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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 .
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 pole of a line is the inversion of its closest point to the circle C, whereas the polar of a point is the converse, namely, a line whose closest point to C is the inversion of the point. The eccentricity of the conic section obtained by reciprocation is the ratio of the distances between the two circles' centers to the radius r of ...
In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. [ 1 ] [ 2 ] The word asymptote is derived from the Greek ἀσύμπτωτος ( asumptōtos ) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". [ 3 ]
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
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 ...
the x-coordinate of a point is the signed distance of its projection onto the line (the foot of the perpendicular segment to the line from that point) to the origin; the y-coordinate is the signed distance from the point to the line, with the sign according to whether the point is on the positive or negative side of the oriented line.
Choose a line (the x-axis) in the hyperbolic plane (with a standardized curvature of -1) and label the points on it by their distance from an origin (x=0) point on the x-axis (positive on one side and negative on the other). For any point in the plane, one can define coordinates x and y by dropping a perpendicular onto the x-axis.