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At one instant let the x,y intercepts be (a,b).The constant sum of ix,y, intercepts is a+ b = c ;
It is a parabola passing through (a+b,0), (0,a+b) tangential to x- and y- axes and remaining in the first quadrant. It has equation sqrt(x)+ sqrt(y)= sqrt(c) .
Method : let f = y-b - x*b /(b-c) the equation to the set of lines where b is a variable parameter. To find envelope or singular solution, eliminate b between f = 0 and partial derivative del f /del b = 0. It is called C-discriminant method in differential calculus. As a practice problem find envelope of y = C x - C^2. It is not a circle, a fact you will appreciate by scontinuing sketching it upto nail number 20.... Hope it helps.
Joseph Ferrara <jpferrara06379@...> wrote:
Joseph Ferrara <jpferrara06379@...> wrote:
Hi,
May I ask a question that has bugged me for some time.
I had a wall planck which was an ordinate and abscissa
consisting of evenly spaced nails embedded in the wood
about eight inches by eight.
Let's say the x axis and the y axis nails are numbered
from the origion 1,2,3,4,5,6,7,8,9...20. Then number
20 on the y axis is connected by a wire to number 1 on
the x axis, 19 on the y to 2 on the x, 18 on the y to
3 on the x, 17 on the y to 4 on the x...etc.
The effect was (very) roughly a quarter circle but
what exactly is the name of the curve that is
generated by this construction? Any mathematics would
be of interest.
Thank you very much.
Joseph Ferrara
> 1. Re: Help Needed
> From: "glnarasimham"
> <glnarasimham@...>
> Message: 1
> Date: Mon, 29 Aug 2005 23:00:55 -0000
> From: "glnarasimham" <glnarasimham@...>
> Subject: Re: Help Needed
>
> --- In curves_surfaces@yahoogroups.com, "aneesh
> venkatraman"
> <aneesh_82@y...> wrote:
> > I am looking for literature on the curve,
> "Alysoid". If anyone has
> > come across any book or a publication in which
> these curves are
> > described, please let me know. It is very urgent.
>
> Same as Catenary, a very familiar curve. From
> Mathworld Wolfram site:
>
> "The word catenary is derived from the Latin word
> for "chain." In
> 1669,Jungius disproved Galileo's claim that the
> curve of a chain
> hanging under gravity would be a parabola (MacTutor
> Archive). The
> curve is also called the alysoid and chainette. The
> equation was
> obtained by Leibniz , Huygens .."
G.L.Narasimham, Ex Advisor and Head, Product Design, Composites Group, Vikram Sarabhai Space Center, Indian Space Research Organization, Trivandrum 695013
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