Here, I've added the draggable complex point q through which there's
a unique equiangular spiral going through (1, 0) and with "collapsed
radius" of 1. By "collapsed radius", I mean the radius of the circle
we get when the ray-intersection angle is a right angle.

It can be seen how for any ray, there's a family of spirals and that
bringing the test point onto the ray selects the spiral that matches
the rectangular-coordinates one.

Next, to find out how we can do the reverse---find the rectangular-
coordinates spiral that matches a given real-power-of-complex-point
spiral.

Also on the back burner: illustrating construction of spirals via
buildup using increasing number of rays, as described in http://
mathworld.wolfram.com/LogarithmicSpiral.html :

Here, I've just used the unit normal vector to thicken the border of
the main triangle to distinguish it from the sub-triangles (thinner,
colored borders).

Even thicker borders, with bigger points for the vertices, also:

If we extend the sides of the triangle, we can make out how the
different regions are related to the way the subtriangles overlap.

If P is in a vertical angle [Is this the right term?] of one of the
vertices, then the exterior subtriangles won't overlap the interior
ones.

If we move P over into the "shadow" of the third side, BC, (imagining
point A to be emitting light beams in all directions), then the third
sub-triangle (yellow)overlaps the triangle, while the second
subtriangle (light blue) remains outside the triangle.

Or, just in terms of the subtriangles, subtriangles two and three
overlap.

I'm not exactly sure why this is working, but I've finally got
overlapping stripes to show how the positive area (in this case, the
yellow triangle outside triangle ABC)

Red, orange, and yellow are used for positive areas:

Green, light blue, and dark blue are used for negative areas:

The the signed areas of any point P with respect to the vertices always
add up to the area of the triangle ABC:

Thought you might be interested in this as an example of illustrating
algebraic geometry with Graphing Calculator. Free GC viewers are
available at http://www.pacifict.com/FreeStuff.html

Simplified the coloring routines. My previous example was needlessly
complicated. Still painfully slow on my iMac, unfortunately. But good
for making illustrations, perhaps, more than for interactive use,
unless you've got one of the latest computers.

Here we can see better how the "code" works.  Each component of the
vector returned by T(A, B, C, P) represents the area of the
sub-triangle formed by the test point P and two of the vertices of the
triangle. That is,

is the area of the first sub-triangle, triangle PAB

is the area of the second sub-triangle, triangle PBC

is the area of the third sub-triangle, triangle PCA

So, as the second sub-triangle "takes over" the whole area of the
triangle, that means that the test point must be approaching the first
vertex.

If the third sub-triangle is the biggest, that means we're near the
second vertex (the "previous" one if the vertices are considered in
cyclic order).

If the first sub-triangle is the biggest, that means we're near the
third vertex (the "previous" one if the vertices are considered in
cyclic order).

If sub-triangle ABP is approaching zero while the other two
sub-triangles are positive, that means the test point P is near side
AB:

If we go across side AB, the triangle APB becomes negative in area,
This is  shown by a green hue, which is unfortunately covered up by the
other triangles. I need to do some kind of cross-hatching here.

If we "flip the triangle over", by making the points in clockwise
order, than the area of the triangle as a whole becomes negative.

Here's an attempt to have striped coloring. Can't get negative areas
yet, and it's too slow to be usable on my iMac.
Cross-hatching with lines might be OK, though.

If it worked, it would show how the areas of the sub-triangles with
respect to an external point add up to the area of the triangle ABC,
since the part that's outside is negative in area!  See Klein,
"Elementary Mathematics from an Advanced Standpoint: Geometry", page 7
for details.

Hi, everyone Im new within you. I m a MSc student. I m researching
Spherical Indicators in Lorentzian Space. Could you have some files
abaout it?

