Geometric calculus extends the geometric algebra to include differentiation and integration including differential geometry and differential forms. ref David Hestenes , Garrett Sobczyk Clifford Algebra to Geometric Calculus, a Unified Language for mathematics and Physics Dordrecht Boston G.Reidel Publ.Co., 1984, ISBN 90 277 2561 6 ref File Geometric Calculus Family Tree.png right 300px thumb Figure 1 from 32 A diagram of the history of Geometric Calculus Given a geometric algebra, the vector derivative is defined as the operator math 1 . Essentially, the vector derivative is defined so that the GA version of Green s theorem is true, math oint A dA nabla f oint dA dx f math and then one can write math nabla f nabla cdot f nabla wedge f math as a geometric product, effectively generalizing Stokes theorem including the differential forms version of it . In math 1D math when A is a curve with endpoints math a math and math b math , then math oint A dA nabla f oint dA dx f math reduces to math int a b dx nabla f int a b dx cdot nabla f int a b df f b f a math or the fundamental theorem of integral calculus. Also developed are the concept of vector manifold and geometric integration theory which generalizes Cartan s differential forms . References reflist differential geometry stub Category Calculus ... more details
Geometric combinatorics is a branch of mathematics in general and combinatorics in particular. It includes a number of subareas such as polyhedral combinatorics the study of Face geometry faces of convex polyhedron convex polyhedra , convex geometry the study of convex set s, in particular combinatorics of their intersections , and discrete geometry , which in turn has many applications to computational geometry . Other important areas include metric geometry of polyhedra , such as the Cauchy s theorem geometry Cauchy theorem on rigidity of convex polytopes. The study of regular polytope s, Archimedean solid s, and kissing number s is also a part of geometric combinatorics. Special polytopes are also considered, such as the permutohedron , associahedron and Birkhoff polytope . Further reading http www.cis.upenn.edu cis610 topics.pdf Topics in Geometric Combinatorics http www.ams.org bookstore?fn 20&arg1 geotopo&item PCMS 13 Geometric Combinatorics , Edited by Ezra Miller and Victor Reiner http scholar.google.co.uk scholar?q 22Combinatorics of Finite Geometries 22 Combinatorics of Finite Geometries Category Combinatorics Category Discrete geometry combin stub bs Geometrijska kombinatorika ... more details
History of Greek art Geometric art is a phase of Greek art , characterised largely by geometric motifs ... . ref cite journal last Snodgrass first Anthony M. title Greek Geometric Art by Bernhard Schweitzer ... 23 jstor 707869 ref Pottery in the Geometric periods Protogeometric period During the Protogeometric ... bands with a few written geometric shapes within, usually concentric cycles or semicircles engraved with a caliper. Early Geometric period In the Early geometric period 900 850 BC the height of the vessels ... design, the most characteristic element of geometric art. Middle geometric period At the Middle geometric ... the handles. Image Eleusis geometric amhora.JPG 200px thumb right Amphora of 8th c.BC from the Archaeological Museum of Eleusis with geometric motifs Late Geometric period While the technique from the Middle Geometric period was still continued at the beginning of 8th century BC some laboratories enriched ... form. This was the first phase of the Late Geometric period 760 700 BC , in which the great ... at a height of 1.50 m and the perfection of their execution, the highest expression of the Greek geometric ... eased, the geometric shapes have become more freely, and areas with animals, birds, scenes of shipwrecks, hunting scenes, themes from mythology or the Homeric epics led geometric pottery into more naturalistic expressions. ref http www.greek thesaurus.gr geometric period art.html Geometric periods of pottery at Greek thesaurus.gr ref One of the characteristic examples of the Late geometric ... style of Corinth distinguished. Geometric motives File Dipylon vase.jpg thumb right Dipylon Vase Vases in the Geometric style are characterized by several horizontal bands about the circumference covering the entire vase. Between these lines the geometric artist used a number of other decorative motifs ... book last Coldstream first John N. title Geometric Greece 900 700 BC publisher Routledge date 1979, 2003 ... period List of Greek vase painters Geometric period National Archaeological Museum of Greece ... more details
