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DATE 2017-03-01

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Key: id Value: 543735

Key: archive_learn Value: 2017-03-01

MESSAGE
DATE 2017-03-21
FROM Ruben Safir
SUBJECT Subject: [Learn] Anyone understand this well
From learn-bounces-at-nylxs.com Tue Mar 21 11:39:52 2017
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6. Cylindrical and spherical coordinates
Recall that in the plane one can use polar coordinates rather than
Cartesian coordinates. In polar coordinates we specify a point using
the distance r from the origin and the angle ? with the x-axis.
In polar coordinates, if a is a constant, then r = a represents a circle
of radius a, centred at the origin, and if ? is a constant, then ? = ?
represents a half ray, starting at the origin, making an angle ?.
Suppose that r = a?, a a constant. This represents a spiral (in
fact, the Archimedes spiral), starting at the origin. The smaller a, the
‘tighter’ the spiral.
By convention, if r is negative, we use this to mean that we point in
the opposite direction to the direction given by ?. Also by convention,
? and ? + 2? represent the same point. We may require r ? 0 and
0 ? ? < 2? and if we are not at the origin, this gives us unique polar
coordinates.
It is straightforward to convert to and from polar coordinates:
x = r cos ?
y = r sin ?,
and
r 2 = x2 + y 2
tan ? = y/x.
For example, what curve does the equation r = 2a cos ? represent?
Well if we multiply both sides by r, then we get
r 2 = 2ar cos ?.
So we get
x2 + y 2 = 2ax.
Completing the square gives
(x ? a)2 + y 2 = a2 .

**********I understand until here*******************

So this is a circle radius a, centred at (a, 0). Polar coordinates can be
very useful when we have circles or lines through the origin, or there is
a lot of radially symmetry.
Instead of using the vectors ?? and j?, in polar coordinates it makes
sense to use orthogonal vectors of unit length, that move as the point
moves (these are called moving frames). At a point P in the plane,
with polar coordinates (r, ?), we use the vector e?r to denote the vector
of unit length pointing in the radial direction:
e?r = cos ??? + sin ?j.
ˆ
1
e?r points in the direction of increasing r. The vector e?? is a unit vector
pointing in the direction of increasing ?. It is orthogonal to e?r and so
in fact
e?? = ? sin ??? + cos ?j.
ˆ
We will call a set of unit vectors which are pairwise orthogonal, an
orthonormal basis if we have two in the plane or three in space.
We want do something similar in space but now there are two choices
beyond Cartesian coordinates. The first just takes polar coordinates
in the xy-plane and throws in the extra variable z. So a point P is
specified by three coordinates, (r, ?, z). r is the distance to the origin,
??
of the projection P ? of P down to the xy-plane, ? is the angle OP ? makes
with the x-axis, so that (r, ?) are just polar coordinates for the point
P ? in the xy-plane, and z is just the height of P from the xy-plane.
x = r cos ?
y = r sin ?
z = z.
Note that the locus r = a, specifies a cylinder in three space. For
this reason we call (r, ?, z) cylindrical coordinates. The locus ? = ?,
specifies a half-plane which is vertical (if we allow r < 0 then we get
the full vertical plane). The locus z = a specifies a horizontal plane,
parallel to the xy-plane.
The locus z = ar specifies a half cone. At height one, the cone has
radius a, so the larger a the more ‘open’ the cone.
The locus z = a? is rather complicated. If we fix the angle, then we
get a line of this height and this angle. The resulting surface is called
a helicoid, and looks a little bit like a spiral staircase.
Again it is useful to write down an orthonormal coordinate frame. In
this case there are three vectors, pointing in the direction of increasing
r, increasing ? and increasing z:
e?r = cos ??? + sin ?j?
e?? = ? sin ??? + cos ?j?
ˆ
e?z = k.
The third coordinate system in space uses two angles and the dis­
tance to the origin, (?, ?, ?). ? is the distance to the origin, ? is the
angle made by the projection of P down to the xy-plane and ? is the
angle the radius vector makes with the z-axis. Typically we use coor­
dinates such that 0 ? z ? ?, 0 ? ? < 2? and 0 ? ? ? ?. To get
from spherical coordinates to Cartesian coordinates, we first convert to
2
cylindrical coordinates,
r = ? sin ?
? = ?
z = ? cos ?.
So, in Cartesian coordinates we get
x = ? sin ? cos ?
y = ? sin ? sin ?
z = ? cos ?.
The locus z = a represents a sphere of radius a, and for this reason
we call (?, ?, ?) cylindrical coordinates. The locus ? = a represents a
cone.
Example 6.1. Describe the region
x2 + y 2 + z 2 ? a2
and
x2 + y2 ? z2,
in spherical coordinates. The first region is the region inside the sphere
of radius,
? ? a.
The second is the region outside a cone. The surface of the cone is
given by z 2 = x2 + y 2. Now one point on this cone is the point (1, 1, 1),
so that this a right-angled cone, and the region is given by
?/4 ? ? ? 3?/4.
So we can describe this region by the inequalities
? ? a
and
?/4 ? ? ? 3?/4.
Finally, let’s write down the moving frame given by spherical coordi­
nates, the one corresponding to increasing ?, increasing ? and increasing
?.
x?? + yj? + zk?
e?? =?
x2 + y 2 + z 2
= sin ? cos ??? + sin ? sin ?ˆ
j + cos ?k.
ˆ
e?? = ? sin ??? + cos ?j.
ˆ
3
To calculate e??, we use the fact that it has unit length and it is
orthogonal to both e?? and e?? . We have
e?? = ±ˆ
e? × e ˆ ?
?
?
?
??
j?
k??
??
= ? sin ?
cos ?
0?? ??
?sin ? cos ? sin ? sin ? cos ??
= cos ? cos ??? + sin ? cos ?j? ? (sin2 ? sin ? + cos2 ? sin ?)k?
= cos ? cos ??? + cos ? sin ?j? ? sin ?k?
Now when ? increases, z decreases. So we want the vector with
negative z-component, which is exactly the last vector we wrote down.

