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The equation in trilinear coordinates x, y, z of any circumconic of a triangle is [1]: p. 192 l y z + m z x + n x y = 0. {\displaystyle lyz+mzx+nxy=0.} If the parameters l, m, n respectively equal the side lengths a, b, c (or the sines of the angles opposite them) then the equation gives the circumcircle .
The modern systematic use of triangulation networks stems from the work of the Dutch mathematician Willebrord Snell, who in 1615 surveyed the distance from Alkmaar to Breda, approximately 72 miles (116 kilometres), using a chain of quadrangles containing 33 triangles in all. Snell underestimated the distance by 3.5%.
The distance from coast to coast was 360 miles (580 km) and this survey line was completed in 1806. [7] The East India Company thought that this project would take about five years, but it took nearly 70 years, well past the Indian Rebellion of 1857 and the end of company rule in India .
Geographical distance or geodetic distance is the distance measured along the surface of the Earth, or the shortest arch length. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic ...
This formula generalizes Heron's formula for the area of a triangle. A triangle may be regarded as a quadrilateral with one side of length zero. From this perspective, as d approaches zero, a cyclic quadrilateral converges into a cyclic triangle (all triangles are cyclic), and Brahmagupta's formula simplifies to Heron's formula.
The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation , it is a special case of a more general formula in spherical trigonometry , the law of haversines , that relates the sides and angles of spherical triangles.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles.According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
In three dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric equations. [18] The equation x 2 + y 2 = r 2 is the equation for any circle centered at the origin (0, 0) with a radius of r.