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In linear algebra, geometry, and trigonometry, the Cayley–Menger determinant is a formula for the content, i.e. the higher-dimensional volume, of a -dimensional simplex in terms of the squares of all of the distances between pairs of its vertices. The determinant is named after Arthur Cayley and Karl Menger.
The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume: =. where is the base' area and is the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apices to the opposite faces are inversely proportional to the areas of ...
Cayley 1841 introduced the modern notation for the determinant using vertical bars. [35] [36] ... The volume of any tetrahedron, given its vertices,,, ...
Tartaglia, from 16th century AD, generalized it to give the volume of tetrahedron from the distances between its 4 vertices. The modern theory of distance geometry began with Arthur Cayley and Karl Menger. [7] Cayley published the Cayley determinant in 1841, [8] which is a special case of
The volume of any tetrahedron that shares three converging edges ... can be computed by means of the Gram determinant. Alternatively, the volume is the norm of the ...
Expanding the determinant, using its alternating and multilinear ... where V is the volume of the tetrahedron. Examples of special points. In the ...
7.2 Volume of a tetrahedron. 7.3 Spherical and hyperbolic geometry. 8 See also. ... The same relation can be expressed using the Cayley–Menger determinant, [3]
This is because the n-dimensional dV element is in general a parallelepiped in the new coordinate system, and the n-volume of a parallelepiped is the determinant of its edge vectors. The Jacobian can also be used to determine the stability of equilibria for systems of differential equations by approximating behavior near an equilibrium point.