Spherical coordinate system- sketching curves for constant parameters

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November 24th 2009, 04:36 PM
choccookies

Spherical coordinate system- sketching curves for constant parameters

x=coshEsinTcosQ y=coshEsinTsinQ z=sinhEcosT

where (E,T,Q) refers to the spherical coordinate system.

a) Describe in words and sketch the curves on which E and T are constant.

b) Describe and sketch the curves on which E and Q are constant.

c) Describe and sketch the SURFACES on which E is constant.
November 25th 2009, 02:29 AM
choccookies

anyone have any ideas? i'm really struggling :(
November 25th 2009, 03:40 AM
shawsend

Quote:

Originally Posted by choccookies

x=coshEsinTcosQ y=coshEsinTsinQ z=sinhEcosT

where (E,T,Q) refers to the spherical coordinate system.

a) Describe in words and sketch the curves on which E and T are constant.

b) Describe and sketch the curves on which E and Q are constant.

c) Describe and sketch the SURFACES on which E is constant.

Hi. That's not spherical coordinates right. That's just a user-defined coordinate system:

$x=\cosh(e)\sin(t)\cos(q)$
$y=\cosh(e)\sin(t)\sin(q)$
$z=\sinh(e)\cos(t)$

and with e and t constant, you have:

$x=k\cos(q)$
$y=k\sin(q)$
$z=c$

and as q ranges in some interval say (-10,10), the x and y coordinates just go round a circle of radius k. and the z coordinate is constant say z=5. Then the first set is the circle a distance c above the x-y plane with a radius of k.
Looks to me anyway.
November 25th 2009, 03:49 AM
choccookies

hi, thanks for your help.
yeah i didnt know how to type the real symbols for the sphericals. Oh also I forgot to put the conditions;

E>= 0, Pi>=T>=0, 2Pi>=Q>=0

How would I draw the surface? If E was constant I;d get
x=sinTcosQ
y=sintTsinQ
z=cosT

This seems too tricky to plot?
November 25th 2009, 03:57 AM
shawsend

Quote:

Originally Posted by choccookies

hi, thanks for your help.
yeah i didnt know how to type the real symbols for the sphericals. Oh also I forgot to put the conditions;

E>= 0, Pi>=T>=0, 2Pi>=Q>=0

How would I draw the surface? If E was constant I;d get
x=sinTcosQ
y=sintTsinQ
z=cosT

This seems too tricky to plot?

Build it up piece by piece. First let $t=\pi/2$ . Then we have:

$x=k\cos(q)$
$y=k\sin(q)$
$z=0$

so that's obviously some curve in the x-y plane right? And the sine and cosine thing means it's a circle with radius k. So draw that one. Now, how about letting t=0? Then do a few more and see what it starts looking like. Also, compare your user-defined coordinate system with the standard spherical coordinate transformations and note the similarities:

$x=\rho\sin(\phi)\cos(\theta)$
$y=\rho\sin(\phi)\sin(\theta)$
$z=\rho\cos(\phi)$

That's kinda' tough for me. Here's what it looks like in Mathematica as e is varied between 0 and 1. Try and get a machine to run the code on it, see what's happening, then try and explain it analytically:

Code:

Manipulate[ ParametricPlot3D[{Cosh[e] Sin[t] Cos[q], Cosh[e] Sin[t] Sin[q], Sinh[e] Cos[t]}, {t, 0, \[Pi]}, {q, 0, 2 \[Pi]}, PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}], {e, 0, 1}]