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116-502: The HEALPix projection is a general class of spherical projections, sharing several key properties, which map the 2- sphere to the Euclidean plane . Any of these can be followed by partitioning (pixelising) the resulting region of the 2-plane. In particular, when one of these projections (the H=4, K=3 HEALPix projection) is followed by a pixelisation of the 2-plane, the result is generally known as

232-456: A Fields medal , made new impacts in mathematics by using topological quantum field theory and string theory to make predictions and provide frameworks for new rigorous mathematics, which has resulted for example in the conjectural mirror symmetry and the Seiberg–Witten invariants . Riemannian geometry studies Riemannian manifolds , smooth manifolds with a Riemannian metric . This is

348-487: A directional derivative of a function from multivariable calculus is extended to the notion of a covariant derivative of a tensor . Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry . This notion can also be defined locally , i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However,

464-447: A vector bundle endomorphism (called an almost complex structure ) It follows from this definition that an almost complex manifold is even-dimensional. An almost complex manifold is called complex if N J = 0 {\displaystyle N_{J}=0} , where N J {\displaystyle N_{J}} is a tensor of type (2, 1) related to J {\displaystyle J} , called

580-595: A concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the first order of approximation . Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry. The notion of

696-756: A continuous curved surface onto a plane. The most well-known map projection is the Mercator projection . This map projection has the property of being conformal . However, it has been criticized throughout the 20th century for enlarging regions further from the equator. To contrast, equal-area projections such as the Sinusoidal projection and the Gall–Peters projection show the correct sizes of countries relative to each other, but distort angles. The National Geographic Society and most atlases favor map projections that compromise between area and angular distortion, such as

812-458: A course of constant bearing is always plotted as a straight line. A normal cylindrical projection is any projection in which meridians are mapped to equally spaced vertical lines and circles of latitude (parallels) are mapped to horizontal lines. The mapping of meridians to vertical lines can be visualized by imagining a cylinder whose axis coincides with the Earth's axis of rotation. This cylinder

928-416: A cylindrical projection (for example) is one which: (If you rotate the globe before projecting then the parallels and meridians will not necessarily still be straight lines. Rotations are normally ignored for the purpose of classification.) Where the light source emanates along the line described in this last constraint is what yields the differences between the various "natural" cylindrical projections. But

1044-608: A general class of spherical projections) is represented by the keyword HPX in the FITS standard for writing astronomical data files. It was approved as part of the official FITS World Coordinate System (WCS) by the International Astronomical Union FITS Working Group on April 26, 2006. The spherical projection combines a cylindrical equal area projection, the Lambert cylindrical equal-area projection , for

1160-540: A given level in the hierarchy are of similar but not identical size. The scheme is good at representing complex shapes because the boundaries are all segments of circles of the sphere . Another alternative hierarchical grid is the Quadrilateralized Spherical Cube . The 12 "base resolution pixels" of H=4, K=3 HEALPix projection may be thought of as the facets of a rhombic dodecahedron . The H=6 HEALPix has similarities to another alternative grid based on

1276-403: A given point, using the scale factor h along the meridian, the scale factor k along the parallel, and the angle θ ′ between them, Nicolas Tissot described how to construct an ellipse that illustrates the amount and orientation of the components of distortion. By spacing the ellipses regularly along the meridians and parallels, the network of indicatrices shows how distortion varies across

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1392-450: A light source at some definite point relative to the globe and projecting its features onto a specified surface. Although most projections are not defined in this way, picturing the light source-globe model can be helpful in understanding the basic concept of a map projection. A surface that can be unfolded or unrolled into a plane or sheet without stretching, tearing or shrinking is called a developable surface . The cylinder , cone and

1508-428: A manifold, as even the notion of a topological space had not been encountered, but he did propose that it might be possible to investigate or measure the properties of the metric of spacetime through the analysis of masses within spacetime, linking with the earlier observation of Euler that masses under the effect of no forces would travel along geodesics on surfaces, and predicting Einstein's fundamental observation of

1624-448: A necessary step in creating a two-dimensional map and is one of the essential elements of cartography. All projections of a sphere on a plane necessarily distort the surface in some way. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. The study of map projections