Sir, Actually i am doing project  RF RANGE FINDER which will calculate
the location and direction of FM radio station only within certain
range u can say if device is placed in ur islamic university it will
tell us location and direction of FM radiostation from my device.It is
hardware based project and interfacing on Laptop

Now
Problem 1
I knew the three different physical path of certain FM radio station
from my device.Now how i will determine the point of intersection of
these three physical paths by using trilateration equations.(Three
paths are must be taken by placing our device at three different
points because we are using trilateration).
& what will be the minimum distance between two points (i.e device
location )
Problem 2
Which trilatertion equation help me in calculating my cyclic error.

>
>
>Date: Tue, 25 Oct 2005 21:06:27 -0700 (PDT)
>   From: Narasimham Gudipaty <glnarasimham@...>
>Subject: Re: A new Clifford torus type surface using a six cordinate triaxial
model
>
>Interesting. Any special properties?(chirality etc.) Seems to be a
self-intersecting surface. PlotPoints->{41,21} increased surface smoothness of
g1 and g2.   3D viewing could be better with RealTime3D.  Regards
>
>Roger Bagula <rlbagulatftn@...> wrote:A new Clifford torus type surface
using a six cordinate triaxial model:
>( Mathematica Notebook)
>x1 = Cos[t]; x2 = Cos[t + 2*Pi/3]; x3 = Cos[t - 2*Pi/3];
>y1 = Cos[p]; y2 = Cos[p + 2*Pi/3]; y3 = Cos[p - 2*Pi/3];
>x = x1*(Sqrt[2] + y1)/(Sqrt[2] + x3*y3)
>y = x2*(Sqrt[2] + y2)/(Sqrt[2] + x3*y3)
>z = 1/(Sqrt[2] + x3*y3)
>g1 = ParametricPlot3D[{x, y, z}, {t, 0, 2*Pi}, {p, 0, Pi}]
>g2 = ParametricPlot3D[{x, y, z}, {t, 0, 2*Pi}, {p, Pi, 2*Pi}]
>Show[{g1, g2}]
>Show[{g1, g2}, ViewPoint -> {-0.998, 0.828, 3.125}
>
The symmetrical version :
x=x1/(Sqrt[2]+y1)
y=x2/(Sqrt[2]+y2)
z=x3/(Sqrt[2]+y3)
is very pretty.
Roger L. Bagula { email: rlbagula@... or rlbagulatftn@... }

11759 Waterhill Road,
Lakeside, Ca. 92040    telephone: 619-561-0814

A new Clifford torus type surface using a six cordinate triaxial model:
( Mathematica Notebook)
x1 = Cos[t]; x2 = Cos[t + 2*Pi/3]; x3 = Cos[t - 2*Pi/3];
y1 = Cos[p]; y2 = Cos[p + 2*Pi/3]; y3 = Cos[p - 2*Pi/3];
x = x1*(Sqrt[2] + y1)/(Sqrt[2] + x3*y3)
y = x2*(Sqrt[2] + y2)/(Sqrt[2] + x3*y3)
z = 1/(Sqrt[2] + x3*y3)
g1 = ParametricPlot3D[{x, y, z}, {t, 0, 2*Pi}, {p, 0, Pi}]
g2 = ParametricPlot3D[{x, y, z}, {t, 0, 2*Pi}, {p, Pi, 2*Pi}]
Show[{g1, g2}]
Show[{g1, g2}, ViewPoint -> {-0.998, 0.828, 3.125}

>Roger L. Bagula { email: rlbagula@...  or  rlbagulatftn@... }
>
>
11759 Waterhill Road,
Lakeside, Ca. 92040    telephone: 619-561-0814

Dear glnarasimham,
Just joined egroup here.
Could you repond to my email at:
  rlbagulatftn@...
with your current email address.

Roger

The natural equation of a given curve is curvature = f(s). Find the
natural equation of an offset curve distant + / - p from it. p is
parallel distance and s arc length.

Hi,

May I ask a question that has bugged me for some time.
I had a wall plack 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

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> There is 1 message in this issue.
>
> Topics in this digest:
>
>       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 .."
>
>
>
>
>
________________________________________________________________________
>
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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 .."

