Re: Circle Laying On Curves

"Narek Saribekyan" <narek.saribekyan@xxxxxxxxx> wrote in message
news:1172681916.585957.261320@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Consider curves e^(-x) and ln(x) and a tangent unit circle. How can I
fins the center coordinates?

The set of points at distance 1 above the curve exp(-x), is exp(-x) + 1.
Similarly, the set of points at distance 1 below the curve ln(x), is ln(x) -
1.

Therefore the center of the circle (x,y) will lie at the intersection of the
curves:
exp(-x) + 1 and ln(x) - 1.

x then must satisfy the equation:

exp(-x) + 1 = ln(x) - 1 (1)

(1) cannot be solved exactly, so you have to use numerical methods. Here I use
a Hyper-Lambert HW function to solve this equation numerically as:

ln(x) - exp(-x) = 2 =>
x*(ln(x)/x - exp(-x)/x) = 2 =>
x*exp(ln(ln(x)/x - exp(-x)/x)) = 2 =>
x = HW(ln(ln(x)/x - exp(-x)/x);2) =>
x =~ 7.393603058

Therefore the circle's origin will be at: (x, ln(x) - 1)
--
I.N. Galidakis
http://ioannis.virtualcomposer2000.com/

.

Relevant Pages

  • Re: SRs velocity addition -- ANY Experimental Evidence?
    ... >> All these curves are typical of ballistic predictions. ... >The FACT is that the periods of ALL Mira type stars, ... They are all very distance dependent. ... one is a giant. ...
    (sci.physics)
  • Re: SRs velocity addition -- ANY Experimental Evidence?
    ... >> All these curves are typical of ballistic predictions. ... >The FACT is that the periods of ALL Mira type stars, ... They are all very distance dependent. ... one is a giant. ...
    (sci.physics.relativity)
  • Re: fast curve similarity needed
    ... For a cuve defined by N points, the distance matrix would be the NxN array of all the pairwise distances. ... The distance matrix is invariant under rotations of the curve, and by normalizing the distances you can make it invariant under uniform scaling as well. ... The functions should have the option to match curves regardless of rotation in 3D space, and be preferably invariant in terms of uniform scale as well. ...
    (comp.graphics.algorithms)
  • Re: RAIB publish report on 2006 Ravenglass & Eskdale derailments
    ... "Parallel straight lines are ... "Two curves are parallel if there is a bijection between the points on the two curves, the bijection preserves the relative ordering of the ... distance between P1 and P2 along the normal is the same for all points on the curves." ... For those unfamiliar with the mathematics of infinities, that doesn't mean that the corresponding points have to be at equal distances from other corresponding points. ...
    (uk.railway)
  • Re: Circle Laying On Curves
    ... Consider curves e^and lnand a tangent unit circle. ... The set of points at distance 1 above the curve exp, ... If we call these loci fand g, then the solution must indeed satisfy the ...
    (sci.math)