In mathematics , specifically differential geometry , a geometric flow is the gradient flow associated to a functional on a manifold which has a geometric interpretation, usually associated with some curvature extrinsic or intrinsic curvature . They can be interpreted as flows on a moduli space for intrinsic flows or a parameter space for extrinsic flows . These are of fundamental interest in the calculus of variations , and include several famous problems and theories. Particularly interesting are their critical point mathematics critical point s. A geometric flow is also called a geometric evolution equation . Examples Extrinsic Extrinsic geometric flows are flows on embedded submanifold s, or more generally immersed submanifold s. In general they change both the Riemannian metric and the immersion. Mean curvature flow , as in soap film s critical points are minimal surface s Willmore flow , as in minimax eversion s of spheres Inverse mean curvature flow Intrinsic Intrinsic geometric flows are flows on the Riemannian metric , independent of any embedding or immersion. Ricci flow , as in the Solution of the Poincar conjecture , and Richard Hamilton professor Richard Hamilton s proof of the Uniformization theorem Calabi flow Yamabe flow Classes of flows Important classes of flows are curvature flows , variational flows which extremelize some functional , and flows arising as solutions to parabolic partial differential equation s. A given flow frequently admits all of these interpretations, as follows. Given an elliptic operator L , the parabolic PDE math u t Lu math yields ... of the flow correspond to critical points of the functional. In the context of geometric flows, the functional ... Bakas, I. title The algebraic structure of geometric flows in two dimensions year 2005 arxiv hep th 0507284 cite journal author Bakas, I. title Renormalization group equations and geometric flows year 2007 arxiv hep th 0702034 DEFAULTSORT Geometric Flow Category Geometric flow ... more details
File Geometric sequences.svg thumb right 300px Diagram illustrating three basic geometric sequences of the pattern ... line represents the Infinite geometric series infinite sum of the sequence, a number that it will forever ... , a geometric progression , also known as a geometric sequence , is a sequence of number ... called the common ratio . For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1 2. The sum of the terms of a geometric progression, or of an initial segment of a geometric progression, is known as a geometric series . Thus, the general form of a geometric sequence is math a, ar, ar 2, ar 3, ar 4, ldots math and that of a geometric series is math a ar ar 2 ar 3 ar 4 cdots math where ... properties The n th term of a geometric sequence with initial value a and common ratio r is given by math a n a ,r n 1 . math Such a geometric sequence also follows the recursive relation math a n r ,a n 1 math for every integer math n geq 1. math Generally, to check whether a given sequence is geometric ... ratio of a geometric series may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. For instance 1, 3, 9, 27, 81, 243, ... is a geometric sequence with common ratio 3. The behaviour of a geometric sequence depends on the value of the common ... positive and negative infinity due to the alternating sign . Geometric sequences with common ratio ... of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression. An interesting property of geometric progressions is that, for any value of the common ratio, any three consecutive terms a , b and c will satisfy the following equation math b 2 ac math Geometric series This section is linked from Time value of money main Geometric series A geometric series is the sum of the numbers ... more details
The geometric mean , in mathematics , is a type of mean or average , which indicates the central tendency or typical value of a set of numbers. A geometric mean is often used when comparing different items ... 2012 ref . For example, the geometric mean can give a meaningful average to compare two companies ... for their financial viability. If an arithmetic mean was used instead of a geometric mean, the financial ... of a geometric mean normalizes the ranges being averaged, so that no range dominates the weighting, and a given percentage change in any of the properties has the same effect on the geometric mean. So, a 20 change in environmental sustainability from 4 to 4.8 has the same effect on the geometric mean as a 20 change in financial viability from 60 to 72. The geometric mean is similar to the arithmetic ... of numbers in the set of the resulting product mathematics product is taken. For instance, the geometric ... 2 × 8 2 4 . As another example, the geometric mean of the three numbers 4, 1, and 1 32 is the cube ..., if the numbers are math x 1, ldots,x n math , the geometric mean math G math satisfies math ... expression states that the log of the geometric mean is the arithmetic mean of the logs of the numbers. The geometric mean can also be understood in terms of geometry . The geometric mean of two numbers ... of a rectangle with sides of lengths a and b . Similarly, the geometric mean of three numbers, a , b ... whose lengths are equal to the three given numbers. The geometric mean applies only to positive numbers. ref The geometric mean only applies to positive numbers in order to avoid taking the root of a negative ... allows 0 which yields a geometric mean of 0 , but may be excluded, as one frequently wishes to take the logarithm of geometric means to convert between multiplication and addition , and one cannot ... human population or interest rates of a financial investment. The geometric mean is also one of the three ... and the geometric mean is always in between see Inequality of arithmetic and geometric means ... more details
