|
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 sp ... [...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: i ... [...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. He ... [...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, consi ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Tetrahedron")