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MathGroup Archive 2001

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Re: tangents and their respective equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg31933] Re: [mg31846] tangents and their respective equations
  • From: "Philippe Dumas" <dumasphi at noos.fr>
  • Date: Thu, 13 Dec 2001 01:08:36 -0500 (EST)
  • References: <200112071056.FAA10550@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Hi Manuel,

Clearly your problem is an ill-posed problem : it is obviously impossible to
derive the equation of one curve given its tangent at one point ! One reason
(among others) is that there are infinitely many curves being tangent to one
line at one particular point.
This said, there is an interesting question related to you (slightly
modified) problem. Namely, given a family of straight lines defined by an
equation f(m,x)=0 depending on a free parameter m (e.g. f(m,x) = 2*m*x -
m^2*y + m -1 = 0), what is the curve being envelopped by this family of
lines (i.e. what is the curve everywhere tangent to one particular line of
the family ) ?

The answer is rather straigthfoeward (at least in principle) : the equation
of the curve is obtained by eliminating the free parameter m between f(m,x)
= 0  and D[f(m,x),m] = 0 where D stands for the partial derivative of f vs.
m. In the given example, the solution is to eliminate m between:

2*m*x - m^2*y + m -1 = 0    (1)
 and
2*x -2*m*y +1 = 0                  (2)

introducing this value in (1)

Obviously one has to take care of problems arising because of divisions by
zero (here at y=0)

I hope this will be helpful.

Philippe Dumas
99, route du polygone
03 88 84 67 80
67100 Strasbourg

----- Original Message -----
From: "Manuel Avalos" <manuel at voicenet.com>
To: mathgroup at smc.vnet.net
Subject: [mg31933] [mg31846] tangents and their respective equations


>
> Hi everyone:
>
> I am trying to find the equation of a curve given its tangent and an
> arbitrary point. Is there a mathematica solution for this?
> For example: If I give you the tangent to a curve: -4+11x   at x =2,
> Is it possible to derive the equation of the curve  from that tangent?
> Thanks for whatever.
> Manuel
>
>



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