Argento s mind and hand attempting something different within the geometric genre . The SoHo Weekly ... No. 10. , 1939 42 However, geometric abstraction cannot only be seen as an invention of 20th century ... figures, is a prime example of this geometric pattern based art, which existed centuries before ... in the architecture of Islamic civilations spanning the 7th century 20th century, geometric patterns ... of geometric abstraction. Selected artists Artists who have worked extensively in geometric ... Abstract Artists References references External links commons cat Geometric abstraction http geoform.net ... Geometric Abstraction. DEFAULTSORT Geometric Abstraction Category Modernism Category Modern art Category ... Geometric abstraction sk Geometrick abstrakcia ... more details
The geometric albedo of an astronomical body is the ratio of its actual brightness at zero Phase angle astronomy phase angle i.e., as seen from the light source to that of an idealized flat, fully reflecting, diffuse reflection diffusively scattering Lambertian disk with the same cross section. Diffuse reflection Diffuse scattering implies that radiation is reflected isotropically with no memory of the location of the incident light source. Zero phase angle corresponds to looking along the direction of illumination. For Earth bound observers this occurs when the body in question is at opposition astronomy opposition and on the ecliptic . The visual geometric albedo refers to the geometric albedo quantity when accounting for only electromagnetic radiation in the visible spectrum . Airless bodies The surface materials regolith s of airless bodies in fact, the majority of bodies in the Solar system are strongly non Lambertian and exhibit the opposition effect , which is a strong tendency to reflect light straight back to its source, rather than scattering light diffusely. The geometric albedo of these bodies can be difficult to determine because of this, as their Bidirectional reflectance distribution function reflectance is strongly peaked for a small range of phase angles near zero ... to zero phase angle to obtain an estimate of the geometric albedo. For very bright, solid, airless ... them a geometric albedo above unity 1.4 in the case of Enceladus . Light is preferentially reflected ..., whereas a Lambertian surface would scatter the radiation much more broadly. The geometric albedo above ... of a plane surface, the geometric albedo is the albedo of the surface when the illumination is provided by a beam of radiation that comes in perpendicular to the surface. Examples The geometric albedo ..., California. references DEFAULTSORT Geometric Albedo Category Observational astronomy Category Radiometry ... geom trico fr Alb do g om trique id Albedo geometrik pt Albedo geom trico simple Geometric albedo sl ... more details
about infinite geometric series finite sums geometric progression File GeometricSquares.svg thumb right ... , a geometric series is a series mathematics series with a constant ratio between successive term ... , , cdots math is geometric, because each successive term can be obtained by multiplying the previous term by 1 2. Geometric series are one of the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role ... series convergence of series. Geometric series are used throughout mathematics, and they have important ... , and finance . Common ratio The terms of a geometric series form a geometric progression , meaning that the ratio of successive terms in the series is constant. The following table shows several geometric ... the series has no sum. See for example Grandi s series 1 &minus 1 1 &minus 1 . Sum The sum of a geometric ... of 2 3 the original size. Consider the sum of the following geometric series math s 1 , , frac 2 ... 1 math , the Geometric progression Geometric series sum of the first n terms of a geometric series ... 3 , , cdots frac 1 1 r , math the left hand side being a geometric series with common ratio r . We ... can prove that the geometric series convergent series converges using the sum formula for a geometric ... main Repeating decimal A repeating decimal can be thought of as a geometric series whose common ... 7 10,000 , , cdots. math The formula for the sum of a geometric series can be used to convert ... of the Parabola Archimedes used the sum of a geometric series to compute the area enclosed ... , , frac 1 64 , , cdots. math This is a geometric series with common ratio nowrap 1 4 and the fractional ... of fractal s, geometric series often arise as the perimeter , area , or volume of a self similarity ... is geometric with constant ratio r 4 9. The first term of the geometric series is a ... main Zeno s paradoxes The convergence of a geometric series reveals that a sum involving an infinite ... more details
