Search results
Results from the WOW.Com Content Network
Two tangent lines to B have no (finite) poles because they pass through the center C of the reciprocation circle C; the polars of the corresponding tangent points on B are the asymptotes of the hyperbola. The two branches of the hyperbola correspond to the two parts of the circle B that are separated by these tangent points.
Considering the pencils of confocal ellipses and hyperbolas (see lead diagram) one gets from the geometrical properties of the normal and tangent at a point (the normal of an ellipse and the tangent of a hyperbola bisect the angle between the lines to the foci). Any ellipse of the pencil intersects any hyperbola orthogonally (see diagram).
If a tangent contains the point (x 0, y 0), off the parabola, then the equation = + = holds, which has two solutions m 1 and m 2 corresponding to the two tangents passing (x 0, y 0). The free term of a reduced quadratic equation is always the product of its solutions.
A ray through the unit hyperbola x 2 − y 2 = 1 at the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis. For points on the hyperbola below the x-axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions).
In particular, the indicatrix of an umbilical point is a circle. For hyperbolic points, where the Gaussian curvature is negative, the intersection will form a hyperbola. Two different hyperbolas will be formed on either side of the tangent plane. These hyperbolas share the same axis and asymptotes.
The precision of hyperbolastic functions in modeling real world problems is somewhat due to their flexibility in their point of inflection. [ 1 ] [ 2 ] These functions can be used in a wide variety of modeling problems such as tumor growth, stem cell proliferation, pharma kinetics, cancer growth, sigmoid activation function in neural networks ...
Feuerbach Hyperbola. In geometry, the Feuerbach hyperbola is a rectangular hyperbola passing through important triangle centers such as the Orthocenter, Gergonne point, Nagel point and Schiffler point. The center of the hyperbola is the Feuerbach point, the point of tangency of the incircle and the nine-point circle. [1]
The tangents at the circular points are given by x ± iy = ± a which have real points of intersection at (± a, 0). So the foci are, in fact, foci in the sense defined by Plücker. [8] The circular points are points of inflection so these are triple foci.