Polar Coordinates and Logarithmic SpiralsDate: 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/ |
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