Hyperbola

A hyperbola is an open 1-dimensional surface produced by finding the set of points where the absolute distances between a point on the hyperbola and the focal points are constant.

The two unconnected sections of the hyperbola are called branches. They are mirror images of each other, and their diagonally opposite arms approach the limit to a line. This line is called an asymptote. The point where the two asymptotes intersect is called the midpoint or center.

As a hyperbola is curved, it is often represented embedded in a plane.

A hyperbola can be defined as a graph to an equation such as $f(x) = \frac{1}{x}$ or ${\displaystyle f(x)= x^2 - y^2 = a^2}$ .

A 1-dimensional shape bounded by two points on one of the branches of the hyperbola is called a hyperbolic segment.

A 2-dimensional shape bounded by two line segments that connect two points on the hyperbola and the midpoint is called a hyperbolic sector.

A hyperbola is an example of a conic section that can be drawn on a plane that intersects a double cone created from two nappes. This plane intersects both nappes and both the bases of two cones that can be created from the nappes.

Structure and Sections[]

Hypervolumes[]

vertex count = $0$
edge length = $\infty l$

Subfacets[]

1 hyperbola (1D)

Equations[]

All points on the surface of a rectangular hyperbola (embedded in a Euclidean plane) can be given by the equation $y = A/x$ , where A is the semi-major axis. All points on a conjugate hyperbola can be given by an equation $\frac {x^2}{a^2} - \frac {y^2}{b^2} = r$ or $\frac {y^2}{x^2} - \frac {y^2}{b^2} = r$ . Conjugate hyperbolas can be defined parametrically using a parameter $t$ by