Parallel CurvesDate: 05/21/97 at 19:27:01 From: Melodie A. Bernhard's students Subject: Geometry We know that line segments can be parallel, but can curves be parallel? We looked up the definition of parallel in our math book and in several others. We still aren't sure. Date: 05/27/97 at 17:03:04 From: Doctor Rob Subject: Re: Geometry Well, that depends! It depends on exactly what the wording is when you define the word "parallel." If you mean that the curves never intersect, then there can be parallel curves (can you think of an example?) If you mean that they intersect "at infinity," but nowhere else, then there can be parallel curves. If you mean that the curves are similar (have the same shape) and never intersect, that is a more difficult problem, but there is the case of concentric circles. A meaning I find interesting is that there is a certain minimum distance between any point on one curve and any point on the other, call it d, and, given any point on one curve, there is a point on the other curve with distance d between them. Again the concentric circle example serves, but now you can create more examples. I hope this helps. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 05/30/97 at 10:38:52 From: matt Subject: Latitude and Parallel Lines Dear Dr. Math, Are the lines of latitude parallel? Thanks! Chris and Matt Teacher's note: We were discussing parallel lines in class today and the students (third grade) got into a discussion about 'what if' scenarios. We learned about longitudinal lines in non-Euclidean geometry, but we would like to know if concentric circles are considered parallel in Euclidean geometry (I'm assuming that they are in non-Euclidean geometry) Thanks in advance for your help! Kim Given Date: 05/31/97 at 08:14:50 From: Doctor Jerry Subject: Re: Latitude and Parallel Lines Hi Chris and Matt Mathematicians tend to be fussy about language; this annoys some people but does save a lot of argument. Euclid talks about lines and by that term means "straight lines." Two lines in a plane are parallel if they are identical or never intersect. In space, it's more complicated. Lines can fail to intersect in space but can still be non-parallel. Can you give an example of this? In space two planes are parallel if they are identical or never intersect. One often speaks of "parallels of latitude." I suppose that the term parallel was used since the planes containing the circles of equal latitude are parallel. The term "lines of latitude" may mean the curves on a map that correspond to the parallels of latitude. Depending on the mathematical theory on which the map is based, the curves corresponding to the parallels of latitude may be parallel lines in the plane of the map (Mercator projection, for example) or curves or lines that intersect (polar projection, for example, in which all lines corresponding to the parallels of latitude meet at the pole). -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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