Hyperbola - Geometrical Constructions

Geometrical Constructions

Similar to the ellipse, a hyperbola can be constructed using a taut thread. A straightedge of length S is attached to one focus F₁ at one of its corners A so that it is free to rotate about that focus. A thread of length L = S - 2a is attached between the other focus F₂ and the other corner B of the straightedge. A sharp pencil is held up against the straightedge, sandwiching the thread tautly against the straightedge. Let the position of the pencil be denoted as P. The total length L of the thread equals the sum of the distances L₂ from F₂ to P and L_B from P to B. Similarly, the total length S of the straightedge equals the distance L₁ from F₁ to P and L_B. Therefore, the difference in the distances to the foci, L₁ − L₂ equals the constant 2a

$L_{1} - L_{2} = \left( S - L_{B} \right) - \left( L - L_{B} \right) = S - L = 2a$

A second construction uses intersecting circles, but is likewise based on the constant difference of distances to the foci. Consider a hyperbola with two foci F₁ and F₂, and two vertices P and Q; these four points all lie on the transverse axis. Choose a new point T also on the transverse axis and to the right of the rightmost vertex P; the difference in distances to the two vertices, QT − PT = 2a, since 2a is the distance between the vertices. Hence, the two circles centered on the foci F₁ and F₂ of radius QT and PT, respectively, will intersect at two points of the hyperbola.

A third construction relies on the definition of the hyperbola as the reciprocation of a circle. Consider the circle centered on the center of the hyperbola and of radius a; this circle is tangent to the hyperbola at its vertices. A line g drawn from one focus may intersect this circle in two points M and N; perpendiculars to g drawn through these two points are tangent to the hyperbola. Drawing a set of such tangent lines reveals the envelope of the hyperbola.

A fourth construction is using the parallelogram method. It is similar to such method for parabola and ellipse construction: certain equally spaced points lying on parallel lines are connected with each other by two straight lines and their intersection point lies on the hyperbola.