Calculations for an equilateral hyperbola segment. An equilateral hyperbola has the equation x²-y²=s, where s is greater than zero. When plotted as a curve, it forms two separate regions, one to the left and one to the right of the y-axis, which are mirror symmetrical to this axis. Here, only the positive region is considered. This, in turn, consists of two arms, which are mirror images of each other with respect to the x-axis. This is due to the two results obtained when solving the above equation for y, .
Enter the shape parameter s (s>0, Unit hyperbola s=1) and the maximal input value a (a>√s) and choose the number of decimal places. Then click Calculate.
Formeln:
Shape parameter, input value a, and chord length have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).
The hyperbola has its vertex at the point (√s;0). For the unit hyperbola, this is (1;0). Thus, the shape parameter s determines not only the form of the hyperbola, that is its curvation, but also its position within the coordinate system. While the position is irrelevant to the geometric properties of the area and chord length, the shape parameter s, together with the input value a, also determines the width of the hyperbolic segment, that is, its extent along the x-axis.
The hyperbola is a type of conic section. Conic sections are formed when a double cone is intersected by a plane. In this context, a double cone refers to two identical right circular cones touching straight point-to-point. A cross-section parallel to the base produces a circle. If the plane intersects the cone obliquely, but intersects only one of the two cones, an ellipse is formed. The greater the inclination of the plane, the more elongated the ellipse becomes. If the cutting plane runs in exactly the same direction as the lateral surface of the cone, a parabolic segment is formed as a limiting case. If the plane is inclined even more steeply and intersects both cones, a hyperbolic segment is formed.
If such an equilateral hyperbola is reflected into a circle, which is centered at the origin of the coordinate system and situated between the two branches of the hyperbola, the result is a Lemniskate of Bernoulli.