1740-504: A new interpretation of Euler's theorem in terms of the principle curvatures, which is the modern form of the equation. The field of differential geometry became an area of study considered in its own right, distinct from the more broad idea of analytic geometry, in the 1800s, primarily through the foundational work of Carl Friedrich Gauss and Bernhard Riemann , and also in the important contributions of Nikolai Lobachevsky on hyperbolic geometry and non-Euclidean geometry and throughout

1856-461: A nondegenerate 2- form ω , called the symplectic form . A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: d ω = 0 . A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism . Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2,

1972-530: A number of mathematical properties which make it efficient for certain computations, e.g. spherical harmonic transforms . In the case of the H=4, K=3 projection, the pixels are squares in the plane (which can be inversely projected back to quadrilaterals with non-geodesic sides on the 2-sphere) and every vertex joins four pixels, with the exception of eight vertices which each join only three pixels. The latitude of transition between equatorial-orthogonal and polar-convergent longitude lines has been selected to allow

2088-416: A parallel of origin (usually written φ 0 ) are often used to define the origin of the map projection. A globe is the only way to represent the Earth with constant scale throughout the entire map in all directions. A map cannot achieve that property for any area, no matter how small. It can, however, achieve constant scale along specific lines. Some possible properties are: Projection construction

2204-475: A single point necessarily involves choosing priorities to reach a compromise. Some schemes use distance distortion as a proxy for the combination of angular deformation and areal inflation; such methods arbitrarily choose what paths to measure and how to weight them in order to yield a single result. Many have been described. The creation of a map projection involves two steps: Some of the simplest map projections are literal projections, as obtained by placing

2320-422: A sphere's surface cannot be represented on a plane without distortion. The same applies to other reference surfaces used as models for the Earth, such as oblate spheroids , ellipsoids , and geoids . Since any map projection is a representation of one of those surfaces on a plane, all map projections distort. The classical way of showing the distortion inherent in a projection is to use Tissot's indicatrix . For

2436-515: A symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange on analytical mechanics and later in Carl Gustav Jacobi 's and William Rowan Hamilton 's formulations of classical mechanics . By contrast with Riemannian geometry, where

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2552-521: A theorem expressing the curvature of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem . Later in the 1700s, the new French school led by Gaspard Monge began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studied surfaces of revolution and envelopes of plane curves and space curves. Several students of Monge made contributions to this same theory, and for example Charles Dupin provided

2668-470: A well-known standard definition of metric and parallelism. In Riemannian geometry , the Levi-Civita connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be spacetime and the bundles and connections are related to various physical fields. From

2784-423: A year later Tullio Levi-Civita and Gregorio Ricci-Curbastro produced their textbook systematically developing the theory of absolute differential calculus and tensor calculus . It was in this language that differential geometry was used by Einstein in the development of general relativity and pseudo-Riemannian geometry . The subject of modern differential geometry emerged from the early 1900s in response to

2900-459: Is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds . It uses the techniques of differential calculus , integral calculus , linear algebra and multilinear algebra . The field has its origins in the study of spherical geometry as far back as antiquity . It also relates to astronomy , the geodesy of the Earth , and later

3016-401: Is a function F : T M → [ 0 , ∞ ) {\displaystyle F:\mathrm {T} M\to [0,\infty )} such that: Symplectic geometry is the study of symplectic manifolds . An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e.,

3132-475: Is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem .) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by

3248-709: Is according to properties of the model they preserve. Some of the more common categories are: Because the sphere is not a developable surface , it is impossible to construct a map projection that is both equal-area and conformal. The three developable surfaces (plane, cylinder, cone) provide useful models for understanding, describing, and developing map projections. However, these models are limited in two fundamental ways. For one thing, most world projections in use do not fall into any of those categories. For another thing, even most projections that do fall into those categories are not naturally attainable through physical projection. As L. P. Lee notes, No reference has been made in

3364-399: Is also affected by how the shape of the Earth or planetary body is approximated. In the following section on projection categories, the earth is taken as a sphere in order to simplify the discussion. However, the Earth's actual shape is closer to an oblate ellipsoid . Whether spherical or ellipsoidal, the principles discussed hold without loss of generality. Selecting a model for a shape of

3480-412: Is given by all the smooth complex projective varieties . CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds . Conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space. Differential topology is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from

