Space curves: Plotting Vector-Valued Functions in Maple Calc IV Lab Week One by Karen Donnelly all rights reserved Initializations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== Load the plots package so that the animate command may be used. Load the VectorCalculus package so that vectors can be defined and we can use the SpaceCurve command. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSS9WZWN0b3JDYWxjdWx1c0dGJA== print(); # input placeholder JSFH Define options for plotting and name opts to save typing in subsequent plotting commands (saves typing): Pkklb3B0c0c2IjYkL0klYXhlc0dGJEkmYm94ZWRHRiQvSSdsYWJlbHNHRiQ3JUkieEdGJEkieUdGJEkiekdGJA== print(); # input placeholder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

The easiest way to plot a curve in space in Maple is to use the SpaceCurve function which is part of theVectorCalculus library. For example, to plot the helix curve 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 for t between 0 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1: LUkrU3BhY2VDdXJ2ZUc2IjYlLUkkPCw+R0YkNiUtSSRjb3NHRiQ2I0kidEdGJC1JJHNpbkdGJEYrRiwvRiw7IiIhLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiRJL1ZlY3RvckNhbGN1bHVzRzYkRjVJKF9zeXNsaWJHRiQ2JCIiJUkjUGlHRjVJJW9wdHNHRiQ= To see how the curve is traced out as t increases from 0 to 4*Pi we can use the animate command applied to the SpaceCurve command -- The syntax for this form of the animate command is "animate( plotcommand, [ list of arguments for the plot command ], variablename = minvalue..maxvalue)" Maple will generate 25 frames equally spaced on the animation parameter variablename for the range = minvalue..maxvalue. For example: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2I1EhRictSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGNC8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GMDYmRjJGNUY4L0Y7USVhdXRvRidGKy8lK2V4ZWN1dGFibGVHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYn print(); # input placeholder LUkoYW5pbWF0ZUc2IjYlSStTcGFjZUN1cnZlR0YkNyUtSSQ8LD5HRiQ2JS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJ0R0YkLUkkc2luR0YtRjBGMS9GMTsiIiFJIlRHRiRJJW9wdHNHRiQvRjc7RjYtSSIqRzYkRi4vSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0dGLTYkIiIlSSNQaUdGLg== %; %; A more complicated example is given by the following knot curve. ( A mathematical knot is a curve that begins and ends at the same point.) QyQ+SSVrbm90RzYiLUkkPCw+R0koX3N5c2xpYkdGJTYlLUkiK0c2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRi1GKDYkLUYrNiQtSSItR0YsNiMtSSIqR0YsNiQiIzUtSSRjb3NHRjE2I0kidEdGJS1GNjYjLUY5NiQiIiMtRj02Iy1GOTYkIiImRj8tRjk2JCIjOi1JJHNpbkdGMTYjLUY5NiRGREY/LUYrNiQtRis2JC1GNjYjLUY5NiRGTC1GPUZPLUY5NiRGOy1GTkY+LUY2NiMtRjk2JEZELUZORkYtRjk2JEY7LUY9NiMtRjk2JCIiJEY/IiIi LUkrU3BhY2VDdXJ2ZUc2IjYlSSVrbm90R0YkL0kidEdGJDsiIiEtSSIqRzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHNiRGLkkoX3N5c2xpYkdGJDYkIiIjSSNQaUdGLkklb3B0c0dGJA== print(); # input placeholder Here we animate the above curve to see show its orientation as it is drawn: LUkoYW5pbWF0ZUc2IjYlSStTcGFjZUN1cnZlR0YkNyVJJWtub3RHRiQvSSJ0R0YkOyIiIUkiVEdGJEklb3B0c0dGJC9GLTtGLC1JIipHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0c2JEY0SShfc3lzbGliR0YkNiQiIiNJI1BpR0Y0 A curve in the plane is just a special case a curve in space. If the curve is in the x-y plane, the k coordinate is 0. Thus the plot of 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 is given by: LUkrU3BhY2VDdXJ2ZUc2IjYlLUkkPCw+R0koX3N5c2xpYkdGJDYlLUkkc2luRzYkJSpwcm90ZWN0ZWRHRig2I0kidEdGJC1JJGNvc0dGLEYuIiIhL0YvO0YyLUkiKkc2JEYtL0krbW9kdWxlbmFtZUdGJEkvVmVjdG9yQ2FsY3VsdXNHRiw2JCIiI0kjUGlHRi1JJW9wdHNHRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= print(); # input placeholder In animated form: LUkoYW5pbWF0ZUc2IjYlSStTcGFjZUN1cnZlR0YkNyUtSSQ8LD5HSShfc3lzbGliR0YkNiUtSSRjb3NHNiQlKnByb3RlY3RlZEdGKjYjSSJ0R0YkLUkkc2luR0YuRjAiIiEvRjE7RjRJIlRHRiRJJW9wdHNHRiQvRjc7RjQtSSIqRzYkRi8vSSttb2R1bGVuYW1lR0YkSS9WZWN0b3JDYWxjdWx1c0dGLjYkIiIjSSNQaUdGLw==

Exercise 39, page 838 Graph and identify the curve: 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 i +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= j + 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kLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= %; 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 This is a _______________________ Exercise 43, page 838: Graph the vector valued function 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 i + 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 j + 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k over the range t = 0 to 4Pi. Then make a conjecture about each of the transformations of the graph without plotting the transformations!! Next, plot to check your answers. (Use the display command as below to see both on the same graph). Correct your conjectures if you made a mistake. Part a) is partially completed for you. Complete part a) and the remaining parts in a similar fashion. JSFH JSFH %; print(); # input placeholder Graph of original function: QyU+SSJQRzYiLUkrU3BhY2VDdXJ2ZUdGJTYmLUkkPCw+R0koX3N5c2xpYkdGJTYlLUkiKkc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHRiVJL1ZlY3RvckNhbGN1bHVzRzYkRjBGKzYkIiIjLUkkY29zR0Y0NiNJInRHRiUtRi42JEY2LUkkc2luR0Y0RjktRi42JEY6IyIiIkY2L0Y6OyIiIS1GLjYkIiIlSSNQaUdGMC9JJmNvbG9yR0YlSSRSRURHRiVJJW9wdHNHRiUhIiJGJA== Part a) Conjecture: The graph of the transformed curve will be the same, except that the x-coordinates (i component) will be shifted by -2. 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 LUkoZGlzcGxheUc2IjYkSSJQR0YkSSNQYUdGJA== Part b) Conjecture: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRIlBGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI1BiRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ== Part c) Conjecture: JSFH 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRIlBGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI1BjRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ== Part d) Conjecture: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRIlBGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI1BkRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ== Part e) Conjecture: JSFH 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEoZGlzcGxheUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYnLUYsNiVRIlBGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRI1BlRidGL0YyLyUrZXhlY3V0YWJsZUdGRUZBRkFGZW5GQQ==

JSFH