Taxobox name Geometric moray image Gymnothorax griseus by Marek Jakubowski.jpg regnum Animalia phylum Chordata classis Actinopterygii ordo Anguilliformes familia Muraenidae genus Gymnothorax species G. griseus binomial Gymnothorax griseus binomial authority Bernard Germain de Lac p de Lac p de , 1803 The geometric moray , Gymnothorax griseus , is a moray eel of the family biology family Muraenidae , found throughout the western Indian Ocean at depths down to 40 m. Its length is up to 65 cm. References FishBase species genus Gymnothorax species griseus month June year 2006 Category Gymnothorax griseus Category Animals described in 1803 af Geometriese bontpaling ca Gymnothorax griseus de Graue Mur ne es Gymnothorax griseus fr Gymnothorax griseus nl Gymnothorax griseus ... more details
In mathematical physics , geometric quantization is a mathematical approach to defining a Quantum mechanics quantum theory corresponding to a given classical theory . It attempts to carry out Quantization physics quantization , for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in. One of the earliest attempts at a natural quantization was Weyl quantization , proposed by Hermann Weyl in 1927. Here, an attempt is made to associate a quantum mechanical observable a self adjoint operator on a Hilbert space with a real valued function on classical phase space . The position and momentum in this phase space are map to the generators .... The geometric quantization procedure falls into the following three steps prequantization, polarization ... to a vector field X . Geometric quantization of Poisson manifolds and symplectic foliations also is developed ... title Geometric Quantization and Quantum Mechanics publisher Springer isbn 0 387 90496 7 url cite book author N.M.J. Woodhouse year 1991 title Geometric Quantization publisher Clarendon Press isbn ..., Gennadi Sardanashvily G. Sardanashvily year 2005 title Geometric and Algebraic Topological Methods ... abs math ph 0208008 William Ritter s review of Geometric Quantization presents a general framework for all problems in physics and fits geometric quantization into this framework http math.ucr.edu home baez quantization.html John Baez s review of Geometric Quantization , by John Baez is short and pedagogical http www.blau.itp.unibe.ch lecturesGQ.ps.gz Matthias Blau s primer on Geometric Quantization ..., N. Roman Roy, Mathematical foundations of geometric quantization, http arxiv.org abs math ph 9904008 arXiv math ph 9904008. Gennadi Sardanashvily G. Sardanashvily , Geometric quantization of symplectic ... more details
Problems of the following type, and their solution techniques, were first studied in the 19th century, and the general topic became known as geometric probability . Buffon s needle What is the chance that a needle dropped randomly onto a floor marked with equally spaced parallel lines will cross one of the lines? What is the mean length of a random chord of a unit circle? cf. Bertrand s paradox probability Bertrand s paradox . What is the chance that three random points in the plane form an acute rather than obtuse triangle? What is the mean area of the polygonal regions formed when randomly oriented lines are spread over the plane? For mathematical development see the concise monograph Solomon. ref cite book author Herbert Solomon title Geometric Probability year 1978 publisher Society for Industrial and Applied Mathematics location Philadelphia, PA ref Since the late 20th century the topic has split into two topics with different emphases. Integral geometry sprang from the principle that the mathematically natural probability models are those that are invariant under certain transformation groups. This topic emphasises systematic development of formulas for calculating expected values associated with the geometric objects derived from random points, and can in part be viewed as a sophisticated branch of multivariate calculus. Stochastic geometry emphasises the random geometrical objects themselves. For instance different models for random lines or for random tessalations of the plane random sets formed by making points of a Poisson process spatial Poisson process be say centers of discs. See also Wendel s theorem References references DEFAULTSORT Geometric Probability Category Geometry Category Probability theory eu Probabilitate geometriko uk ... more details
In probability theory and statistics , the geometric distribution is either of two discrete probability ... 0, 1, 2, 3, ... nowrap Which of these one calls the geometric distribution is a matter of convention and convenience. Probability distribution two name Geometric type mass pdf image File geometric pmf.svg 450px cdf image File geometric cdf.svg 450px parameters ... different geometric distributions should not be confused with each other. Often, the name shifted geometric distribution is adopted for the former one distribution of the number X however, to avoid ... for k 1, 2, 3, .... The above form of geometric distribution is used for modeling the number of trials until the first success. By contrast, the following form of geometric distribution is used for modeling ... 