈ Li + nvi to be the closest point (in the sense of passage times) to xi ; assuming u ...
11 Neutral Geometry III (Comparing geometries we`ve studied)
... triangle with vertices at A(0, 0), B(1, 1), and C(−1, 1), together with the 2-2-4 right triangle with vertices at D(3, 0), E(5, 0), and F (3, 2). Clearly we can apply 4DEF to 4ABC with D placed atop point A, side DE along AB and F on the same side of this common line as C. So does E coincide with B? ...
... triangle with vertices at A(0, 0), B(1, 1), and C(−1, 1), together with the 2-2-4 right triangle with vertices at D(3, 0), E(5, 0), and F (3, 2). Clearly we can apply 4DEF to 4ABC with D placed atop point A, side DE along AB and F on the same side of this common line as C. So does E coincide with B? ...
Hyperbolic geometry 2 1
... Direct isometries of form z ֏ pz (p > 0) are dilations centred at (0 , 0) . Note that they fix two points on the boundary, namely the origin and ∞ , but don’t fix any point in H2 . In general, direct isometries that fix two points on the boundary (and no points in H2 ) are called hyperbolic isometri ...
... Direct isometries of form z ֏ pz (p > 0) are dilations centred at (0 , 0) . Note that they fix two points on the boundary, namely the origin and ∞ , but don’t fix any point in H2 . In general, direct isometries that fix two points on the boundary (and no points in H2 ) are called hyperbolic isometri ...
Introduction to Teichmüller Spaces
... of S and β a seam. Let (X, f ) ∈ T1,1 . As shown in Figure 4, then the map f identifies α with a curve (also called) α in X. Let ` = `X (α) be the length of the unique geodesic in X in the homotopy class of α. As seen on the right side of the figure, in hyperbolic geometry, there exists a unique arc ...
... of S and β a seam. Let (X, f ) ∈ T1,1 . As shown in Figure 4, then the map f identifies α with a curve (also called) α in X. Let ` = `X (α) be the length of the unique geodesic in X in the homotopy class of α. As seen on the right side of the figure, in hyperbolic geometry, there exists a unique arc ...
Explaining Data in High-Dimensional Space
... correspond to the eigen vectors and values of its scatter matrix. Direction of thinness: A short axis defines a direction in which the data does not extend. If V is a zero-valued eigen vector, then it defines a constraint for any data point x: Vx=0 ...
... correspond to the eigen vectors and values of its scatter matrix. Direction of thinness: A short axis defines a direction in which the data does not extend. If V is a zero-valued eigen vector, then it defines a constraint for any data point x: Vx=0 ...
08. Non-Euclidean Geometry 1. Euclidean Geometry
... • Euclidean geometry is "flat". Spherical geometry is "positively curved". ...
... • Euclidean geometry is "flat". Spherical geometry is "positively curved". ...
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).