Search results
Results From The WOW.Com Content Network
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 ...
The area of a hyperbolic triangle is given by its defect in radians multiplied by R 2, which is also true for all convex hyperbolic polygons. [2] Therefore all hyperbolic triangles have an area less than or equal to R 2 π. The area of a hyperbolic ideal triangle in which all three angles are 0° is equal to this maximum.
The unit hyperbola finds applications where the circle must be replaced with the hyperbola for purposes of analytic geometry. A prominent instance is the depiction of spacetime as a pseudo-Euclidean space. There the asymptotes of the unit hyperbola form a light cone.
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.
Given the above general parametrization of the hyperbola in Cartesian coordinates, the eccentricity can be found using the formula in Conic section#Eccentricity in terms of coefficients. The center ( x c , y c ) {\displaystyle (x_{c},y_{c})} of the hyperbola may be determined from the formulae
Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...
In geometry, two circles are said to be orthogonal if their respective tangent lines at the points of intersection are perpendicular (meet at a right angle). A straight line through a circle's center is orthogonal to it, and if straight lines are also considered as a kind of generalized circles , for instance in inversive geometry , then an ...
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 of P.