0, 1, 2, 3, .... In either case, the sequence of probabilities is a geometric ... and is a geometric distribution with p 1 6. Moments and cumulants The expected value of a geometrically ... subsets of the set of points where they converge. Parameter estimation For both variants of the geometric ... s 1 p 1 . end align math Like its continuous analogue the exponential distribution , the geometric ... observed. The die one throws or the coin one tosses does not have a memory of these failures. The geometric ... , the geometric distribution X with parameter p 1 is the one with the maximum entropy probability distribution largest entropy . The geometric distribution of the number Y of failures before ... distribution indecomposable . Golomb coding is the optimal prefix code for the geometric discrete distribution. Related distributions The geometric distribution Y is a special case of the negative ... k sup k . Then math sum k 1 infty k ,X k math has a geometric distribution taking values in the set ... is the continuous analogue of the geometric distribution. If X is an exponentially distributed ... links planetmath reference id 3456 title Geometric distribution http mathworld.wolfram.com GeometricDistribution.html ... more details
In algebraic geometry , the geometric genus is a basic birational invariant p sub g sub of algebraic varieties and complex manifold s. Definition The geometric genus can be defined for non singular complex projective varieties and more generally for complex manifold s as the Hodge number h sup n ,0 sup equal to h sup 0, n sup by Serre duality , that is, the dimension of the Canonical bundle General case canonical linear system . In other words for a variety V of complex dimension n it is the number of linearly independent holomorphic n differential form forms to be found on V . ref Danilov & Shokurov 1998 , Google books quote id mU6ciaFCC1IC page 53 text geometric genus p. 53 ref This definition, as the dimension of H sup 0 sup V ,&Omega sup n sup then carries over to any base field mathematics field , when &Omega is taken to be the sheaf of K hler differential s and the power is the top exterior power , the canonical bundle canonical line bundle . The geometric genus is the first invariant p sub g sub P sub 1 sub of a sequence of invariants P sub n sub called the plurigenera . The case of curves In the case of complex varieties, the complex loci of non singular curves are Riemann surfaces . The algebraic definition of genus agrees with the genus of a surface topological notion . On a nonsingular curve, the canonical line bundle has degree 2g 2 . The notion of genus features prominently in the statement of the Riemann Roch theorem see also Riemann Roch theorem for algebraic curves and of the Riemann Hurwitz formula . If C is an irreducible and smooth hypersurface in the Algebraic geometry of projective spaces projective plane cut out by a polynomial equation of degree d , then its normal line bundle is the Serre twisting sheaf math mathcal O d math , so by the adjunction ... O d C mathcal O d 3 C math . Genus of singular varieties The definition of geometric genus is carried over classically to singular curves C , by decreeing that p sub g sub C is the geometric genus of the normalization ... more details
No footnotes date October 2011 A geometric program GP is an optimization mathematics optimization problem of the form Minimize math f 0 x math subject to math f i x leq 1, quad i 1, dots,m math math h i x 1, quad i 1, dots,p math where math f 0, dots,f m math are posynomials and math h 1, dots,h p math are monomials. In the context of geometric programming unlike all other disciplines , a monomial is defined as a function math f mathbb R n to mathbb R math with math mathrm dom f mathbb R n math defined as math f x c x 1 a 1 x 2 a 2 cdots x n a n math where math c 0 math and math a i in mathbb R math . GPs have numerous application, such as components sizing in Integrated circuit IC design ref http www.stanford.edu boyd papers opamp.html ref and parameter estimation via logistic regression in statistics. The maximum likelihood estimator in logistic regression is a GP. Convex form Geometric programs are not in general convex optimization problems, but they can be transformed to convex problems by a change of variables and a transformation of the objective and constraint functions. In particular, defining math y i log x i math , the monomial math f x c x 1 a 1 cdots x n a n mapsto e a T y b math , where math b log c math . Similarly, if math f math is the posynomial math f x sum k 1 K c k x 1 a 1k cdots x n a nk math then math f x sum k 1 K e a k T y b k math , where math a k a 1k , dots,a nk math and math b k log c k math . After the change of variables, a posynomial becomes a sum of exponentials of affine functions. References cite book author Richard J. Duffin coauthors Elmor L. Peterson, Clarence Zener title Geometric Programming publisher John Wiley and Sons date 1967 pages 278 isbn 0471223700 External links S. Boyd, S. J. Kim, L. Vandenberghe, and A. Hassibi, http www.stanford.edu boyd gp tutorial.html A Tutorial on Geometric Programming S. Boyd, S. J. Kim, D. Patil, and M. Horowitz http www.stanford.edu boyd gp digital ckt.html Digital Circuit Optimization via Geometric ... more details
