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Polar Coordinates and Logarithmic Spirals

Date: 04/25/2004 at 23:02:15
From: Phil
Subject: a question on the equation for logarithmic spirals

The polar equation for a logarithmic spiral is given by r = ae^bq , 
where a and b are constants and r and q are the coordinates.  My 
problem is that a logarithmic spiral has more than one r value for 
each q value.

Since this equation uses the polar coordinate system, q obviously has
to be between 0 and 360 degrees, and if you were to plug all of these
numbers in, you would get one value for r for each.  However, in a
drawing of a logarithmic spiral, if you look at say a forty-five
degree angle, it will intersect the spiral at MORE THAN ONE PLACE,
which means each q has more than one r.  I do not understand how this
is possible.

I have just started to learn about polar coordinates.  Am I missing
something?



Date: 04/26/2004 at 12:49:42
From: Doctor Peterson
Subject: Re: a question on the equation for logarithmic spirals

Hi, Phil.

Although for some purposes you would want to restrict the angle in 
polar coordinates to [0, 360), so that each point has a unique pair 
of coordinates, that is not a requirement.  Every point P corresponds 
to infinitely many angles q.  Every pair (r,q) that satisfies your 
equation represents a point on the graph, and that gives the complete 
spiral.  There is only one r for each q, but infinitely many q's for 
each direction.

Here is one of many discussions on the web (which I found using 
Google to look for the phrases "polar coordinates" and "not unique") 
that point out this feature of polar coordinates:

  Polar Coordinates
    http://www.lhs.logan.k12.ut.us/~rweeks/trig/polar.htm 

  Note: r is the "directed distance" from the origin or pole.  If
  r < 0, then the point is plotted 180  from where it normally
  would be plotted.  Because of this and the fact that there are
  an infinite number of angles coterminal with each other, the
  polar form of the coordinates of a point are NOT unique, unlike
  cartesian coordinates (or rectangular coordinates) which ARE
  unique.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 04/26/2004 at 21:36:04
From: Phil
Subject: Thank you (a question on the equation for logarithmic spirals)

Thanks for your help, Doctor Peterson.  Your explanation of a point
having infinite values for q made sense after I thought about it,
since 1 degree obviously would equal 361 degrees, 721 degrees, 1081
degrees, and so on.  It was just that I had read on a site that q was
restricted between 0 and 360 degrees.  I also had not heard of what to
do if r was less than zero.  You were a load of help.  Thanks again.



Date: 04/27/2004 at 08:14:25
From: Doctor Peterson
Subject: Re: Thank you (a question on the equation for logarithmic
spirals)

Hi, Phil.

You're welcome.

Generally, if you want to label an existing point, you will choose 
the principal angle, between 0 and 360 (or perhaps -180 to 180 in 
some cases); but you should recognize that that is only one possible 
name for it.  It's when you plot an equation that you have to be sure 
to allow all angles, and not stop at 360.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Equations, Graphs, Translations
High School Functions

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