Search results
Results From The WOW.Com Content Network
Hyperbola: the midpoints of parallel chords lie on a line. Hyperbola: the midpoint of a chord is the midpoint of the corresponding chord of the asymptotes. The midpoints of parallel chords of a hyperbola lie on a line through the center (see diagram). The points of any chord may lie on different branches of the hyperbola.
In the Cartesian plane, these pairs lie on a hyperbola, and when the double sum is fully expanded, there is a bijection between the terms of the sum and the lattice points in the first quadrant on the hyperbolas of the form xy = k, where k runs over the integers 1 ≤ k ≤ n: for each such point (x,y), the sum contains a term g(x)h(y), and ...
The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.
A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points (a, 1/a) and (b, 1/b) on the rectangular hyperbola xy = 1, or the corresponding region when this hyperbola is re-scaled and its orientation is altered by a rotation leaving the center at the origin, as with the unit hyperbola.
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).
This can be shown by taking the points X and Y to the standard points [::] and [::] by a projective transformation, in which case the pencils of lines correspond to the horizontal and vertical lines in the plane, and the intersections of corresponding lines to the graph of a function, which (must be shown) is a hyperbola, hence a conic, hence ...
[2]: 400 This definition is analogous to the definition of a planar hyperbola. The one-dimensional wave equation: = is an example of a hyperbolic equation. The two-dimensional and three-dimensional wave equations also fall into the category of hyperbolic PDE.
In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes.A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.