Other uses In mathematics , geometric topology is the study of manifold s and maps between them, particularly embedding s of one manifold into another. Topics Main List of geometric topology topics Some examples of topics in geometric topology are Orientable manifold orientability , handle decomposition s, local flatness , and the planar and higher dimensional Jordan Sch nflies theorem Sch nflies theorem s. In all dimensions, the fundamental group of a manifold is a very important invariant, and determines much of the structure in dimensions 1, 2 and 3, the possible fundamental groups are restricted, while in every dimension 4 and above every finitely presented group is the fundamental group of a manifold note that it is sufficient to show this for 4 and 5 dimensional manifolds, and then to take products with spheres to get higher ones . In low dimensional topology Surfaces 2 manifolds 3 manifold s 4 manifold s each have their own theory, where there are some connections. Knot theory is the study of the Three dimensional space 3 dimensional embedding s of circles 1 dimension into 3. In high dimensional topology, characteristic classes are a basic invariant, and surgery theory is a key theory. Low dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions every surface admits a constant curvature metric geometrically, it has one of 3 possible geometries positive curvature spherical, zero curvature flat, negative curvature hyperbolic and the geometrization conjecture now theorem in 3 dimensions every 3 manifold can be cut into pieces ... Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 ... homotopy theory. See also Category Maps of manifolds List of geometric topology topics References R.B. Sher and R.J. Daverman 2002 , Handbook of Geometric Topology , North Holland. ISBN 0 444 82432 4 DEFAULTSORT Geometric Topology Category Geometric topology da Geometrisk topologi de Geometrische Topologie ... more details
A geometric spanner or a k spanner graph or a k spanner was initially introduced as a weighted graph over a set of points as its vertices which for every pair of vertices has a path between them of weight at most k times the spatial distance between these points for a fixed k . The parameter k is called the stretch factor or dilation factor of the spanner. ref citation first1 Giri last1 Narasimhan first2 Michiel last2 Smid title Geometric Spanner Networks publisher Cambridge University Press year 2007 isbn 0521815134 . ref In computational geometry , the concept was first discussed by L.P. Chew in 1986, ref citation first L. Paul last Chew contribution There is a planar graph almost as good as the complete graph title Proc. 2nd Annual Symposium on Computational Geometry year 1986 pages 169 177 doi 10.1145 10515.10534 . ref although the term spanner was not used in the original paper. The notion of graph spanner s has been known in graph theory k spanners are spanning subgraph s of graphs with similar dilation property, where distances between graph vertices are defined in Glossary of graph theory Distance graph theoretical terms . Therefore geometric spanners are graph spanners of complete graph s graph embedding embedded in the plane with edge weights equal to the distances between the embedded vertices in the corresponding metric. Spanners may be used in computational geometry for solving some proximity problems . They have also found applications in other areas, such as in motion planning , in telecommunication network s network reliability, optimization of roaming in mobile network s, etc. Research Chew s main result was that for a set of points in the plane there is a triangulation of this pointset such that for any two points there is a path along the edges of the triangulation ... Computing Geometric Minimum Dilation Graphs is NP Hard editor1 last Kaufmann editor1 first ... reflist Category Geometric algorithms Category Geometric graphs ... more details
1173955498.pdf ref It is one of the rarest tortoise species in the world Diet The geometric tortoise ... The geometric tortoise has lost 97 of its habitat , and only 2000 to 3000 individuals exist today. It is threatened ... geometric tortoise.htm Library.thinkquest.org entry Baard, E. H. W. 1989. The Ecology and Conservation Status of the Geometric Tortoise Psammobates geometricus Preliminary Results Jour. Herp. Ass. Afr. 36 72 72 Baard, E. H. W. 1991. A Review of the Taxonomic History of and some Literature on the Geometric ..., age at maturity and sexual dimorphism in the geometric tortoise, Psammobates geometricus Jour. Herp ... of the Geometric Tortoise, Psammobates geometricus, in South Africa Herpetological Journal 3 ... more details
A geometric lathe was used for making ornamental patterns on the plates used in printing bank notes and postage stamps. It is sometimes called a Guilloch lathe. It was developed early in the nineteenth century when efforts were introduced to combat forgery, and is an adaptation of an ornamental turning lathe. The lathe was able to generate intersecting and interlacing patterns of fine lines in various shapes, which were almost impossible to forge by hand engraving. They were used by many national Mints. Further reading Peter Bower, Economic warfare Banknote Forgery as a deliberate weapon , and Maureen Greenland, Compound plate printing and nineteenth century bank notes, in Virginia Hewitt, ed. The Banker s Art Studies in paper money , pp 46 63, and pp 84 87, The British Museum Press, 1995, ISBN 0 7141 0879 0 See also Security printing External links http ornamentalturning.net Ornamental Turning Category Money forgery Category Automatic lathes mechanically automated metalworking stub ... more details
