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Ordinary trigonometry studies triangles in the Euclidean plane .There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions [broken anchor], definitions via differential equations [broken anchor], and definitions using functional equations.
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, a 0-dimensional simplex is a point, a 1-dimensional simplex is a line segment,
Generalization for arbitrary triangles, green area = blue area Construction for proof of parallelogram generalization. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure ...
Fig. 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.. In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles.
In theoretical physics, twistor theory was proposed by Roger Penrose in 1967 [1] as a possible path [2] to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics. Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge.
That this function is a distance function means firstly that it is symmetric in and , secondly that the distance between and itself is zero, and otherwise the distance between and must be positive, and lastly that the triangle inequality holds, meaning that the length of one leg of a triangle xyz cannot exceed the sum of the lengths of the ...
The Jacobian matrix is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. [9] [10] A further generalization for a function between Banach spaces is the Fréchet derivative.
In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit space M/G where M is a manifold (or a theory), and G is a group of its isometries (or symmetries) — not necessarily all of them. In string theory, these symmetries do not have to have a geometric interpretation.