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A rotation of the original hyperbola by results in a rectangular hyperbola entirely in the second and fourth quadrants, with the same asymptotes, center, semi-latus rectum, radius of curvature at the vertices, linear eccentricity, and eccentricity as for the case of + rotation, with equation =, >,
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 ...
The asymptotes of an algebraic curve in the affine plane are the lines that are tangent to the projectivized curve through a point at infinity. [13] For example, one may identify the asymptotes to the unit hyperbola in this manner.
The quadrature of the hyperbola is the evaluation of the area of a hyperbolic sector. It can be shown to be equal to the corresponding area against an asymptote. The quadrature was first accomplished by Gregoire de Saint-Vincent in 1647 in Opus geometricum quadrature circuli et sectionum coni. As expressed by a historian,
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 endpoints (,) of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length b . Denoting the semi-major axis length (distance from the center to a vertex) as a , the semi-minor and semi-major axes' lengths appear in the equation of the ...
A plane with slope less than 1 (1 is the slope of the asymptotes of the generating hyperbola) intersects either in an ellipse or in a point or not at all, A plane with slope equal to 1 containing the origin (midpoint of the hyperboloid) does not intersect H 2 {\displaystyle H_{2}} ,
A parabola, being tangent to the line at infinity, would have its center being a point on the line at infinity. Hyperbolas intersect the line at infinity in two distinct points and the polar lines of these points are the asymptotes of the hyperbola and are the tangent lines to the hyperbola at these points of infinity.