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Chapter 2: Space Curves
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Section 2.1: PositionVector Representation
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Essentials


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A plane curve defined parametrically can be graphed with the plot command; in ${\mathrm{\ℝ}}^{3}$, a curve defined parametrically can be graphed with the spacecurve command from the plots package. However, these commands are amongst the oldest of the graphing commands in Maple and do not make provision for graphing ancillary items along the curves. These ancillary items include points, and various sets of vectors defined along the curve.

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Hence, the examples below will illustrate the use of two newer and more robust commands found in the VectorCalculus packages. Since the Student VectorCalculus package is most lenient with respect to declaring coordinate systems and coordinatevariable names, this package turns out to be the simpler one to use.

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The two commands are PositionVector and PlotPositionVector, the first of which defines a curve as a position vector; the second, graphs the position vector as a curve. Although the Context Panel system can change a "free" vector to a position vector, there is as yet no provision for implementing the PlotPositionVector command interactively. Hence, in all the examples below, these two commands are executed explicitly.

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The argument of the PositionVector command is a list of expressions that define the curve parametrically. An optional argument can be used to declare the coordinate system in which these expressions are to be interpreted. The position vector exists only in Cartesian coordinates, do any expressions that are deemed to be in an alternate system are converted to Cartesian coordinates. The command returns a column vector.

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The PlotPositionVector command does not behave like the early graphing commands in Maple where options are added as a sequence of equations. For this command, options are grouped by category, and appear as lists of equations. For example, for the ordinary plot command, the option "scaling = constrained" is added directly as an additional argument. In the PlotPositionVector command, it must be added as a suboption to the "curveoptions" option. This is illustrated in Example 2.1.2, Example 2.1.3, and Example 2.1.4.${}$

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Examples


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Example 2.1.1

For $t\in \left[3\,3\right]$, graph the plane curve defined parametrically by $x\left(t\right)\=1\+{t}^{2}$, $y\left(t\right)\=1{t}^{3}$. On the graph, indicate the points where $t\=3\,0\,1$, and 2.

Example 2.1.2

For $\mathrm{\θ}\in \left[0\,4\mathrm{pi;}\right]$, graph the plane curve defined parametrically by $x\left(\mathrm{\θ}\right)\=\mathrm{\θ}\mathrm{cos}\left(\mathrm{theta;}\right)\mathrm{sin}\left(\mathrm{theta;}\right)$, $y\left(\mathrm{\θ}\right)\=\mathrm{\θ}\mathrm{sin}\left(\mathrm{theta;}\right)plus;\mathrm{cos}\left(\mathrm{theta;}\right)$. On the graph, indicate the points where $\mathrm{\θ}\=0\,\mathrm{\π}$, and $3\mathrm{pi;}$.

Example 2.1.3

For $t\in \left[0\,4\mathrm{pi;}\right]$, graph the plane curve defined parametrically by $x\left(t\right)\={e}^{t\/5}\mathrm{cos}\left(t\right)$, $y\left(t\right)\={e}^{t\/5}\mathrm{sin}\left(t\right)$. On the graph, indicate the points where $t\=0\,\mathrm{\π}$, and $3\mathrm{pi;}$.

Example 2.1.4

For $t\in \left[0\,3\mathrm{pi;}\right]$, graph the helix defined parametrically by $x\left(t\right)\=\mathrm{cos}\left(t\right)$, $y\left(t\right)\=\mathrm{sin}\left(t\right)$, $z\left(t\right)\=t\/3$. On the graph, indicate the points where $t\=\mathrm{\π}$, and $2\mathrm{pi;}$.

Example 2.1.5

For $t\in \left[1\,5\right]$, graph the space curve defined parametrically by $x\left(t\right)\=t$, $y\left(t\right)\=\mathrm{ln}\left(t\right)$, $z\left(t\right)\={e}^{t\/3}$. On the graph, indicate the points where $t\=1\,3$, and $4$.



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