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If , , and are three points on one line, , , and are three points on another line, and meets at , meets at , and meets at , then the three points , , and are collinear. Pappus's hexagon theorem is self-dual.The incidence graph of the configuration corresponding to the theorem is the Pappus graph.

The south pole is the point on a sphere with minimum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the south pole given by the point in the southern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole and takes on the value at the south pole.

If the three straight lines joining the corresponding vertices of two triangles and all meet in a point (the perspector), then the three intersections of pairs of corresponding sides lie on a straight line (the perspectrix). Equivalently, if two triangles are perspective from a point, they are perspective from a line.The 10 lines and 10 3-line intersections form a configuration sometimes called Desargues' configuration.Desargues' theorem is self-dual.

The north pole is the point on a sphere with maximum -coordinate for a given coordinate system. For a rotating sphere like the Earth, the natural coordinate system is defined by the rotation axis, with the north pole given by the point in the northern hemisphere that is farthest from the equator (i.e., midplane of the sphere).The coordinate in spherical coordinates is measured from the north pole.

The perspective image of an infinite checkerboard. It can be constructed starting from any triangle , where and form the near corner of the floor, and is the horizon (left figure). If is the corner tile, the lines and must be parallel to and respectively. This means that in the drawing they will meet and at the horizon, i.e., at point and point respectively (right figure). This property, of course, extends to the two bunches of perpendicular lines forming the grid.The adjacent tile (left figure) can then be determined by the following conditions: 1. The new vertices and lie on lines and respectively. 2. The diagonal meets the parallel line at the horizon . 3. The line passes through . Similarly, the corner-neighbor of (right figure) can be easily constructed requiring that: 1. Point lie on . 2. Point lie on the common diagonal of the two tiles. 3. Line pass through . Iterating the above procedures will yield the complete picture. This construction shows..

A path, also known as a rhumb line, which cuts a meridian on a given surface at any constant angle but a right angle. If the surface is a sphere, the loxodrome is a spherical spiral. The loxodrome is the path taken when a compass is kept pointing in a constant direction. It is a straight line on a Mercator projection or a logarithmic spiral on a polar projection (Steinhaus 1999, pp. 218-219). The loxodrome is not the shortest distance between two points on a sphere.

The latitude of a point on a sphere is the elevation of the point from the plane of the equator. The latitude is related to the colatitude (the polar angle in spherical coordinates) by . More generally, the latitude of a point on an ellipsoid is the angle between a line perpendicular to the surface of the ellipsoid at the given point and the plane of the equator (Snyder 1987).The equator therefore has latitude , and the north and south poles have latitude , respectively. Latitude is also called geographic latitude or geodetic latitude in order to distinguish it from several subtly different varieties of authalic latitudes.The shortest distance between any two points on a sphere is the so-called great circle distance, which can be directly computed from the latitudes and longitudes of the two points...

The polar angle on a sphere measured from the north pole instead of the equator. The angle in spherical coordinates is the colatitude. It is related to the latitude by .

In homogeneous coordinates, the first positive quadrant joins with by "points" , and is mapped onto the hyperbolic line by the correspondence . Now define(1)Let be any bounded linear transformation of a Banach space that maps a closed convex cone of onto itself. Then the -norm of is defined by(2)for pairs with finite . Birkhoff's inequality then states that if the transform of under has finite diameter under , then(3)(Birkhoff 1957).

A projection matrix is an square matrix that gives a vector space projection from to a subspace . The columns of are the projections of the standard basis vectors, and is the image of . A square matrix is a projection matrix iff .A projection matrix is orthogonal iff(1)where denotes the adjoint matrix of . A projection matrix is a symmetric matrix iff the vector space projection is orthogonal. In an orthogonal projection, any vector can be written , so(2)An example of a nonsymmetric projection matrix is(3)which projects onto the line .The case of a complex vector space is analogous. A projection matrix is a Hermitian matrix iff the vector space projection satisfies(4)where the inner product is the Hermitian inner product. Projection operators play a role in quantum mechanics and quantum computing.Any vector in is fixed by the projection matrix for any in . Consequently, a projection matrix has norm equal to one, unless ,(5)Let be a -algebra. An element..

There are a number of algebraic equations known as the icosahedral equation, all of which derive from the projective geometry of the icosahedron. Consider an icosahedron centered , oriented with -axis along a fivefold () rotational symmetry axis, and with one of the top five edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest icosahedral equation is defined by projecting the vertices of the icosahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the height above..

The tetrahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider a tetrahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top three edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest tetrahedral equation is defined by projecting the vertices of the tetrahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not..

The octahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider an octahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top four edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest octahedral equation is defined by projecting the vertices of the octahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the..

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