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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 ...
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 ...
For example, cocompact lattices in the orthogonal or unitary group preserving a form of signature (,) are hyperbolic. A further generalisation is given by groups admitting a geometric action on a CAT(k) space , when k {\displaystyle k} is any negative number. [ 3 ]
Indeed, hyperbolic angle corresponds to area of a sector in the plane with its "unit circle" given by {(,): =}. The contracted unit hyperbola { cosh a + j sinh a : a ∈ R } {\displaystyle \{\cosh a+j\sinh a:a\in \mathbb {R} \}} of the split-complex plane has only half the area in the span of a corresponding hyperbolic sector.
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.}
For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass M of gas with changing volume will have variable density δ = M / V, and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the ...
More generally, given a vector space V (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form (⋅, ⋅), the special orthogonal Lie algebra consists of tracefree endomorphisms which are skew-symmetric for this form ((,) + (,) =). Over a field of characteristic 2 we consider instead the alternating ...
The origin and all events on the light cone are self-orthogonal. When a time event and a space event evaluate to zero under the bilinear form, then they are hyperbolic-orthogonal. This terminology stems from the use of conjugate hyperbolas in the pseudo-Euclidean plane: conjugate diameters of these hyperbolas are hyperbolic-orthogonal.