In the mathematical field of numerical ordinary differential equations , a geometric integrator is a numerical method that preserves geometric properties of the exact Vector field Flow curves flow of a differential equation. Pendulum example We can motivate the study of geometric integrators by considering the motion of a simple pendulum pendulum . Assume that we have a pendulum whose bob has mass math m 1 math and whose rod is massless of length math ell 1 math . Take the acceleration due to gravity to be math g 1 math . Denote by math q t math the angular displacement of the rod from the vertical, and by math p t math the pendulum s momentum. The Hamiltonian mechanics Hamiltonian of the system, the sum of its kinetic energy kinetic and potential energy potential energies, is math H q,p T p U q frac 1 2 p 2 cos q, math which gives Hamilton s equations math dot q, dot p p, sin q . , math It is natural to take the configuration space math Q math of all math q math to be the unit circle math mathbb S 1 math , so that math q,p math lies on the cylinder math mathbb S 1 times mathbb R math . However, we will take math q,p in mathbb R 2 math , simply because math q,p math space is then easier to plot. Define math z t q t ,p t mathrm T math and math f z p, sin q mathrm T math . Let us experiment by using some simple numerical methods to integrate this system. As usual, we select a constant step size, math h math , and for an aribtrary non negative integer math k math we write math z k z kh math . We use the following methods. math z k 1 z k hf z k , math Euler method explicit Euler ... . The implicit midpoint rule has similar geometric properties. To summarize the pendulum example shows ... preserving, just as the exact flow is they are two examples of geometric in fact, symplectic ... References references Ernst Hairer, Christian Lubich and Gerhard Wanner, Geometric Numerical Integration ... Press, 2005. ISBN 0 521 77290 7. DEFAULTSORT Geometric Integrator Category Numerical differential ... more details
In Classical mechanics classical and quantum mechanics , the geometric phase , Pancharatnam Berry phase named after S. Pancharatnam and Michael Berry physicist Sir Michael Berry , Pancharatnam phase or most commonly Berry phase , is a Phase waves phase acquired over the course of a Period physics cycle , when the system is subjected to cyclic adiabatic process quantum mechanics adiabatic process es, which results from the geometrical properties of the parameter space of the Hamiltonian quantum mechanics Hamiltonian . The phenomenon was first discovered in 1956, ref cite journal author S. Pancharatnam title Generalized Theory of Interference, and Its Applications. Part I. Coherent Pencils journal Proc. Indian Acad. Sci. A volume 44 pages 247 262 year 1956 ref and rediscovered in 1984. ref cite journal author M. V. Berry journal Proc. R. Soc. Lond. A title Quantal Phase Factors Accompanying Adiabatic Changes volume 392 issue 1802 pages 45 57 year 1984 doi 10.1098 rspa.1984.0023 bibcode 1984RSPSA.392...45B ref It can be seen in the Aharonov Bohm effect and in the conical intersection of potential energy surface s. In the case of the Aharonov Bohm effect, the adiabatic parameter is the magnetic ... analogue of the Berry phase is known as the Hannay angle . Theory In general the geometric phase ... of geometric phases The Foucault pendulum One of the easiest examples is the Foucault pendulum . An easy explanation in terms of geometric phases is given by von Bergmann and von Bergmann ref cite journal ... changes of parameters. The stochastic pump effect can be interpreted in terms of a geometric phase ... Christian and Kazimir Wanelik title Resource Letter GPP 1 Geometric Phases in Physics journal Am ... Geometric Phase in Optics http departments.colgate.edu physics faculty EGalvez articles PreprintRflash.pdf ... paths , J. Chem. Phys. 117, 7405 2002 . Frank Wilczek and Alfred Shapere, Geometric Phases in Physics ... Wiley year 2006 isbn 0471778915 DEFAULTSORT Geometric Phase Category Quantum mechanics Category ... more details
In geometry , a group of isometry isometries of hyperbolic space is called geometrically finite if it has a well behaved fundamental domain . A hyperbolic manifold is called geometrically finite if it can be described in terms of geometrically finite group mathematics groups . Geometrically finite polyhedra A convex polyhedron C in hyperbolic space is called geometrically finite if its closure overline C in the conformal compactification of hyperbolic space has the following property For each point x in overline C , there is a neighborhood U such that all faces of overline C meeting U also pass through x harv Ratcliffe 1994 loc 12.4 . For example, every polyhedron with a finite number of faces is geometrically finite. In hyperbolic space of dimension at most 2, every geometrically finite polyhedron has a finite number of sides, but there are geometrically finite polyhedra in dimensions 3 and above with infinitely many