Plots and coordinates
- Thread starter Dale10101
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Bob Brown MSEE
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Plot b(a)=1/(sin(a)-cos(a))
seems to be a reasonable request.
Click on it.
Unless the problem specifies a coordinate system, Cartesian is assumed.
Sometimes people specify the intended coordinate system, by using variables' names reserved for that coordinate system.
(x,y,z) => Cartesian
(r, theta) => Polar
seems to be a reasonable request.
Click on it.
Unless the problem specifies a coordinate system, Cartesian is assumed.
Sometimes people specify the intended coordinate system, by using variables' names reserved for that coordinate system.
(x,y,z) => Cartesian
(r, theta) => Polar
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Thanks
Thanks for your response Bob, Wolfram alpha is very cool.
What I have trying to get at however is a confirmation (and this might once again be obvious to all but moi) that, given a plot of points in space, changing the coordinate system describing those points does not change the plot. I had thought that when changing an equation from, say Cartesian to Polar coordinates the change in the geometry of of the Polar coordinate system would project the plot into a new geometry as well, like when you project an image through a fish-eye lens. That does not seem to be the case (?).
Thanks for your response Bob, Wolfram alpha is very cool.
What I have trying to get at however is a confirmation (and this might once again be obvious to all but moi) that, given a plot of points in space, changing the coordinate system describing those points does not change the plot. I had thought that when changing an equation from, say Cartesian to Polar coordinates the change in the geometry of of the Polar coordinate system would project the plot into a new geometry as well, like when you project an image through a fish-eye lens. That does not seem to be the case (?).
HallsofIvy
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If you have a wire running from a power pole to your house, you can get a number of different numerical values for its length, depending on whether you measure in feet, yards, meters, etc. But you understand, don't you, that changing your measuring system does NOT change its length?!
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OK
I do understand that but in this case I was thinking that a change in coordinates would change the plot, i.e. the power line would look different if viewed through something called a "Polar equation lens" as opposed to a "Rectangular equation lens". The length of the power line would, of course not change, but the measurements taken from each frame would be numerically different but inter-convertible using the same formulas used in transforming the coordinates.
I believe you are confirming that I was wrong, that a "change of coordinates" is a transformation derived exactly not to produce a change in the plot, i.e. that a line, parabola, circle etc appears the same when plotted as a rectangular equation within its coordinate system as a polar equation within its coordinate system. That is, given an unlabeled plot you would not know which equation produced it.
The other type of transformation, say the one produced by a magnifying glass and described mathematically would be a transformation but not a coordinate transformation, or maybe it would but be distinguishable by context (?).
All rather obvious when you think in terms of using different coordinate systems developed to describe something like a power line but not so obvious when you are looking at the algebra of an equation transformation and not thinking of its application - easy to get lost in the trees. Thank you for your help,
(BTW, I have transformed and plotted several equations in both rectangular and polar form in there respective coordinate systems so I am not trying to concoct some sort of gotcha question ... its just that I find the result a personal epiphany ... simple fellow that I am.)
I do understand that but in this case I was thinking that a change in coordinates would change the plot, i.e. the power line would look different if viewed through something called a "Polar equation lens" as opposed to a "Rectangular equation lens". The length of the power line would, of course not change, but the measurements taken from each frame would be numerically different but inter-convertible using the same formulas used in transforming the coordinates.
I believe you are confirming that I was wrong, that a "change of coordinates" is a transformation derived exactly not to produce a change in the plot, i.e. that a line, parabola, circle etc appears the same when plotted as a rectangular equation within its coordinate system as a polar equation within its coordinate system. That is, given an unlabeled plot you would not know which equation produced it.
The other type of transformation, say the one produced by a magnifying glass and described mathematically would be a transformation but not a coordinate transformation, or maybe it would but be distinguishable by context (?).
All rather obvious when you think in terms of using different coordinate systems developed to describe something like a power line but not so obvious when you are looking at the algebra of an equation transformation and not thinking of its application - easy to get lost in the trees. Thank you for your help,
(BTW, I have transformed and plotted several equations in both rectangular and polar form in there respective coordinate systems so I am not trying to concoct some sort of gotcha question ... its just that I find the result a personal epiphany ... simple fellow that I am.)
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Ah! I see my confusion
When one converts from rectangular to polar coordinates one has the choice of creating a graph using the rectangular unit vectors x and y or the polar unit vectors r and theta.
I was mentally switching from rectangular to polar unit vectors when I changed from rectangular to polar coordinates, thus the power line goes from an arc along the x-axis to an arc wrapped around the unit circle. That was the major confusion in my mind ... how a change of coordinates seemed to change the plot of the points being described.
When one converts from rectangular to polar coordinates one has the choice of creating a graph using the rectangular unit vectors x and y or the polar unit vectors r and theta.
I was mentally switching from rectangular to polar unit vectors when I changed from rectangular to polar coordinates, thus the power line goes from an arc along the x-axis to an arc wrapped around the unit circle. That was the major confusion in my mind ... how a change of coordinates seemed to change the plot of the points being described.
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