--
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that Brooklyn, like Atlantis, reaches mythological
proportions in the mind of the world - RI Safir 1998
http://www.mrbrklyn.com

DRM is THEFT - We are the STAKEHOLDERS - RI Safir 2002
http://www.nylxs.com - Leadership Development in Free Software
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  1. 2017-03-02 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] neutal networks and pacman
  2. 2017-03-02 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Ultrametric networks: a new tool for phylogenetic analysis
  3. 2017-03-05 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] cost in evolution
  4. 2017-03-09 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] CatScan fossil files
  5. 2017-03-10 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] neural Networks and Quantum Mechanics
  6. 2017-03-12 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] I just found a GREAT video on partial derivatives
  7. 2017-03-13 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] Contact DOJ and thell them to blow it out their ass
  8. 2017-03-14 Ruben Safir <ruben-at-mrbrklyn.com> Re: [Learn] CatScan fossil files
  9. 2017-03-14 Ramon Nagesan <ramon.nagesan-at-gmail.com> Re: [Learn] CatScan fossil files
  10. 2017-03-16 Ruben Safir <ruben-at-mrbrklyn.com> Re: [Learn] is this up
  11. 2017-03-16 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] hang out is down
  12. 2017-03-16 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] is this working
  13. 2017-03-16 Charlie Gonzalez <itcharlie-at-gmail.com> Subject: [Learn] Registration for The Perl Conference 2017 is now open!!
  14. 2017-03-16 From: "soledad.esteban" <soledad.esteban-at-icp.cat> Subject: [Learn] [dinosaur] Advanced course Geometric Morphometrics in R,
  15. 2017-03-17 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Fwd: [dinosaur] Advanced course Geometric Morphometrics in
  16. 2017-03-17 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] good news !
  17. 2017-03-18 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] circles
  18. 2017-03-20 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Alice
  19. 2017-03-20 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] hough transform - Lecture 9
  20. 2017-03-21 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Anyone understand this well
  21. 2017-03-21 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Fwd: [dinosaur] Digital mapping of dinosaurian tracksites
  22. 2017-03-22 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] ODBASE 2017 - The 16th International Conference on
  23. 2017-03-24 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Decision Tree
  24. 2017-03-24 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Genetic Modification with decent
  25. 2017-03-27 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] Fwd: Re: hough transform - Lecture 9
  26. 2017-03-27 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] MOOCS
  27. 2017-03-27 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Peter Novig learning and on line teaching
  28. 2017-03-28 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] computations
  29. 2017-03-29 Christopher League <league-at-contrapunctus.net> Re: [Learn] This is hard to understand what the logic is here
  30. 2017-03-29 Ruben Safir <ruben-at-mrbrklyn.com> Re: [Learn] This is hard to understand what the logic is here
  31. 2017-03-29 Christopher League <league-at-contrapunctus.net> Re: [Learn] This is hard to understand what the logic is here
  32. 2017-03-29 Ruben Safir <mrbrklyn-at-panix.com> Re: [Learn] This is hard to understand what the logic is here
  33. 2017-03-29 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] perseptors
  34. 2017-03-29 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] This is hard to understand what the logic is here
  35. 2017-03-29 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] This is hard to understand what the logic is here
  36. 2017-03-30 Ruben Safir <mrbrklyn-at-panix.com> Re: [Learn] c arrays
  37. 2017-03-30 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] c arrays
  38. 2017-03-30 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] random weights
  39. 2017-03-30 Ruben Safir <mrbrklyn-at-panix.com> Subject: [Learn] randomize with commentary
  40. 2017-03-31 Ruben Safir <ruben-at-mrbrklyn.com> Subject: [Learn] Computational Paleo

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