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The vectors z 1 and z 2 in the complex number plane, and w 1 and w 2 in the hyperbolic number plane are said to be respectively Euclidean orthogonal or hyperbolic orthogonal if their respective inner products [bilinear forms] are zero. [3] The bilinear form may be computed as the real part of the complex product of one number with the conjugate ...
Hyperbolic lines are then either half-circles orthogonal to B or rays perpendicular to B. The length of an interval on a ray is given by logarithmic measure so it is invariant under a homothetic transformation ( x , y ) ↦ ( λ x , λ y ) , λ > 0. {\displaystyle (x,y)\mapsto (\lambda x,\lambda y),\quad \lambda >0.}
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.
Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces , families of functions are used to form an orthogonal basis , such as in the contexts of orthogonal polynomials , orthogonal ...
Split-complex numbers z and w are said to be hyperbolic-orthogonal if z, w = 0. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle.
Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces , families of functions are used to form an orthogonal basis , such as in the contexts of orthogonal polynomials , orthogonal ...
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci F 1 {\displaystyle F_{1}} and F 2 {\displaystyle F_{2}} are generally taken to be fixed at − a {\displaystyle -a} and + a {\displaystyle +a} , respectively, on the x ...
The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups , since the condition of preserving a form can be expressed as an equality of matrices.