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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 ...
Therefore, in hyperbolic geometry the ratio of a circle's circumference to its radius is always strictly greater than , though it can be made arbitrarily close by selecting a small enough circle. If the Gaussian curvature of the plane is −1 then the geodesic curvature of a circle of radius r is: 1 tanh ( r ) {\displaystyle {\frac {1 ...
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 .
In Euclidean plane geometry, a tangent line to a circle is a line that touches the circle at exactly one point, never entering the circle's interior. Tangent lines to circles form the subject of several theorems , and play an important role in many geometrical constructions and proofs .
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 hyperbolic geometry, a hypercycle, hypercircle or equidistant curve is a curve whose points have the same orthogonal distance from a given straight line (its axis). Given a straight line L and a point P not on L , one can construct a hypercycle by taking all points Q on the same side of L as P , with perpendicular distance to L equal to that ...
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes.A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.
This equation is called the canonical form of a hyperbola, because any hyperbola, regardless of its orientation relative to the Cartesian axes and regardless of the location of its center, can be transformed to this form by a change of variables, giving a hyperbola that is congruent to the original (see below).