2d Curves by Wassenaar J.

By Wassenaar J.

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Degenerate hyperbola: the hyperbola has been degenerated to his asymptotes • adjungated hyperbola: given a hyperbola a second one is added, with the same asymptotes, but axes switched. An example of this relationship is formed by a pair of hyperbolas x y = 1 or x y = −1 Some relations with other curves: • the hyperbola is the isoptic of the parabola • the hyperbola can be seen as a sinusoidal spiral • depending on the center of inversion and the form of the hyperbola, as polar inverse the following curves can be obtained: ♦ lemniscate (of Bernoulli) (center) ♦ (right) strophoid (vertex) ♦ trisectrix of MacLaurin (vertex) • the pedal curve (focus) is the circle • a pedal (center) of the rectangular hyperbola is the lemniscate • its isoptic is the circle The Greek Euklid and Aristaeus wrote about the general hyperbola, although they confined the curve to one branch.

9) Davis 1981 p. 100. 10) Ku−klux is derived from the Greek word for circle: kyklos. ellipse conic section last updated: 2003−05−20 The word ellipse is derived from the Greek word elleipsis (defect). The same defect do we find in the linguistic ellipse: omitting a part of a sentence, in situations that the part can be thought easily. ' an linguistic ellipse. Another fine quality of an ellipse is that for each point the sum of distances to the foci is constant. This gives us an easy way to construct an ellipse: fasten a rope in the two foci, and draw the curve with a pencil tight to the (strained) cord.

Now the line OO' through pole O and fixed point O' is no longer perpendicular to the construction line l. The parameter a = Π/2 for the right strophoid can now also have other values, in Cartesian coordinates: The curve is a pedal of the parabola, and a special case of the general strophoid. Sometimes the oblique strophoid itself is called the 'general strophoid'. resonance curve cubic last updated: 2002−07−13 A well−known parameter equation for this curve is the following: Resonance occurs, when an external oscillation is exerted on a system, with a frequency in the neighborhood of a certain resonance frequency.

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