sides. For example, in Euclidean space R sup n sup of dimension n 2 there is a polyhedron P with an infinite number of sides. The upper half plane model of n 1 dimensional hyperbolic space in R sup n 1 sup projects to R sup n sup , and the inverse image of P under this projection is a geometrically finite polyhedron with an infinite number of sides. A geometrically finite polyhedron has only a finite number of cusps, and all but finitely many sides meet one of the cusps. Geometrically finite groups A discrete group G of isometries of hyperbolic space is called geometrically finite if it has a fundamental domain C that is convex, geometrically finite, and exact every face is the intersection of C and gC for some g G harv Ratcliffe 1994 loc 12.4 . In hyperbolic spaces of dimension at most 3, every exact, convex, fundamental polyhedron for a geometrically finite group has only a finite number of sides, but in dimensions 4 and above there are examples with an infinite number of sides harv Ratcliffe 1994 loc theorem 12.4.6 . In hy ... more details
In computer aided design , Geometric Description Language GDL is the programming language of ArchiCAD library part s. GSM is the file format of these CAD objects. Area of usage These objects are similar to blocks in AutoCAD , but unlike blocks, these are parametric, and the 2D and 3D features are connected, so in any view one can get the correct visualization for example a side view on the section, top view on plan, and perspective in the 3D view . GDL scripts define an ArchiCAD library part in its main roles, these are 3D model, 3D model projected to section elevation or to 2D plan, 2D plan view, user interface display and behaviour and listing quantities. All versions of the ArchiCAD contain their own default libraries, also objects like furniture, windows, doors, trees, people, cars, construction elements, etc. There are several commercial websites for selling high detailed intelligent parametric objects, which are perhaps better and more variable than the default libraries, for example there are a lot of environmental objects like plants, people or garden elements, which are not included in the original software. Licensing GDL is a free technology although ArchiCAD itself is a commercial software , it is allowed and possible to develop GDL based object libraries using free tools like Graphisoft LP XMLConverter and Graphisoft GDL Web Plug In. Technical information The GDL programming language is fundamentally BASIC like. It has the same control flow statements and variable logic. In 2D and 3D in GDL, all the model elements are linked to a local right handed coordinate system . For placing an element in the desired position, you have to move the coordinate system to the desired position and orientation , then generate the element itself. Every movement, rotation or stretching ... typed programming languages Category Domain specific programming languages es Geometric Description Language fr Geometric Description Language it Geometric Description Language lv GDL hu Geometric ... more details
Refimprove date December 2009 2D computer graphics is the computer based generation of digital image s&mdash mostly from two dimensional models such as 2Dgeometricmodel s, text, and digital images and by techniques ... were landmark developments in the field. 2D graphics techniques 2D graphics models may combine 2Dgeometricmodelgeometricmodel s also called vector graphics , digital image s also called raster graphics ... , or similar device of 2D computer graphics primitives. These editors generally provide 2Dgeometric ... Sprite computer graphics sprite s left and masks right 2D computer graphics are mainly used in applications ... of a document based on 2D computer graphics techniques can be much smaller than the corresponding ... as graphics file format 2D graphic files . 2D computer graphics started in the 1950s, based ... can be modified and manipulated by two dimensional Transformation geometry geometric transformation .... Such non standard orientations are rarely used in mathematics but are common in 2D computer ... that is the same in all directions. The result of uniform scaling is Similarity geometry similar in the geometric ... an object by a Vector geometric vector v v sub x sub , v sub y sub , v sub z sub , each point ... coordinates . To scale an object by a Vector geometric vector v v sub x sub , v sub y sub ... the pixel colors directly, but most will rely on some 2D graphics library and or the machine s graphics ... and angle paint a simple geometric shape , such as a triangle defined by three corners, or a circle ... is linked from Layer main Layers digital image editing The models used in 2D computer graphics ..., shadow s, Reflection physics reflection , refraction , etc. However, they usually can model ... anti aliasing of complex drawings and provide a sound model for certain techniques such as mitered joints ... based model, the target image is produced by painting or pasting each layer, in order of decreasing ... as soon as it is produced by the rendering procedure. Layers that consist of complex geometric ... more details
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