Search results
Results from the WOW.Com Content Network
This property provides a possibility to construct points of a hyperbola if the asymptotes and one point are given. This property of a hyperbola is an affine version of the 4-point-degeneration of Pascal's theorem. [13]
The unit hyperbola is blue, its conjugate is green, and the asymptotes are red. In geometry, the unit hyperbola is the set of points (x,y) in the Cartesian plane that satisfy the implicit equation = In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an alternative radial length
For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves, [14] although they also make sense when defined in this way for curves over an arbitrary field. [15] A plane curve of degree n intersects its asymptote at most at n−2 other points, by Bézout's theorem, as ...
The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At = the asymptotes are at right angles. With > the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion ...
If P 0 is taken to be the point (1, 1), P 1 the point (x 1, 1/x 1), and P 2 the point (x 2, 1/x 2), then the parallel condition requires that Q be the point (x 1 x 2, 1/x 1 1/x 2). It thus makes sense to define the hyperbolic angle from P 0 to an arbitrary point on the curve as a logarithmic function of the point's value of x. [1] [2]
The asymptotic directions are the same as the asymptotes of the hyperbola of the Dupin indicatrix through a hyperbolic point, or the unique asymptote through a parabolic point. [1] An asymptotic direction is a direction along which the normal curvature is zero: take the plane spanned by the direction and the surface's normal at that point. The ...
The image of the Kiepert hyperbola under the isogonal transformation is the Brocard axis of triangle which is the line joining the symmedian point and the circumcenter. Let P {\displaystyle P} be a point in the plane of a nonequilateral triangle A B C {\displaystyle ABC} and let p {\displaystyle p} be the trilinear polar of P {\displaystyle P ...
Then whichever hyperbola (A) or (B) is used, the operation is an example of a hyperbolic involution where the asymptote is invariant. Hyperbolically orthogonal lines lie in different sectors of the plane, determined by the asymptotes of the hyperbola, thus the relation of hyperbolic orthogonality is a heterogeneous relation on sets of lines in ...