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Pohlke's Theorem
Pohlke's theorem is the fundamental theorem of axonometry. It was established 1853 by the German painter and teacher of descriptive geometry Karl Wilhelm Pohlke. The first proof of the theorem was published 1864 by the German mathematician Hermann Amandus Schwarz, who was a student of Pohlke. Therefore the theorem is sometimes called theorem of Pohlke and Schwarz, too. The theorem *Three arbitrary line sections \overline O\overline U,\overline O\overline V,\overline O\overline W in a plane originating at point \overline O, which are not contained in a line, can be considered as the parallel projection of three edges OU,OV,OW of a cube. For a mapping of a unit cube, one has to apply an additional scaling either in the space or in the plane. Because a parallel projection and a scaling preserves ratios one can map an arbitrary point P=(x,y,z) by the axonometric procedure below. Pohlke's theorem can be stated in terms of linear algebra as: *Any affine mapping of the 3-dimensional s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axonometry
Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled parallel projection of the object. In general, the resulting parallel projection is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane), which in this context is called an orthogonal axonometry. In technical drawing and in architecture, axonometric perspective is a form of two-dimensional representation of three-dimensional objects whose goal is to preserve the impression of volume or relief. Sometimes also called rapid perspective or artificial perspective, it differs from conical perspective and does not represent what the eye actually sees: in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axonometry
Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled parallel projection of the object. In general, the resulting parallel projection is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane), which in this context is called an orthogonal axonometry. In technical drawing and in architecture, axonometric perspective is a form of two-dimensional representation of three-dimensional objects whose goal is to preserve the impression of volume or relief. Sometimes also called rapid perspective or artificial perspective, it differs from conical perspective and does not represent what the eye actually sees: in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Schwarz
Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of Moses Mendelssohn). Schwarz and Kummer had six children, including his daughter Emily Schwarz. Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wilhelm Weierstrass persuaded him to change to mathematics. He received his Ph.D. from the Universität Berlin in 1864 and was advised by Kummer and Weierstrass. Between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic. From 1875 he worked at Göttingen University, dealing with the subjects of complex analysis, differential geometry and the calculus of variations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Pohlke
Karl Wilhelm Pohlke (28 January 1810 in Berlin – 27 November 1876 in Berlin) was a German painter who established an important geometric statement, which is fundamental for axonometric projections. The statement is called Pohlke's theorem. Life Karl Wilhelm Pohlke was taught painting by Wilhelm Hensel at the Königlich Preussischen Akademie der Künste in Berlin and participated in his first exhibition there in 1832. After finishing his studies he earned his living for some years painting landscapes and teaching perspective drawing privately. In 1835 Pohlke went to France and improved his abilities at the École des Beaux-Arts with Louis Étienne Watelet and Léon Cogniet. In 1843 he went to Italy. After 10 years he returned in 1845 to Berlin, where he got 1849 at the Königlichen Bauakademie an appointment as lecturer and in 1860 was promoted to Professor for Descriptive Geometry and Perspective. Between 1860 and 1876 he published a textbook on descriptive geometry, cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lhuilier
Simon Antoine Jean L'Huilier (or L'Huillier) (24 April 1750 in Geneva – 28 March 1840 in Geneva) was a Swiss mathematician of French Huguenot descent. He is known for his work in mathematical analysis and topology, and in particular the generalization of Euler's formula for planar graphs. He won the mathematics section prize of the Berlin Academy of Sciences for 1784 in response to a question on the foundations of the calculus. The work was published in his 1787 book ''Exposition elementaire des principes des calculs superieurs''. (A Latin version was published in 1795.) Although L'Huilier won the prize, Joseph Lagrange, who had suggested the question and was the lead judge of the submissions, was disappointed in the work, considering it "the best of a bad lot." Lagrange would go on to publish his own work on foundations. L'Huilier and Cauchy L'Huilier introduced the abbreviation "lim" for limit that reappeared in 1821 in Cours d'Analyse by Augustin Louis Cauchy, who woul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrilateral
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons (e.g. pentagon). Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices A, B, C and D is sometimes denoted as \square ABCD. Quadrilaterals are either simple (not self-intersecting), or complex (self-intersecting, or crossed). Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ''ABCD'' add up to 360 degrees of arc, that is :\angle A+\angle B+\angle C+\angle D=360^. This is a special case of the ''n''-gon interior angle sum formula: ''S'' = (''n'' − 2) × 180°. All non-self-crossing quadrilaterals tile the plane, b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthographic Projection
Orthographic projection (also orthogonal projection and analemma) is a means of representing Three-dimensional space, three-dimensional objects in Two-dimensional space, two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface. The obverse of an orthographic projection is an oblique projection, which is a parallel projection in which the projection lines are ''not'' orthogonal to the projection plane. The term ''orthographic'' sometimes means a technique in multiview projection in which principal axes or the planes of the subject are also parallel with the projection plane to create the ''primary views''. If the principal planes or axes of an object in an orthographic projection are ''not'' parallel with the projection plane, the depiction is called ''axonometric'' or an ''auxiliary views''. (''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descriptive Geometry
Descriptive geometry is the branch of geometry which allows the representation of three-dimensional objects in two dimensions by using a specific set of procedures. The resulting techniques are important for engineering, architecture, design and in art. The theoretical basis for descriptive geometry is provided by planar geometric projections. The earliest known publication on the technique was "Underweysung der Messung mit dem Zirckel und Richtscheyt", published in Linien, Nuremberg: 1525, by Albrecht Dürer. Italian architect Guarino Guarini was also a pioneer of projective and descriptive geometry, as is clear from his ''Placita Philosophica'' (1665), ''Euclides Adauctus'' (1671) and ''Architettura Civile'' (1686—not published until 1737), anticipating the work of Gaspard Monge (1746–1818), who is usually credited with the invention of descriptive geometry. Gaspard Monge is usually considered the "father of descriptive geometry" due to his developments in geometric pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Similarity (geometry)
In Euclidean geometry, two objects are similar if they have the same shape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or reducing), possibly with additional translation (geometry), translation, rotation (mathematics), rotation and reflection (mathematics), reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruence (geometry), congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other. If two angles of a triangle h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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