3596-445: Is given by φ): In the first case (Mercator), the east-west scale always equals the north-south scale. In the second case (central cylindrical), the north-south scale exceeds the east-west scale everywhere away from the equator. Each remaining case has a pair of secant lines —a pair of identical latitudes of opposite sign (or else the equator) at which the east-west scale matches the north-south-scale. Normal cylindrical projections map

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3712-473: Is intimately linked to the development of geometry more generally, of the notion of space and shape, and of topology , especially the study of manifolds . In this section we focus primarily on the history of the application of infinitesimal methods to geometry, and later to the ideas of tangent spaces , and eventually the development of the modern formalism of the subject in terms of tensors and tensor fields . The study of differential geometry, or at least

3828-410: Is not limited to perspective projections, such as those resulting from casting a shadow on a screen, or the rectilinear image produced by a pinhole camera on a flat film plate. Rather, any mathematical function that transforms coordinates from the curved surface distinctly and smoothly to the plane is a projection. Few projections in practical use are perspective. Most of this article assumes that

3944-657: Is one which defines some notion of size, distance, shape, volume, or other rigidifying structure. For example, in Riemannian geometry distances and angles are specified, in symplectic geometry volumes may be computed, in conformal geometry only angles are specified, and in gauge theory certain fields are given over the space. Differential geometry is closely related to, and is sometimes taken to include, differential topology , which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on

4060-417: Is primarily about the characterization of their distortions. There is no limit to the number of possible map projections. More generally, projections are considered in several fields of pure mathematics, including differential geometry , projective geometry , and manifolds . However, the term "map projection" refers specifically to a cartographic projection. Despite the name's literal meaning, projection

4176-423: Is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i.e. a notion of parallel transport . An important example is provided by affine connections . For a surface in R , tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has

4292-417: Is the Levi-Civita connection of g {\displaystyle g} . In this case, ( J , g ) {\displaystyle (J,g)} is called a Kähler structure , and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold . A large class of Kähler manifolds (the class of Hodge manifolds )

4408-452: Is the Riemannian symmetric spaces , whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry . Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite . A special case of this is a Lorentzian manifold , which is

4524-467: Is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics and physical gauge theories which underpin the standard model of particle physics . Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli spaces of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as

4640-477: Is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields . Beside the structure theory there is also the wide field of representation theory . Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. Gauge theory

4756-456: Is wrapped around the Earth, projected onto, and then unrolled. By the geometry of their construction, cylindrical projections stretch distances east-west. The amount of stretch is the same at any chosen latitude on all cylindrical projections, and is given by the secant of the latitude as a multiple of the equator's scale. The various cylindrical projections are distinguished from each other solely by their north-south stretching (where latitude

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4872-521: The Bernoulli brothers , Jacob and Johann made important early contributions to the use of infinitesimals to study geometry. In lectures by Johann Bernoulli at the time, later collated by L'Hopital into the first textbook on differential calculus , the tangents to plane curves of various types are computed using the condition d y = 0 {\displaystyle dy=0} , and similarly points of inflection are calculated. At this same time

4988-482: The Collignon projection in polar areas. The term "conic projection" is used to refer to any projection in which meridians are mapped to equally spaced lines radiating out from the apex and circles of latitude (parallels) are mapped to circular arcs centered on the apex. When making a conic map, the map maker arbitrarily picks two standard parallels. Those standard parallels may be visualized as secant lines where

5104-529: The Euler–Lagrange equations and the first theory of the calculus of variations , which underpins in modern differential geometry many techniques in symplectic geometry and geometric analysis . This theory was used by Lagrange , a co-developer of the calculus of variations, to derive the first differential equation describing a minimal surface in terms of the Euler–Lagrange equation. In 1760 Euler proved

5220-462: The Euler–Lagrange equations describing the equations of motion of certain physical systems in quantum field theory , and so their study is of considerable interest in physics. The apparatus of vector bundles , principal bundles , and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle . Loosely speaking, this structure by itself

5336-447: The Mercator projection as a way of mapping the Earth. Mercator had an understanding of the advantages and pitfalls of his map design, and in particular was aware of the conformal nature of his projection, as well as the difference between praga , the lines of shortest distance on the Earth, and the directio , the straight line paths on his map. Mercator noted that the praga were oblique curvatur in this projection. This fact reflects

5452-541: The Nijenhuis tensor (or sometimes the torsion ). An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas . An almost Hermitian structure is given by an almost complex structure J , along with a Riemannian metric g , satisfying the compatibility condition An almost Hermitian structure defines naturally a differential two-form The following two conditions are equivalent: where ∇ {\displaystyle \nabla }

