Definition 1. A hyperbola is the set of all points in the Euclidean plane such that the absolute value of the difference of the distances from P to two fixed points and is a constant positive number. Here and are called the focal points, or the foci, of the hyperbola. The midpoint of the line segment is called the centre of the hyperbola.
Figure 1: Points on hyperbola |
Figure 1 shows two fixed points and points of the hyperbola in the arbitrary distance from the first focus. Here we have
If a point moves on a hyperbola, the difference always has the constant value.
See Animation 1.
Figure 2: Definition of hyperbola |
Consider the hyperbola in Figure 2 with centre and foci and . The distance between the centre and either focus or is called the linear eccentricity of the hyperbola and it is denoted by . Notice that the hyperbola is symmetric about the line through and . Let and be the points where the line intersects the hyperbola. Centre bisects the line segment and hyperbola is symmetric also with respect to the line through and perpendicular to the line . The line is called the transverse (major) axis, and the perpendicular is called the imaginery (conjugate) axis of the hyperbola. Let and be the points on this line located in the distance from the centre of the hypebola. The four points are called the vertices of the hyperbola. Let denotes the length of the line segment and is length of the line segment . The numbers and are called the semitransverse (semimajor) axis and the semiconjugate (semiminor) axis of the hyperbola.
Hence is the distance between foci and called eccentricity .
Applying the Pythagorean theorem to the right triangle we find that
which is called the geometric equation of the hyperbola.
See animation 2.
Rectangular KLMN with sides passing through vertices and parallel to the axes of hyperbola is called characteristic rectangular of a hyperbola. Diagonals of this rectangular are asymptotes, two lines passing through centre of hyperbola and approaching the hyperbola two branches as the distance of its points is getting very large. Asymptotes are also defined as tangent lines to hyperbola at infinity.
Focal circle of hyperbola is he circle with centre in the centre of hyperbola and radius equal to eccentricity intersecting asymptotes in the vertices of the characteristic rectangular.
Lines passing through the arbitrary point of the hyperbola and either focus or are called focal lines of the point . Line segments and located in the focal lines of the point indicate the distances of the hyperbola point to the foci, the difference of which is constant and it equals to .
Focal lines of the point form two angles with the vertices in their common point on the hyperbola - exterior angle in which the transverse (major) vertices and are located, and interior angle, as illustrated in the Figure 3.
Figure 3: Focal lines |
Any line segment MN determined by endpoints M, N on the hyperbola is called the chord of the hyperbola. A focal chord is passing through a focus perpendicularly to the transverse (major) axis of the hyperbola.