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com

Hi,

May I ask a question that has bugged me for some time.
I had a wall plack 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

--- curves_surfaces@yahoogroups.com wrote:

> ------------------------ Yahoo! Groups Sponsor
> --------------------~-->
> Get fast access to your favorite Yahoo! Groups. Make
> Yahoo! your home page
>
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>
--------------------------------------------------------------------~->
>
>
> There is 1 message in this issue.
>
> Topics in this digest:
>
>       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 .."
>
>
>
>
>
________________________________________________________________________
>
________________________________________________________________________
>
>
>
>
------------------------------------------------------------------------
> Yahoo! Groups Links
>
>
>     curves_surfaces-unsubscribe@yahoogroups.com
>
>
>
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>
>
>
>
>

--- 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 .."

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.

--- In SingleVariableCalculus@yahoogroups.com, "mcyberliver"
<mcyberliver@y...> wrote:




Our group is devoted to the study of Single Variable Calculus.
Topics open for discussion include: limits and continuity,
the derivative, transcendental functions, analysis of functions and
their graphs, applications of the derivative, integration,
applications of the definite integral, hyperbolic functions, methods
of integration, infinite series, and analytic geometry.


http://groups.yahoo.com/group/SingleVariableCalculus/
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Added a Mathematica notebook and Graphing Calc file to the Kuen surface:
http://xahlee.org/surface/kuen/kuen.html

try to plot it in your fav plotter!
quite a fascinating surface.

  Xah

btw, excuse my ignorance, what exactly is a soliton?

   Xah

On Apr 24, 2005, at 5:55 AM, Narasimham Gudipaty wrote:

Breather breathtaking indeed !!
 
Allow me acomment about related  Three soliton surface with K= -1. One
sees K > 0 areas in the peripheral regions of the surface. I had sent a
messsage before also, to Prof. Richard Palais. I doubt if it would give
a flat 3D Plot. 
 
Best Regards.

xah lee <xah@...> wrote:
> just made an update to the breather surface, with graphing calc file &
> Mathematica file.
>
> http://www.xahlee.org/surface/breather_p/breather_p.html
>
>   Xah
>
>   ☄


G.L.Narasimham, Ex Advisor and Head, Product Design, Composites Group,
Vikram Sarabhai Space Center, Indian Space Research Organization,
Trivandrum 695013


   ☄

> Better to say Reflector dishes instead of reflecting disks.

I took your advice.

  > All these dishes have essentially a parabolic meridian,...

if that means the surface is a surface of revolution. If so, the focus
must be in the center. But that TV disk has the receiving bar a bit
below the center, thus prompted me to suspect that the surface is not
parabolic. i.e. not a surface of revolution exactly.

  Xah


---------
Better to say Reflector dishes instead of reflecting disks.

The TV dish receiver is not necessarily parabolic, since the focus is
below the center of the dish as we can clearly see in the photo.

Not clear what this means. All these dishes have essentially a parabolic
meridian, truncated at various depths. Dish designs have focal length f
/ Dish diameter d ratio ( known as f/d in the industry ) between say 0.5
for shallow dishes to .25 for deep dishes terminated at latus rectum,
normal value is around 0.3.

Regards
Narasimham

Xah Lee <xah@...> wrote:
just created this page:
http://xahlee.org/Whirlwheel_dir/reflecting_disks/reflecting_disks.html

a photo gallery of reflecting disks. (radio dishes, solar dishes)

Xah

excuse my ignorance...(for, this is really a difficult subject :) what
exactly is a soliton?

  Xah




Breather breathtaking indeed !!

Allow me acomment about related  Three soliton surface with K= -1. One
sees K > 0 areas in the peripheral regions of the surface. I had sent a
messsage before also, to Prof. Richard Palais. I doubt if it would give
a flat 3D Plot.