5568-501: The Poincaré conjecture . During this same period primarily due to the influence of Michael Atiyah , new links between theoretical physics and differential geometry were formed. Techniques from the study of the Yang–Mills equations and gauge theory were used by mathematicians to develop new invariants of smooth manifolds. Physicists such as Edward Witten , the only physicist to be awarded

5684-663: The Robinson projection and the Winkel tripel projection . Many properties can be measured on the Earth's surface independently of its geography: Map projections can be constructed to preserve some of these properties at the expense of others. Because the Earth's curved surface is not isometric to a plane, preservation of shapes inevitably requires a variable scale and, consequently, non-proportional presentation of areas. Similarly, an area-preserving projection can not be conformal , resulting in shapes and bearings distorted in most places of

5800-565: The Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds

5916-621: The curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré–Birkhoff theorem , conjectured by Henri Poincaré and then proved by G.D. Birkhoff in 1912. It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then

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6032-447: The equivalence principle a full 60 years before it appeared in the scientific literature. In the wake of Riemann's new description, the focus of techniques used to study differential geometry shifted from the ad hoc and extrinsic methods of the study of curves and surfaces to a more systematic approach in terms of tensor calculus and Klein's Erlangen program, and progress increased in the field. The notion of groups of transformations

6148-400: The icosahedron . Map projection In cartography , a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane . In a map projection, coordinates , often expressed as latitude and longitude , of locations from the surface of the globe are transformed to coordinates on a plane. Projection is

6264-401: The orthogonality between the osculating circles of a plane curve and the tangent directions is realised, and the first analytical formula for the radius of an osculating circle, essentially the first analytical formula for the notion of curvature , is written down. In the wake of the development of analytic geometry and plane curves, Alexis Clairaut began the study of space curves at just

6380-399: The 1600s. Around this time there were only minimal overt applications of the theory of infinitesimals to the study of geometry, a precursor to the modern calculus-based study of the subject. In Euclid 's Elements the notion of tangency of a line to a circle is discussed, and Archimedes applied the method of exhaustion to compute the areas of smooth shapes such as the circle , and

6496-531: The 1860s, and Felix Klein coined the term non-Euclidean geometry in 1871, and through the Erlangen program put Euclidean and non-Euclidean geometries on the same footing. Implicitly, the spherical geometry of the Earth that had been studied since antiquity was a non-Euclidean geometry, an elliptic geometry . The development of intrinsic differential geometry in the language of Gauss was spurred on by his student, Bernhard Riemann in his Habilitationsschrift , On

6612-463: The Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system. Complex differential geometry is the study of complex manifolds . An almost complex manifold is a real manifold M {\displaystyle M} , endowed with a tensor of type (1, 1), i.e.

6728-454: The Earth involves choosing between the advantages and disadvantages of a sphere versus an ellipsoid. Spherical models are useful for small-scale maps such as world atlases and globes, since the error at that scale is not usually noticeable or important enough to justify using the more complicated ellipsoid. The ellipsoidal model is commonly used to construct topographic maps and for other large- and medium-scale maps that need to accurately depict

6844-482: The HEALPix pixelisation, which is widely used in physical cosmology for maps of the cosmic microwave background . This pixelisation can be thought of as mapping the sphere to twelve square facets (diamonds) on the plane followed by the binary division of these facets into pixels, though it can be derived without using the projection. The associated software package HEALPix implements the algorithm. The HEALPix projection (as

6960-494: The above definitions to cylinders, cones or planes. The projections are termed cylindric or conic because they can be regarded as developed on a cylinder or a cone, as the case may be, but it is as well to dispense with picturing cylinders and cones, since they have given rise to much misunderstanding. Particularly is this so with regard to the conic projections with two standard parallels: they may be regarded as developed on cones, but they are cones which bear no simple relationship to

7076-448: The age of 16. In his book Clairaut introduced the notion of tangent and subtangent directions to space curves in relation to the directions which lie along a surface on which the space curve lies. Thus Clairaut demonstrated an implicit understanding of the tangent space of a surface and studied this idea using calculus for the first time. Importantly Clairaut introduced the terminology of curvature and double curvature , essentially

