Curves
Parabolish fncts.
– ellipse, parabalo, hyperbola are all part of conic
I. Elliptic
II. Parabolic
– general equation is> ax^2 + bx + c
– for envelope special use has unit parabola defined by y = x^2, however x^ nother values is no longer parabola…
Focus/ directrix
Focus – point defined by blue (a, b) points.
Directrix – violet c line
Distance from focus to any point on parabola is same like distance from this particular point to directrix.
Respectively (green line from left point): distance from any point on parabola is the same like distance from this point to directrix.
IV Hyperbolic
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Unit hyperbola
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.
– are based on same principle like trigonometric fnct, but instead of unit circle they are based on unit hyperbola.
– instead if sin you have sinh etc.
– it is used for example in beam free vibration calculation – see Beams.
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– sinh is of course not in the Grid, but can be written with euler no –
-Sinh and Cosh
Sinh
– valid for odd parts
Sinh x = e^x – e^-x/ 2 = e^2x – 1/ 2x^x
Hyperbolic fncts can also be written by natural log.
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