Space curves: Plotting Vector-Valued Functions in Maple Calc IV Lab Week One by Karen Donnelly all rights reserved Load the plots library so that the spacecurve and animate command may be used: JSFH QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi print(); # input placeholder Define options for plotting and name opts to save typing in subsequent plotting commands: Pkklb3B0c0c2IjYkL0klYXhlc0dGJEkmYm94ZWRHRiQvSSdsYWJlbHNHRiQ3JUkieEdGJEkieUdGJEkiekdGJA== print(); # input placeholder JSFH

The easiest way to plot a curve in space in Maple is to use the spacecurve function which is part of the plots library. For example, to plot the helix curve 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 for t between 0 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRJSZwaTtGJy8lJ2l0YWxpY0dGPkY1RjVGK0Y1: LUkrc3BhY2VjdXJ2ZUc2IjYlNyUtSSRjb3NHRiQ2I0kidEdGJC1JJHNpbkdGJEYpRiovRio7IiIhLCRJI1BpRyUqcHJvdGVjdGVkRyIiJUklb3B0c0dGJA== An visually interesting plot can be seen if you replace spacecurve with the command: LUkpdHViZXBsb3RHNiI2JTclLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiNJInRHRiQtSSRzaW5HRilGLEYtL0YtOyIiISwkSSNQaUdGKiIiJUklb3B0c0dGJA== 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 space curve 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 LUkoYW5pbWF0ZUc2IjYlSStzcGFjZWN1cnZlR0YkNyU3JS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJ0R0YkLUkkc2luR0YrRi5GLy9GLzsiIiFJIlRHRiRJJW9wdHNHRiQvRjU7RjQsJEkjUGlHRiwiIiU= %; %; 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.) QyU+SSVrbm90RzYiNyUsKC1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjSSJ0R0YlISM1LUYpNiMsJEYuIiImISIjLUkkc2luR0YqNiMsJEYuIiIjIiM6LCgtRilGNyEjOi1GNkYtIiM1LUY2RjFGNCwkLUYpNiMsJEYuIiIkRj8hIiItSStzcGFjZWN1cnZlR0YlNiVGJC9GLjsiIiEsJEkjUGlHRitGOUklb3B0c0dGJQ== print(); # input placeholder Here we animate the above curve to see show its orientation as it is drawn: LUkoYW5pbWF0ZUc2IjYlSStzcGFjZWN1cnZlR0YkNyVJJWtub3RHRiQvSSJ0R0YkOyIiIUkiVEdGJEklb3B0c0dGJC9GLTtGLCwkSSNQaUclKnByb3RlY3RlZEciIiM= A curve in the plane is just a special case a curve in space. If in the x-y plane, the k coordinate is 0. Thus the plot of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2KUYrLUYjNictRiw2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEidEYnRjZGOS8lK2V4ZWN1dGFibGVHRkRGQEZARmhuRkAtRj02LVEiPUYnRkBGQkZFRkdGSUZLRk1GTy9GUlEsMC4yNzc3Nzc4ZW1GJy9GVUZeby1GIzYpRistRiM2KEYrLUYjNictRiw2JVEkY29zRicvRjdGREZARjxGVkZobkZALUY9Ni1RMSZJbnZpc2libGVUaW1lcztGJ0ZARkJGRUZHRklGS0ZNRk9GUUZULUYsNiVRImlGJ0Y2RjlGaG5GQC1GPTYtUSIrRidGQEZCRkVGR0ZJRktGTUZPL0ZSUSwwLjIyMjIyMjJlbUYnL0ZVRmRwLUYjNihGKy1GIzYnLUYsNiVRJHNpbkYnRmlvRkBGPEZWRmhuRkBGam8tRiw2JVEiakYnRjZGOUZobkZARitGaG5GQEYrRmhuRkBGK0ZobkZA is given by: LUkrc3BhY2VjdXJ2ZUc2IjYlNyUtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0kidEdGJC1JJGNvc0dGKUYsIiIhL0YtO0YwLCRJI1BpR0YqIiIjSSVvcHRzR0Yk JSFH print(); # input placeholder In animated form: LUkoYW5pbWF0ZUc2IjYlSStzcGFjZWN1cnZlR0YkNyU3JS1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjSSJ0R0YkLUkkc2luR0YrRi4iIiEvRi87RjJJIlRHRiRJJW9wdHNHRiQvRjU7RjIsJEkjUGlHRiwiIiM=

Exercise 39, page 838 Graph and identify the curve: 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 i +LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= j + 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kJSFH %; 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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!! Then, plot to check your answers. 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= %; print(); # input placeholder Graph of original function: 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 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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 Part b) Conjecture: 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 Part c) Conjecture: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 Part d) Conjecture: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEjUGRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSVXSEFURicvRjBGPUY5LUY2Ni1RIjpGJ0Y5RjtGPkZARkJGREZGRkhGUkZURjVGKy8lK2V4ZWN1dGFibGVHRj1GOQ== Part e) Conjecture: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=