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7192-403: The beginning and through the middle of the 19th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with

7308-494: The central meridian. Therefore, meridians are equally spaced along a given parallel. On a pseudocylindrical map, any point further from the equator than some other point has a higher latitude than the other point, preserving north-south relationships. This trait is useful when illustrating phenomena that depend on latitude, such as climate. Examples of pseudocylindrical projections include: The HEALPix projection combines an equal-area cylindrical projection in equatorial regions with

7424-407: The cone intersects the globe—or, if the map maker chooses the same parallel twice, as the tangent line where the cone is tangent to the globe. The resulting conic map has low distortion in scale, shape, and area near those standard parallels. Distances along the parallels to the north of both standard parallels or to the south of both standard parallels are stretched; distances along parallels between

7540-414: The development of quantum field theory and the standard model of particle physics . Outside of physics, differential geometry finds applications in chemistry , economics , engineering , control theory , computer graphics and computer vision , and recently in machine learning . The history and development of differential geometry as a subject begins at least as far back as classical antiquity . It

7656-476: The development of the modern theory, including Jean-Louis Koszul who introduced connections on vector bundles , Shiing-Shen Chern who introduced characteristic classes to the subject and began the study of complex manifolds , Sir William Vallance Douglas Hodge and Georges de Rham who expanded understanding of differential forms , Charles Ehresmann who introduced the theory of fibre bundles and Ehresmann connections , and others. Of particular importance

7772-443: The distinction between the two subjects). Differential geometry is also related to the geometric aspects of the theory of differential equations , otherwise known as geometric analysis . Differential geometry finds applications throughout mathematics and the natural sciences . Most prominently the language of differential geometry was used by Albert Einstein in his theory of general relativity , and subsequently by physicists in

7888-445: The equator and not a meridian. Pseudocylindrical projections represent the central meridian as a straight line segment. Other meridians are longer than the central meridian and bow outward, away from the central meridian. Pseudocylindrical projections map parallels as straight lines. Along parallels, each point from the surface is mapped at a distance from the central meridian that is proportional to its difference in longitude from

8004-411: The equatorial regions of the sphere and a pseudocylindrical equal area projection, an interrupted Collignon projection , for the polar regions. At a given level in the hierarchy the pixels are of equal area (which is done by bisecting the square in the case of the H=4, K=3 projection) and their centers lie on a discrete number of circles of latitude, with equal spacing on each circle. The scheme has

8120-496: The first set of intrinsic coordinate systems on a surface, beginning the theory of intrinsic geometry upon which modern geometric ideas are based. Around this time Euler's study of mechanics in the Mechanica lead to the realization that a mass traveling along a surface not under the effect of any force would traverse a geodesic path, an early precursor to the important foundational ideas of Einstein's general relativity , and also to

8236-579: The folding of the projection into a perfect cube — "cubing the sphere"; indeed in this way the Arctic Circle becomes a square. The pixelisation related to the H=4, K=3 projection has become widely used in cosmology for storing and manipulating maps of the cosmic microwave background . Gaia mission uses HEALPix as the basis for source identification. An alternative hierarchical grid is the Hierarchical Triangular Mesh (HTM). The pixels at

8352-406: The foundational contributions of many mathematicians, including importantly the work of Henri Poincaré on the foundations of topology . At the start of the 1900s there was a major movement within mathematics to formalise the foundational aspects of the subject to avoid crises of rigour and accuracy, known as Hilbert's program . As part of this broader movement, the notion of a topological space

8468-433: The geoid are used to project maps from. Other regular solids are sometimes used as generalizations for smaller bodies' geoidal equivalent. For example, Io is better modeled by triaxial ellipsoid or prolated spheroid with small eccentricities. Haumea 's shape is a Jacobi ellipsoid , with its major axis twice as long as its minor and with its middle axis one and half times as long as its minor. See map projection of

8584-429: The globe never preserves or optimizes metric properties, so that possibility is not discussed further here. Tangent and secant lines ( standard lines ) are represented undistorted. If these lines are a parallel of latitude, as in conical projections, it is called a standard parallel . The central meridian is the meridian to which the globe is rotated before projecting. The central meridian (usually written λ 0 ) and