Best Regards.

xah lee <xah@...> wrote:
just made an update to the breather surface, with graphing calc file &
Mathematica file.

http://www.xahlee.org/surface/breather_p/breather_p.html

   Xah

   ☄


G.L.Narasimham, Ex Advisor and Head, Product Design, Composites Group,
Vikram Sarabhai Space Center, Indian Space Research Organization,
Trivandrum 695013

yes i remember seeing that in Nature store.

I thought i saw it on wikipedia too... but anyway they have a good
article on mirage
http://en.wikipedia.org/wiki/Mirage

   Xah

On Apr 23, 2005, at 1:29 PM, Hop David wrote:


There was a toy that had a virtual pig floating in air. There was an
actual pig at the base of the toy and his virtual image was at the focus
of a parabolic mirror.  I believe the toy had two parabolic mirrors
sharing a focus.


Xah Lee wrote:
> just created this page:
> http://xahlee.org/Whirlwheel_dir/reflecting_disks/reflecting_disks.html
>
> a photo gallery of reflecting disks. (radio dishes, solar dishes)
>
> Xah
>
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--
Hop David
http://clowder.net/hop/index.html





Yahoo! Groups Links








   ☄

just created this page:
http://xahlee.org/Whirlwheel_dir/reflecting_disks/reflecting_disks.html

a photo gallery of reflecting disks. (radio dishes, solar dishes)

Xah

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just made an update to the breather surface, with graphing calc file &
Mathematica file.

http://www.xahlee.org/surface/breather_p/breather_p.html

  Xah

  ☄

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There was a toy that had a virtual pig floating in air. There was an
actual pig at the base of the toy and his virtual image was at the focus
of a parabolic mirror.  I believe the toy had two parabolic mirrors
sharing a focus.


Xah Lee wrote:
> just created this page:
> http://xahlee.org/Whirlwheel_dir/reflecting_disks/reflecting_disks.html
>
> a photo gallery of reflecting disks. (radio dishes, solar dishes)
>
> Xah
>
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--
Hop David
http://clowder.net/hop/index.html

just created this page:
  http://xahlee.org/Whirlwheel_dir/reflecting_disks/reflecting_disks.html

a photo gallery of reflecting disks. (radio dishes, solar dishes)

  Xah

just made an update to the breather surface, with graphing calc file &
Mathematica file.

http://www.xahlee.org/surface/breather_p/breather_p.html

   Xah

   ☄

interesting curve.

i also find it interesting to know its application or how it came up.

   Xah
   xah@...
∑ http://xahlee.org/





On Apr 11, 2005, at 10:58 PM, Narasimham Gudipaty wrote:

An intersting 2D locus, however, I don't know its name. 
 
Perhaps the parameteric form is simpler. It is not difficult to derive
a relation between
 th and an auxiliary angle om(ega). In Mathematica :


   a=1; b=.36; om=ArcTan[Cos[th],Sin[th] (b/a)^2];

cs=Cos[th];sn=Sin[th];CS=Cos[om];SN=Sin[om];

x= a cs; y= b sn; X=a (cs + SN)/2 ; Y= b (sn - CS)/2;

ell= ParametricPlot[{x,y},{th,0,2 Pi}]; anon=
ParametricPlot[{X,Y},{th,0,2Pi}];

Show[ell,anon];
But, what is its application, or , how does it come about ? Depending
on that you could choose its name.
 
Best Regards
 
G.L.Narasimham

garciacapitan <pacoga@...> wrote:

I am new in this group. I am a Spanish math teacher in a secondary
school and I have a question about a curve.

Let P move on the ellipse x^2/a^2+y^2/b^2 = 1 and let Q be one of
points such that QOP is a right angle.

   What is the locus of the midpoint M of chord PQ?

I have found that the equation of this locus is

(a^2 + b^2)*(b^2*X^2 + a^2*Y^2)^2 == a^2*b^2*(b^4*X^2 + a^4*Y^2)

But, what is the mame of this curve?

You can find a picture at

http://garciacapitan.auna.com/problemas/lugar1

Best regards,

Francisco Javier García Capitán
http://garciacapitan.auna.com


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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