polar coordinate curvesThere are two ways of plotting a polar coordinate curve 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, but you must invoke the 1-1 option (or scaling=constrained) to see circles as circles. Consider this circle: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Either we plot 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 as a parametrized curve as usual, substituting for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as a function of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMg== :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or we use the polar coordinate option which assumes the 1-1 option and plots LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRJyYjOTUyO0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHRj5GP0Y/RkFGPw== versus \316\270: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or enter the expression for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNiUtRiw2JlEnJiM5NTI7RicvRjBGNEYyL0Y2USdub3JtYWxGJ0YyRkFGMkZBRjJGQQ== and from the context sensitive menu choose Plotbuilder, then 2d polar plot, which adds the polar coordinate grid (it won't work for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiPUYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRIjJGJ0YyRjxGMkY8, too simple):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIn5GJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkctSSNtaUdGJDYlUSRzaW5GJy8lJ2l0YWxpY0dGOEYvLUkobWZlbmNlZEdGJDYkLUYjNiUtRks2JVEidEYnL0ZPUSV0cnVlRicvRjBRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdGOEYvRi9GZm5GLw==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State_088171406_1852_193139816415_0
If the curve is complicated enough, one may need to increase the number of points plotted with the option numpoints = 200 (play with the number, whose default is 50). None of our examples here need this.Try this parametrized plot again for the polar coordinate 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Here is an interesting curve. How do we figure out its period as a closed curve?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 question.the answer (ignore this, unless you want to see some cool math videos)The radial oscillation period is found by 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The angular period is of course LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSNtaUdGJDYmUScmIzk2MDtGJy8lJ2l0YWxpY0dGMUYvRjJGL0Yy, so we need to find the common period LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiYtRi82JlEnY29tbW9uRidGMkY1RjhGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0Y1L0Y5USdub3JtYWxGJw== such that an integer number of each of the two periods fits into the common period. This means both the radius and angle return to the same [equivalent] values at the end of the common period.
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Here is an animation of the curve as it is traced out. You have to click on it to get the video controls on the tool bar.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Looks like here we still don't need to worry about the plot being jagged, numpoints is not needed.
If we change the frequency to an irrational number, there is no common period and the figure continues to rotate around the origin each time with a slight advance until the entire circular region will be covered eventually.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 we wait a bit longer...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't that amazing?10.3.67 nephroid of Freeth?If we divide theta by an integer, we multiply the period by that integer to trace out the full curve (multivalued as a function of the physical angular directions between 0 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSNtaUdGJDYmUScmIzk2MDtGJy8lJ2l0YWxpY0dGMUYvRjJGL0Yy). The animate curve helps us see how it is traced out. For this case there are two loops starting from the origin and letting LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEnJiM5NTI7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGMS8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGMkY0 increase by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEifkYnRi9GMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSNtaUdGJDYmUSUmcGk7RicvJSdpdGFsaWNHRjFGL0YyRi9GMg== which cross over each other because of the negative values of the radius, making this interesting pattern below.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Looks like the horizontal axis labels are screwed up in Maple 2019.-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6'-I#miGF$6%Q%withF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I(mfencedGF$6$-F#6%-F,6%Q&plotsF'F/F2/%+executableGQ&falseF'/F3Q'normalF'F@-I#moGF$6-Q":F'F@/%&fenceGF?/%*separatorGF?/%)stretchyGF?/%*symmetricGF?/%(largeopGF?/%.movablelimitsGF?/%'accentGF?/%'lspaceGQ,0.2777778emF'/%'rspaceGFVF=F@-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF'6&-I#miGF$6%Q-animatecurveF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I(mfencedGF$6$-F#65-I#mnGF$6$Q"1F'/F3Q'normalF'-I#moGF$6-Q"+F'F>/%&fenceGQ&falseF'/%*separatorGFF/%)stretchyGFF/%*symmetricGFF/%(largeopGFF/%.movablelimitsGFF/%'accentGFF/%'lspaceGQ,0.2222222emF'/%'rspaceGFU-F;6$Q"2F'F>-FA6-Q"~F'F>FDFGFIFKFMFOFQ/FTQ&0.0emF'/FWFin-F,6%Q$sinF'/F0FFF>-F66$-F#6&-I&mfracGF$6(F:-F#6$FXF>/%.linethicknessGF=/%+denomalignGQ'centerF'/%)numalignGF\p/%)bevelledGFFFen-F,6%Q(&theta;F'F^oF>F>F>-FA6-Q",F'F>FD/FHF1FIFKFMFOFQFhn/FWQ,0.3333333emF'Fap-FA6-Q"=F'F>FDFGFIFKFMFOFQ/FTQ,0.2777778emF'/FWF^q-F;6$Q"0F'F>-FA6-Q#..F'F>FDFGFIFKFMFOFQFSFjn-F;6$Q"4F'F>Fen-F,6%Q%&pi;F'F^oF>Fdp-F,6%Q'coordsF'F/F2Fjp-F,6%Q&polarF'F^oF>F>F>/%+executableGFFF>%!GM7R0
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