8700-417: The hypotheses which lie at the foundation of geometry . In this work Riemann introduced the notion of a Riemannian metric and the Riemannian curvature tensor for the first time, and began the systematic study of differential geometry in higher dimensions. This intrinsic point of view in terms of the Riemannian metric, denoted by d s 2 {\displaystyle ds^{2}} by Riemann,

8816-507: The inventor of intrinsic differential geometry. In his fundamental paper Gauss introduced the Gauss map , Gaussian curvature , first and second fundamental forms , proved the Theorema Egregium showing the intrinsic nature of the Gaussian curvature, and studied geodesics, computing the area of a geodesic triangle in various non-Euclidean geometries on surfaces. At this time Gauss was already of

8932-610: The lack of a metric-preserving map of the Earth's surface onto a flat plane, a consequence of the later Theorema Egregium of Gauss . The first systematic or rigorous treatment of geometry using the theory of infinitesimals and notions from calculus began around the 1600s when calculus was first developed by Gottfried Leibniz and Isaac Newton . At this time, the recent work of René Descartes introducing analytic coordinates to geometry allowed geometric shapes of increasing complexity to be described rigorously. In particular around this time Pierre de Fermat , Newton, and Leibniz began

9048-498: The land surface. Auxiliary latitudes are often employed in projecting the ellipsoid. A third model is the geoid , a more complex and accurate representation of Earth's shape coincident with what mean sea level would be if there were no winds, tides, or land. Compared to the best fitting ellipsoid, a geoidal model would change the characterization of important properties such as distance, conformality and equivalence . Therefore, in geoidal projections that preserve such properties,

9164-405: The level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point p , a hyperplane distribution is determined by a nowhere vanishing 1-form α {\displaystyle \alpha } , which is unique up to multiplication by a nowhere vanishing function: A local 1-form on M is a contact form if

9280-399: The map has at least two fixed points. Contact geometry deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A contact structure on a (2 n + 1) -dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with

9396-505: The map. Many other ways have been described of showing the distortion in projections. Like Tissot's indicatrix, the Goldberg-Gott indicatrix is based on infinitesimals, and depicts flexion and skewness (bending and lopsidedness) distortions. Rather than the original (enlarged) infinitesimal circle as in Tissot's indicatrix, some visual methods project finite shapes that span a part of

9512-568: The map. Data sets are geographic information; their collection depends on the chosen datum (model) of the Earth. Different datums assign slightly different coordinates to the same location, so in large scale maps, such as those from national mapping systems, it is important to match the datum to the projection. The slight differences in coordinate assignation between different datums is not a concern for world maps or those of large regions, where such differences are reduced to imperceptibility. Carl Friedrich Gauss 's Theorema Egregium proved that

9628-415: The map. Each projection preserves, compromises, or approximates basic metric properties in different ways. The purpose of the map determines which projection should form the base for the map. Because maps have many different purposes, a diversity of projections have been created to suit those purposes. Another consideration in the configuration of a projection is its compatibility with data sets to be used on

9744-429: The map. For example, a small circle of fixed radius (e.g., 15 degrees angular radius ). Sometimes spherical triangles are used. In the first half of the 20th century, projecting a human head onto different projections was common to show how distortion varies across one projection as compared to another. In dynamic media, shapes of familiar coastlines and boundaries can be dragged across an interactive map to show how

9860-427: The mapped graticule would deviate from a mapped ellipsoid's graticule. Normally the geoid is not used as an Earth model for projections, however, because Earth's shape is very regular, with the undulation of the geoid amounting to less than 100 m from the ellipsoidal model out of the 6.3 million m Earth radius . For irregular planetary bodies such as asteroids , however, sometimes models analogous to

9976-411: The mathematical basis of Einstein's general relativity theory of gravity . Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric , that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M {\displaystyle M}

10092-413: The natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms . Beside Lie algebroids , also Courant algebroids start playing a more important role. A Lie group is a group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which

10208-621: The notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma } for the Christoffel symbols, both coming from G in Gravitation . Élie Cartan helped reformulate the foundations of the differential geometry of smooth manifolds in terms of exterior calculus and the theory of moving frames , leading in the world of physics to Einstein–Cartan theory . Following this early development, many mathematicians contributed to

10324-434: The notion of principal curvatures later studied by Gauss and others. Around this same time, Leonhard Euler , originally a student of Johann Bernoulli, provided many significant contributions not just to the development of geometry, but to mathematics more broadly. In regards to differential geometry, Euler studied the notion of a geodesic on a surface deriving the first analytical geodesic equation , and later introduced

