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De Gua's Theorem
__NOTOC__ In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces: A_^2 = A_^2+A_^2+A_^2 Generalizations The Pythagorean theorem and de Gua's theorem are special cases () of a general theorem about ''n''-simplices with a right-angle corner, proved by P. S. Donchian and H. S. M. Coxeter in 1935. This, in turn, is a special case of a yet more general theorem by Donald R. Conant and William A. Beyer (1974), which can be stated as follows. Let ''U'' be a measurable subset of a ''k''-dimensional affine subspace of \mathbb^n (so k \le n). For any subset I \subseteq \ with exactly ''k'' elements, let U_I be the orthogonal projection of ''U'' onto the linear span of e_, \ldots, e_, where I = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in ,∞to each set in \R^n or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in \R^n is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is not necessarily an integer. These measures are fundamenta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heron's Formula
In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century engineer Heron of Alexandria (or Hero) who proved it in his work ''Metrica'', though it was probably known centuries earlier. Example Let be the triangle with sides , and . This triangle’s semiperimeter is :s=\frac=\frac=16 and so the area is : \begin A &= \sqrt = \sqrt\\ &= \sqrt = \sqrt = 24. \end In this example, the side lengths and area are integers, making it a Heronian triangle. However, Heron's formula works equally well in cases where one or more of the side lengths are not integers. Alternate expressions Heron's formula can also be written in terms of just the side lengths instead of using the semiperimeter, in several ways, :\begin A &=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfrac\sqrt \\ mu&=\tfra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bivector
In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalar (mathematics), scalars and Euclidean vector, vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector can be thought of as being of degree two. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotation (mathematics), rotations in any number of dimensions, and are a useful tool for classifying such rotations. They are also used in physics, tying together a number of otherwise unrelated quantities. Bivectors are generated by the exterior product on vectors: given two vectors a and b, their exterior product is a bivector, as is the sum of any bivectors. Not all bivectors can be generated as a single exterior product. More precisely, a bivecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projected Area
Projected area is the two dimensional area measurement of a three-dimensional object by projecting its shape on to an arbitrary plane. This is often used in mechanical engineering and architectural engineering related fields, specifically hardness testing, axial stress, wind pressures, and terminal velocity. The geometrical definition of a projected area is: "the rectilinear parallel projection In three-dimensional geometry, a parallel projection (or axonometric projection) is a projection of an object in three-dimensional space onto a fixed plane, known as the '' projection plane'' or ''image plane'', where the '' rays'', known as ' ... of a surface of any shape onto a plane". This translates into the equation: A_\text = \int_ \cos \, dA where A is the original area, and \beta is the angle between the normal to the local plane and the line of sight to the surface A. For basic shapes the results are listed in the table below. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Area
In 3-dimensional geometry and vector calculus, an area vector is a vector combining an area quantity with a direction, thus representing an ''oriented area'' in three dimensions. Every bounded surface in three dimensions can be associated with a unique area vector called its vector area. It is equal to the surface integral of the surface normal, and distinct from the usual (scalar) surface area. Vector area can be seen as the three dimensional generalization of signed area in two dimensions. Definition For a finite planar surface of scalar area and unit normal , the vector area is defined as the unit normal scaled by the area: \mathbf = \mathbfS For an orientable surface composed of a set of flat facet areas, the vector area of the surface is given by \mathbf = \sum_i \mathbf_i S_i where is the unit normal vector to the area . For bounded, oriented curved surfaces that are sufficiently well-behaved, we can still define vector area. First, we split the surface into in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathematics was central to his method of inquiry, and he connected the previously separate fields of geometry and algebra into analytic geometry. Descartes spent much of his working life in the Dutch Republic, initially serving the Dutch States Army, later becoming a central intellectual of the Dutch Golden Age. Although he served a Protestant state and was later counted as a deist by critics, Descartes considered himself a devout Catholic. Many elements of Descartes' philosophy have precedents in late Aristotelianism, the revived Stoicism of the 16th century, or in earlier philosophers like Augustine. In his natural philosophy, he differed from the schools on two major points: first, he rejected the splitting of corporeal substance into mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johann Faulhaber
Johann Faulhaber (5 May 1580 – 10 September 1635) was a German mathematician. Born in Ulm, Faulhaber was a trained weaver who later took the role of a surveyor of the city of Ulm. He collaborated with Johannes Kepler and Ludolph van Ceulen. Besides his work on the fortifications of cities (notably Basel and Frankfurt), Faulhaber built water wheels in his home town and geometrical instruments for the military. Faulhaber made the first publication of Henry Briggs's Logarithm in Germany. He is also credited with the first printed solution of equal temperament.Date,name,ratio,cents: from equal temperament monochord tables p55-p78; J. Murray Barbour ''Tuning and Temperament'', Michigan State University Press 1951 He died in Ulm. Faulhaber's major contribution was in calculating the sums of powers of integers. Jacob Bernoulli makes references to Faulhaber in his ''Ars Conjectandi''. Works * See also * Faulhaber's formula In mathematics, Faulhaber's formula, named after the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles De Tinseau D'Amondans
Charles-Marie-Thérèse-Léon de Tinseau d'Amondans de Gennes (1748-1822) was a military engineer and mathematician from France in the 18th century. Life and work Charles Tinseau was the sixth son (from seven) of Marie-Nicolas Tinseau, ''seigneur'' de Gennes, and Jeanne Petramand de Velay, a noble family in the Franche-Comté. He entered in the ''École du Génie'' at Mézières (the Military School of Artillery of France) in 1769 and he graduated in 1771. Gaspard Monge, his professor of mathematics, interested him in mathematics., Complete Dictionary of Scientific Biography. However, he followed his military career achieving the rank of General Brigadier. In the school he knew the future naturalist Justin Girod-Chantrans, born at Besançon like himself. In 1772 he presented two papers in the Acadèmie Royal des Sciences (published 1774). The more influential of the two papers was ''Sur quelques propriétés des solides renfermés par des surfaces composées des lignes droite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley–Menger Determinant
In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a n-dimensional simplex in terms of the squares of all of the distances between pairs of its vertices. The determinant is named after Arthur Cayley and Karl Menger. The pairwise distance polynomials between ''n'' points in a real Euclidean space are Euclidean invariants that are associated via the Cayley-Menger relations.Sitharam, Meera; St. John, Audrey; Sidman, Jessica. ''Handbook of Geometric Constraint Systems Principles''. Boca Raton, FL: CRC Press. These relations served multiple purposes such as generalising Heron's Formula, computing the content of a ''n''-dimensional simplex, and ultimately determining if any real symmetric matrix is a Euclidean distance matrix in the field of Distance geometry. History Karl Menger was a young geometry professor at the University of Vienna and Arthur Cayley was a British mathematician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. Ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
