A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between two points located at latitude and longitude of and on a sphere of radius , convert spherical coordinates to Cartesian coordinates using(1)(Note that the latitude is related to the colatitude of spherical coordinates by , so the conversion to Cartesian coordinates replaces and by and , respectively.) Now find the angle between and using the dot product,(2)(3)(4)The great circle distance is then(5)For the Earth, the equatorial radius is km, or 3963 (statute) miles. Unfortunately, the flattening..
The altitudes of a triangle are the Cevians that are perpendicular to the legs opposite . The three altitudes of any triangle are concurrent at the orthocenter (Durell 1928). This fundamental fact did not appear anywhere in Euclid's Elements.The triangle connecting the feet of the altitudes is known as the orthic triangle.The altitudes of a triangle with side length , , and and vertex angles , , have lengths given by(1)(2)(3)where is the circumradius of . This leads to the beautiful formula(4)Other formulas satisfied by the altitude include(5)where is the inradius, and(6)(7)(8)where are the exradii (Johnson 1929, p. 189). In addition,(9)(10)(11)where is again the circumradius.The points , , , and (and their permutations with respect to indices; left figure) all lie on a circle, as do the points , , , and (and their permutations with respect to indices; right figure).Triangles and are inversely similar.Additional properties involving..
A vector is formally defined as an element of a vector space. In the commonly encountered vector space (i.e., Euclidean n-space), a vector is given by coordinates and can be specified as . Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector is often called a two-vector, an -dimensional vector is often called an n-vector, and so on.Vectors can be added together (vector addition), subtracted (vector subtraction) and multiplied by scalars (scalar multiplication). Vector multiplication is not uniquely defined, but a number of different types of products, such as the dot product, cross product, and tensor direct product can be defined for pairs of vectors.A vector from a point to a point is denoted , and a vector may be denoted , or more commonly, . The point is often called the "tail" of the vector, and is called the vector's "head." A vector with unit length is called a unit vector..
A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration.Geodesics preserve a direction on a surface (Tietze 1965, pp. 26-27) and have many other interesting properties. The normal vector to any point of a geodesic arc lies along the normal to a surface at that point (Weinstock 1974, p. 65).Furthermore, no matter how badly a sphere is distorted, there exist an infinite number of closed geodesics on it. This general result, demonstrated in the early 1990s, extended earlier work by Birkhoff, who proved in 1917 that there exists at least one closed geodesic on a distorted sphere, and Lyusternik and Schnirelmann, who proved in 1923 that..
A quantity which gives the inclination of a curve or line with respect to another curve or line. For a line in the -plane making an angle with the x-axis, the slope is a constant given by(1)where and are changes in the two coordinates over some distance.For a plane curve specified as , the slope is(2)for a curve specified parametrically as , the slope is(3)where and , for a curve specified as , the slope is(4)and for a curve given in polar coordinates as , the slope is(5)(Lawrence 1972, pp. 8-9).It is meaningless to talk about the slope of a curve in three-dimensional space unlessthe slope with respect to what is specified.J. Miller has undertaken a detailed study of the origin of the symbol to denote slope. The consensus seems to be that it is not known why the letter was chosen. One high school algebra textbook says the reason for is unknown, but remarks that it is interesting that the French word for "to climb" is "monter."..
A method of computing the determinant of a square matrix due to Charles Dodgson (1866) (who is more famous under his pseudonym Lewis Carroll). The method is useful for hand calculations because, for an integer matrix, all entries in submatrices computed along the way must also be integers. The method is also implemented efficiently in a parallel computation. Condensation is also known as the method of contractants (Macmillan 1955, Lotkin 1959).Given an matrix, condensation successively computes an matrix, an matrix, etc., until arriving at a matrix whose only entry ends up being the determinant of the original matrix. To compute the matrix (), take the connected subdeterminants of the matrix and divide them by the central entries of the matrix, with no divisions performed for . The matrices arrived at in this manner are the matrices of determinants of the connected submatrices of the original matrices.For example, the first condensation..
Given a map from a space to a space and another map from a space to a space , a lift is a map from to such that . In other words, a lift of is a map such that the diagram (shown below) commutes. If is the identity from to , a manifold, and if is the bundle projection from the tangent bundle to , the lifts are precisely vector fields. If is a bundle projection from any fiber bundle to , then lifts are precisely sections. If is the identity from to , a manifold, and a projection from the orientation double cover of , then lifts exist iff is an orientable manifold.If is a map from a circle to , an -manifold, and the bundle projection from the fiber bundle of alternating n-forms on , then lifts always exist iff is orientable. If is a map from a region in the complex plane to the complex plane (complex analytic), and if is the exponential map, lifts of are precisely logarithms of ...