10440-476: The opinion that the standard paradigm of Euclidean geometry should be discarded, and was in possession of private manuscripts on non-Euclidean geometry which informed his study of geodesic triangles. Around this same time János Bolyai and Lobachevsky independently discovered hyperbolic geometry and thus demonstrated the existence of consistent geometries outside Euclid's paradigm. Concrete models of hyperbolic geometry were produced by Eugenio Beltrami later in

10556-403: The plane are all developable surfaces. The sphere and ellipsoid do not have developable surfaces, so any projection of them onto a plane will have to distort the image. (To compare, one cannot flatten an orange peel without tearing and warping it.) One way of describing a projection is first to project from the Earth's surface to a developable surface such as a cylinder or cone, and then to unroll

10672-402: The projection distorts sizes and shapes according to position on the map. Another way to visualize local distortion is through grayscale or color gradations whose shade represents the magnitude of the angular deformation or areal inflation. Sometimes both are shown simultaneously by blending two colors to create a bivariate map . To measure distortion globally across areas instead of at just

10788-468: The projection surface into a flat map. The most common projection surfaces are cylindrical (e.g., Mercator ), conic (e.g., Albers ), and planar (e.g., stereographic ). Many mathematical projections, however, do not neatly fit into any of these three projection methods. Hence other peer categories have been described in the literature, such as pseudoconic, pseudocylindrical, pseudoazimuthal, retroazimuthal, and polyconic . Another way to classify projections

10904-602: The proof of the Atiyah–Singer index theorem . The development of complex geometry was spurred on by parallel results in algebraic geometry , and results in the geometry and global analysis of complex manifolds were proven by Shing-Tung Yau and others. In the latter half of the 20th century new analytic techniques were developed in regards to curvature flows such as the Ricci flow , which culminated in Grigori Perelman 's proof of

11020-424: The purposes of mapping the shape of the Earth. Implicitly throughout this time principles that form the foundation of differential geometry and calculus were used in geodesy , although in a much simplified form. Namely, as far back as Euclid 's Elements it was understood that a straight line could be defined by its property of providing the shortest distance between two points, and applying this same principle to

11136-417: The restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on H p at each point. If the distribution H can be defined by a global one-form α {\displaystyle \alpha } then this form is contact if and only if the top-dimensional form is a volume form on M , i.e. does not vanish anywhere. A contact analogue of

11252-479: The same period the development of projective geometry . Dubbed the single most important work in the history of differential geometry, in 1827 Gauss produced the Disquisitiones generales circa superficies curvas detailing the general theory of curved surfaces. In this work and his subsequent papers and unpublished notes on the theory of surfaces, Gauss has been dubbed the inventor of non-Euclidean geometry and

11368-490: The scales and hence in the distortions: map distances from the central point are computed by a function r ( d ) of the true distance d , independent of the angle; correspondingly, circles with the central point as center are mapped into circles which have as center the central point on the map. The mapping of radial lines can be visualized by imagining a plane tangent to the Earth, with the central point as tangent point. Differential geometry Differential geometry

11484-427: The sphere. In reality, cylinders and cones provide us with convenient descriptive terms, but little else. Lee's objection refers to the way the terms cylindrical , conic , and planar (azimuthal) have been abstracted in the field of map projections. If maps were projected as in light shining through a globe onto a developable surface, then the spacing of parallels would follow a very limited set of possibilities. Such

11600-423: The standard parallels are compressed. When a single standard parallel is used, distances along all other parallels are stretched. Conic projections that are commonly used are: Azimuthal projections have the property that directions from a central point are preserved and therefore great circles through the central point are represented by straight lines on the map. These projections also have radial symmetry in

11716-494: The study of hyperbolic geometry by Lobachevsky . The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space , and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds . A geometric structure

11832-435: The study of plane curves and the investigation of concepts such as points of inflection and circles of osculation , which aid in the measurement of curvature . Indeed, already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition d 2 y = 0 {\displaystyle d^{2}y=0} indicates the existence of an inflection point. Shortly after this time

11948-416: The study of the geometry of smooth shapes, can be traced back at least to classical antiquity . In particular, much was known about the geometry of the Earth , a spherical geometry , in the time of the ancient Greek mathematicians. Famously, Eratosthenes calculated the circumference of the Earth around 200 BC, and around 150 AD Ptolemy in his Geography introduced the stereographic projection for

