Do every problem. For full credit, be sure to show all your work. The

... Instructions: Do every problem. For full credit, be sure to show all your work. The point is to show me that you know HOW to do the problems, not that you can get the right answer, possibly by accident. ...

... Instructions: Do every problem. For full credit, be sure to show all your work. The point is to show me that you know HOW to do the problems, not that you can get the right answer, possibly by accident. ...

Applied Math Seminar The Geometry of Data Spring 2015

... that as data is spread into high dimensions, the distance between points becomes large and the corresponding density very low and difficult to estimate. In order to avoid this issue, one imagines that only the data representation is high dimensional but the data actually lies along curved low-dimens ...

... that as data is spread into high dimensions, the distance between points becomes large and the corresponding density very low and difficult to estimate. In order to avoid this issue, one imagines that only the data representation is high dimensional but the data actually lies along curved low-dimens ...

Lab 3

... So far, we know that in hyperbolic geometries, triangles' angles add up to less than 180 degrees, and in elliptic geometries, triangles' angles add up to more than 180 degrees. In both cases, the difference between that sum and 180 degrees gives the area of the triangle, up to a constant. For the fo ...

... So far, we know that in hyperbolic geometries, triangles' angles add up to less than 180 degrees, and in elliptic geometries, triangles' angles add up to more than 180 degrees. In both cases, the difference between that sum and 180 degrees gives the area of the triangle, up to a constant. For the fo ...

Statistical Analysis of Shapes of Curves and Surfaces

... I will present examples of generating Bayesian inferences from image data. The case of analyzing shapes of surfaces, e.g. facial surfaces, is much more diﬃcult. Our approach is to represent a surface as an indexed collection of curves and to extend ideas from curve analysis to perform surface analys ...

... I will present examples of generating Bayesian inferences from image data. The case of analyzing shapes of surfaces, e.g. facial surfaces, is much more diﬃcult. Our approach is to represent a surface as an indexed collection of curves and to extend ideas from curve analysis to perform surface analys ...

11.2-11.3 – Surface area for pyramids, cones, prisms

... 2011 Pearson-Prentice Hall Geometry text, pages 713 & 714 ...

... 2011 Pearson-Prentice Hall Geometry text, pages 713 & 714 ...

Honors Geometry Chapter 12 You Can . . . Find the ratio of surface

... Find the ratio of surface areas or volumes of two different solids with equal width and height Find the surface area and volume of a prism, pyramid, cylinder, cone, and/or sphere Use proportions to find missing linear measurements, areas, and/or volumes of similar solids (12.7) Find the volu ...

... Find the ratio of surface areas or volumes of two different solids with equal width and height Find the surface area and volume of a prism, pyramid, cylinder, cone, and/or sphere Use proportions to find missing linear measurements, areas, and/or volumes of similar solids (12.7) Find the volu ...

Differential geometry of surfaces in Euclidean space

... us assume the n-dimensional space Rn whose coordinates will be labelled with capital Roman indices, xA . Consider now an m-dimensional (m ≤ n) surface Σ embedded in Rn . It can be parameterized by a set of m “curvilinear” coordinates, denoted using Greek indices, y µ ; the surface is defined by givi ...

... us assume the n-dimensional space Rn whose coordinates will be labelled with capital Roman indices, xA . Consider now an m-dimensional (m ≤ n) surface Σ embedded in Rn . It can be parameterized by a set of m “curvilinear” coordinates, denoted using Greek indices, y µ ; the surface is defined by givi ...

Slide 1

... of area K and an area III. The areas I+II+III+K = 2π The areas of Lunes A,B and C sum to 3K+I+II+III = 2 (A+B+C). Subtracting the first equation from the second yields 2K=2(A+B+C)-2π, or K = (A+B+C)-π. ...

... of area K and an area III. The areas I+II+III+K = 2π The areas of Lunes A,B and C sum to 3K+I+II+III = 2 (A+B+C). Subtracting the first equation from the second yields 2K=2(A+B+C)-2π, or K = (A+B+C)-π. ...

Internal geometry of surfaces

... If the sum of the interior angles α and β is less than 180◦ , the two straight lines, produced infinitely, meet on that side. ...

... If the sum of the interior angles α and β is less than 180◦ , the two straight lines, produced infinitely, meet on that side. ...

Einstein memorial lecture.

... 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 geodesics, as curves which roll out to straight lines, coincides with the ...

... 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 geodesics, as curves which roll out to straight lines, coincides with the ...

First Page

... Abstract. We study a geometric free-boundary problem for a bicrystal in which a grain boundary is attached at a groove root to the exterior surface of the bicrystal. Mathematically, this geometric problem couples motion by mean curvature of the grain boundary with surface diﬀusion of the exterior su ...

... Abstract. We study a geometric free-boundary problem for a bicrystal in which a grain boundary is attached at a groove root to the exterior surface of the bicrystal. Mathematically, this geometric problem couples motion by mean curvature of the grain boundary with surface diﬀusion of the exterior su ...