12064-403: The surface into a plane. While the first step inevitably distorts some properties of the globe, the developable surface can then be unfolded without further distortion. Once a choice is made between projecting onto a cylinder, cone, or plane, the aspect of the shape must be specified. The aspect describes how the developable surface is placed relative to the globe: it may be normal (such that

12180-442: The surface of the Earth leads to the conclusion that great circles , which are only locally similar to straight lines in a flat plane, provide the shortest path between two points on the Earth's surface. Indeed, the measurements of distance along such geodesic paths by Eratosthenes and others can be considered a rudimentary measure of arclength of curves, a concept which did not see a rigorous definition in terms of calculus until

12296-413: The surface to be mapped is that of a sphere. The Earth and other large celestial bodies are generally better modeled as oblate spheroids , whereas small objects such as asteroids often have irregular shapes. The surfaces of planetary bodies can be mapped even if they are too irregular to be modeled well with a sphere or ellipsoid. Therefore, more generally, a map projection is any method of flattening

12412-432: The surface's axis of symmetry coincides with the Earth's axis), transverse (at right angles to the Earth's axis) or oblique (any angle in between). The developable surface may also be either tangent or secant to the sphere or ellipsoid. Tangent means the surface touches but does not slice through the globe; secant means the surface does slice through the globe. Moving the developable surface away from contact with

12528-407: The term cylindrical as used in the field of map projections relaxes the last constraint entirely. Instead the parallels can be placed according to any algorithm the designer has decided suits the needs of the map. The famous Mercator projection is one in which the placement of parallels does not arise by projection; instead parallels are placed how they need to be in order to satisfy the property that

12644-422: The triaxial ellipsoid for further information. One way to classify map projections is based on the type of surface onto which the globe is projected. In this scheme, the projection process is described as placing a hypothetical projection surface the size of the desired study area in contact with part of the Earth, transferring features of the Earth's surface onto the projection surface, then unraveling and scaling

12760-458: The volumes of smooth three-dimensional solids such as the sphere, cones, and cylinders. There was little development in the theory of differential geometry between antiquity and the beginning of the Renaissance . Before the development of calculus by Newton and Leibniz , the most significant development in the understanding of differential geometry came from Gerardus Mercator 's development of

12876-411: The whole Earth as a finite rectangle, except in the first two cases, where the rectangle stretches infinitely tall while retaining constant width. A transverse cylindrical projection is a cylindrical projection that in the tangent case uses a great circle along a meridian as contact line for the cylinder. See: transverse Mercator . An oblique cylindrical projection aligns with a great circle, but not

12992-482: The work of Riemann , the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium , to the effect that Gaussian curvature is an intrinsic invariant. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic. However, there

13108-692: Was Hermann Weyl who made important contributions to the foundations of general relativity, introduced the Weyl tensor providing insight into conformal geometry , and first defined the notion of a gauge leading to the development of gauge theory in physics and mathematics . In the middle and late 20th century differential geometry as a subject expanded in scope and developed links to other areas of mathematics and physics. The development of gauge theory and Yang–Mills theory in physics brought bundles and connections into focus, leading to developments in gauge theory . Many analytical results were investigated including

13224-611: Was developed by Sophus Lie and Jean Gaston Darboux , leading to important results in the theory of Lie groups and symplectic geometry . The notion of differential calculus on curved spaces was studied by Elwin Christoffel , who introduced the Christoffel symbols which describe the covariant derivative in 1868, and by others including Eugenio Beltrami who studied many analytic questions on manifolds. In 1899 Luigi Bianchi produced his Lectures on differential geometry which studied differential geometry from Riemann's perspective, and

13340-461: Was distilled in by Felix Hausdorff in 1914, and by 1942 there were many different notions of manifold of a combinatorial and differential-geometric nature. Interest in the subject was also focused by the emergence of Einstein's theory of general relativity and the importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced

13456-416: Was the development of an idea of Gauss's about the linear element d s {\displaystyle ds} of a surface. At this time Riemann began to introduce the systematic use of linear algebra and multilinear algebra into the subject, making great use of the theory of quadratic forms in his investigation of metrics and curvature. At this time Riemann did not yet develop the modern notion of

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