Spatial Data Coordinates and Map Projections

... a surface, what three types of “oids” are used to provide different levels of “surface geometry” detail? Dent Figure 2.1 (c) Geoid – earth-like (best 3-D model of earth) (b) Ellipsoid – oblate sphere (oblate means somewhat squashed), better for map making (a) Spheroid – sphere-like (simplest model), ...

... a surface, what three types of “oids” are used to provide different levels of “surface geometry” detail? Dent Figure 2.1 (c) Geoid – earth-like (best 3-D model of earth) (b) Ellipsoid – oblate sphere (oblate means somewhat squashed), better for map making (a) Spheroid – sphere-like (simplest model), ...

Entropy Euclidean Axioms (Postulates) Parallel Postulate Curved

... S can also be written as S = N kB log(W) N: the number of particles in the system kB: the Boltzmann constant W: the number of possible states in the system ...

... S can also be written as S = N kB log(W) N: the number of particles in the system kB: the Boltzmann constant W: the number of possible states in the system ...

a) See the second attach b) Two teams, one from tower A and

... two rays are. We could put two points A and B on the two rays, and the angle doesn’t ...

... two rays are. We could put two points A and B on the two rays, and the angle doesn’t ...

Math 53 Symmetry and Tiling

... Use the templates to tape together one black heptagon and two white hexagons at each vertex. Hints: I ...

... Use the templates to tape together one black heptagon and two white hexagons at each vertex. Hints: I ...

Branches of differential geometry

... approximation. Various concepts based on length, such the arc length of curves, area of plane regions, and volume of solids all admit natural analogues in Riemannian geometry. The notion of a directional derivative of a function from the multivariable calculus is extended in Riemannian geometry to t ...

... approximation. Various concepts based on length, such the arc length of curves, area of plane regions, and volume of solids all admit natural analogues in Riemannian geometry. The notion of a directional derivative of a function from the multivariable calculus is extended in Riemannian geometry to t ...

Non-Euclidean Geometry

... •Cylindrical surface Euclidean theorems continue to hold. •Model of Riemann’s non Euclidean geometry: spherical surface. ...

... •Cylindrical surface Euclidean theorems continue to hold. •Model of Riemann’s non Euclidean geometry: spherical surface. ...

MATH 301 Survey of Geometries Homework Problems – Week 5

... 7.4 Suppose that r and r0 are geodesic lines or line segments that intersect a common geodesic l at respective points P and P 0 . If the corresponding angles made by r with l at P and r0 with l at P 0 are equal, we say that r and r0 are parallel transports of each other along l. (You should imagine ...

... 7.4 Suppose that r and r0 are geodesic lines or line segments that intersect a common geodesic l at respective points P and P 0 . If the corresponding angles made by r with l at P and r0 with l at P 0 are equal, we say that r and r0 are parallel transports of each other along l. (You should imagine ...

Euclid`s Postulates We have been playing with Non

... We have been playing with Non-Euclidean geometry for a while without discussing the underlying reason why all these strange geometries can exist. It all began with a guy named Euclid. On a dark and stormy night. . . (sorry, wrong story). A long time ago, in a galaxy far, far away. . . (again, wrong ...

... We have been playing with Non-Euclidean geometry for a while without discussing the underlying reason why all these strange geometries can exist. It all began with a guy named Euclid. On a dark and stormy night. . . (sorry, wrong story). A long time ago, in a galaxy far, far away. . . (again, wrong ...

Hyperbolic Spaces

... Lines can be drawn in hyperbolic space that are parallel (do not intersect). Actually, many lines can be drawn parallel to a given line through a given point. ...

... Lines can be drawn in hyperbolic space that are parallel (do not intersect). Actually, many lines can be drawn parallel to a given line through a given point. ...

File

... The environment we live in is 3-dimensional and geometry is the natural language to express concepts and relationships of space. We give names to help us identify shapes in space (3D) and in a plane (2D). We classify objects according to certain attributes. We discover properties and relationships w ...

... The environment we live in is 3-dimensional and geometry is the natural language to express concepts and relationships of space. We give names to help us identify shapes in space (3D) and in a plane (2D). We classify objects according to certain attributes. We discover properties and relationships w ...

Euclidean vs Non-Euclidean Geometry

... two-dimensional plane that are both perpendicular to a third line: ...

... two-dimensional plane that are both perpendicular to a third line: ...

Lecture 6

... • There exists at least one pair of equidistant straight lines. • Through a given point not on a given straight line, and not on that straight line produced, no more than one parallel straight line can be drawn. (Playfair, late 18th century) ...

... • There exists at least one pair of equidistant straight lines. • Through a given point not on a given straight line, and not on that straight line produced, no more than one parallel straight line can be drawn. (Playfair, late 18th century) ...

zero and infinity in the non euclidean geometry

... • "It is this similarity between the whole and its parts, even infinitesimal ones, that makes us consider this curve of von Koch as a line truly marvelous among all. If it were gifted with life, it would not be possible to destroy it without annihilating it whole, for it would be continually reborn ...

... • "It is this similarity between the whole and its parts, even infinitesimal ones, that makes us consider this curve of von Koch as a line truly marvelous among all. If it were gifted with life, it would not be possible to destroy it without annihilating it whole, for it would be continually reborn ...

In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric.Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. On the other hand extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear Euler–Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding.