plot a curve or surface defined by a position Vector - Maple Help

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VectorCalculus[PlotPositionVector] - plot a curve or surface defined by a position Vector

 Calling Sequence PlotPositionVector(pv) PlotPositionVector(pv, r) PlotPositionVector(pv,r, options1) PlotPositionVector(pv, r1,r2) PlotPositionVector(pv,r1,r2, options2)

Parameters

 pv - 'Vector'(algebraic); the position Vector representing a curve or surface r - range or name=range; the range of the parameter of the curve r1 - name=range; the range of one of the parameters of the surface r2 - name=range; the range of one of the parameters of the surface options1 - (optional) curve options, equation(s) of the form keyword = value, where keyword is either 'points', 'vectorfield', 'vectorfieldoptions', 'pvdiff', 'diffoptions', 'tangent', 'tangentoptions', 'normal', 'normaloptions', 'binormal', 'binormaloptions' or 'curveoptions' options2 - (optional) surface options, equation(s) of the form keyword = value, where keyword is either 'points', 'pointoptions', , 'coordcurve', 'curveoptions', 'pvdiff', 'diffoptions', 'normal', 'normaloptions', 'normalfield', 'normalfieldoptions', 'normalorientation', 'vectorfield', 'vectorfieldoptions', or 'surfaceoptions'

Description

 • The PlotPositionVector command plots a curve or surface defined by a two or three dimensional position Vector. For a curve, the plot can be in two or three dimensional space.
 • The first argument pv is a position Vector. The components of the position Vector represent the parametric description of the curve or surface in Cartesian coordinates. For more information about position Vectors, see PositionVector.
 – If the position Vector has one parameter it is assumed to represent a curve.
 – If the position Vector has two parameters and three components it is assumed to represent a surface. If the position Vector has two parameters and two components it is assumed to represent unevaluated curves.
 – If the position Vector has no indeterminates, a single Vector rooted at the origin is plotted.
 • Given a position Vector with one parameter representing a curve, the range r is specified in the form param = a..b or a..b. Given a position Vector with two parameters and three dimensions representing a surface, the ranges r1 and r2 are specified in the form param1=a..b and param2=c..d respectively. Given a position Vector with two parameters and two components, one parameter has to be assigned a specific value and the other must be a range of the form param = a..b.
 • The options1 and options2 arguments control extra structures that can be added to the plot of the curve and surface as well as providing plot options for specific elements of the plot. Additional plot options, as described on the plot/option and plot3d/option help pages, that are applicable may also be provided.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$

Position Vectors

 > $\mathrm{pv1}:=\mathrm{PositionVector}\left(\left[1,2,3\right],{\mathrm{cartesian}}_{x,y,z}\right)$
 ${\mathrm{pv1}}{:=}\left[\begin{array}{r}{1}\\ {2}\\ {3}\end{array}\right]$ (1)
 > $\mathrm{PlotPositionVector}\left(\mathrm{pv1}\right)$
 • Specify the range of the parameter.
 > $\mathrm{R1}:=\mathrm{PositionVector}\left(\left[p,{p}^{2}\right],{\mathrm{polar}}_{r,t}\right)$
 ${\mathrm{R1}}{:=}\left[\begin{array}{c}{p}{}{\mathrm{cos}}{}\left({{p}}^{{2}}\right)\\ {p}{}{\mathrm{sin}}{}\left({{p}}^{{2}}\right)\end{array}\right]$ (2)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R1},p=1..2\right)$
 > $\mathrm{R2}:=\mathrm{PositionVector}\left(\left[v,v\right],{\mathrm{polar}}_{r,\mathrm{θ}}\right)$
 ${\mathrm{R2}}{:=}\left[\begin{array}{c}{v}{}{\mathrm{cos}}{}\left({v}\right)\\ {v}{}{\mathrm{sin}}{}\left({v}\right)\end{array}\right]$ (3)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},0..3\mathrm{π}\right)$

Display tangent vectors on the curve.

 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},v=0..3\mathrm{π},\mathrm{tangent}=\mathrm{true}\right)$

Evaluate a vector field on the curve.

 > $\mathrm{VF1}:=\mathrm{VectorField}\left(⟨-x,-y⟩,{\mathrm{cartesian}}_{x,y}\right)$
 ${\mathrm{VF1}}{:=}{-}{x}{\stackrel{{_}}{{e}}}_{{x}}{-}{y}{\stackrel{{_}}{{e}}}_{{y}}$ (4)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},v=0..3\mathrm{π},\mathrm{vectorfield}=\mathrm{VF1}\right)$

Display the tangent, principal normal and binormal vectors on a curve.

 > $\mathrm{R3}:=\mathrm{PositionVector}\left(\left[1,\frac{\mathrm{π}}{2}+\mathrm{arctan}\left(\frac{1t}{2}\right),t\right],\mathrm{spherical}\right)$
 ${\mathrm{R3}}{:=}\left[\begin{array}{c}\frac{{2}{}{\mathrm{cos}}{}\left({t}\right)}{\sqrt{{{t}}^{{2}}{+}{4}}}\\ \frac{{2}{}{\mathrm{sin}}{}\left({t}\right)}{\sqrt{{{t}}^{{2}}{+}{4}}}\\ {-}\frac{{t}}{\sqrt{{{t}}^{{2}}{+}{4}}}\end{array}\right]$ (5)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R3},t=0..4\mathrm{π}\right)$
 > $\mathrm{PlotPositionVector}\left(\mathrm{R3},t=0..4\mathrm{π},\mathrm{tangent}=\mathrm{true},\mathrm{normal}=\mathrm{true},\mathrm{binormal}=\mathrm{true}\right)$
 > $\mathrm{R4}:=\mathrm{PositionVector}\left(\left[1,p,p\right],{\mathrm{cylindrical}}_{r,p,s}\right)$
 ${\mathrm{R4}}{:=}\left[\begin{array}{c}{\mathrm{cos}}{}\left({p}\right)\\ {\mathrm{sin}}{}\left({p}\right)\\ {p}\end{array}\right]$ (6)

Specify points on the curve for plotting other structures.

 > $\mathrm{PlotPositionVector}\left(\mathrm{R4},p=0..4\mathrm{π},\mathrm{tangent}=\mathrm{true},\mathrm{points}=\left[\mathrm{π}\right]\right)$
 > $\mathrm{PlotPositionVector}\left(\mathrm{R4},p=0..4\mathrm{π},\mathrm{tangent}=\mathrm{true},\mathrm{normal}=\mathrm{true},\mathrm{binormal}=\mathrm{true},\mathrm{points}=\left[\mathrm{π},\frac{\mathrm{π}}{2}\right]\right)$

Two dimensional position Vectors with two parameters.

 > $\mathrm{pv2}:=\mathrm{PositionVector}\left(\left[p,q\right],{\mathrm{polar}}_{r,t}\right)$
 ${\mathrm{pv2}}{:=}\left[\begin{array}{c}{p}{}{\mathrm{cos}}{}\left({q}\right)\\ {p}{}{\mathrm{sin}}{}\left({q}\right)\end{array}\right]$ (7)

Provide numeric value of one of the parameters.

 > $\mathrm{PlotPositionVector}\left(\mathrm{pv2},p=1..2,q=\frac{\mathrm{π}}{2}\right)$
 > $\mathrm{PlotPositionVector}\left(\mathrm{pv2},p=2,q=0..\mathrm{π}\right)$

The commands to create the plots from the Plotting Guide are

 > $\mathrm{PlotPositionVector}\left(\mathrm{R2},v=0..3\mathrm{π},\mathrm{tangent}=\mathrm{true},\mathrm{pvdiff}=\left[v\right]\right)$
 > $\mathrm{VF2}:=\mathrm{VectorField}\left(⟨r,0,0⟩,{\mathrm{spherical}}_{r,p,t}\right)$
 ${\mathrm{VF2}}{:=}\left({r}\right){\stackrel{{_}}{{e}}}_{{r}}$ (8)
 > $\mathrm{PlotPositionVector}\left(\mathrm{R3},t=0..4\mathrm{π},\mathrm{vectorfield}=\mathrm{VF2},\mathrm{vectornum}=6\right)$

Surfaces

 > $\mathrm{S1}:=\mathrm{PositionVector}\left(\left[t,\frac{v}{\sqrt{1+{t}^{2}}},\frac{vt}{\sqrt{1+{t}^{2}}}\right],{\mathrm{cartesian}}_{x,y,z}\right)$
 ${\mathrm{S1}}{:=}\left[\begin{array}{c}{t}\\ \frac{{v}}{\sqrt{{{t}}^{{2}}{+}{1}}}\\ \frac{{v}{}{t}}{\sqrt{{{t}}^{{2}}{+}{1}}}\end{array}\right]$ (9)
 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3\right)$

Specify points on the surface.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{points}=\left[\left[0,0\right],\left[1,1\right],\left[2,2\right],\left[3,3\right]\right]\right)$

Visualize tangent vectors along coordinate curves.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{pvdiff}=\left[\left[t\right],\left[v\right]\right]\right)$
 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨x,y,0⟩,{\mathrm{cartesian}}_{x,y,z}\right),\mathrm{vectorgrid}=\left[3,3\right]\right)$

Visualize the normal field of a given vector field.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S1},t=-3..3,v=-3..3,\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨x,y,0⟩,{\mathrm{cartesian}}_{x,y,z}\right),\mathrm{normalfield}=\mathrm{true}\right)$

 > $\mathrm{S2}:=\mathrm{PositionVector}\left(\left[1,p,q\right],{\mathrm{toroidal}}_{r,\mathrm{φ},\mathrm{θ}}\right)$
 ${\mathrm{S2}}{:=}\left[\begin{array}{c}\frac{{\mathrm{sinh}}{}\left({p}\right){}{\mathrm{cos}}{}\left({q}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\\ \frac{{\mathrm{sinh}}{}\left({p}\right){}{\mathrm{sin}}{}\left({q}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\\ \frac{{\mathrm{sin}}{}\left({1}\right)}{{\mathrm{cosh}}{}\left({p}\right){-}{\mathrm{cos}}{}\left({1}\right)}\end{array}\right]$ (10)

Display a coordinate curve on a surface.

 > $\mathrm{PlotPositionVector}\left(\mathrm{S2},p=0..2\mathrm{π},q=0..2\mathrm{π},\mathrm{coordcurve}=\left[p=\frac{\mathrm{π}}{2},\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨r,0,0⟩,{\mathrm{spherical}}_{r,\mathrm{φ},\mathrm{θ}}\right)\right]\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{PlotPositionVector}\left(\mathrm{S2},p=0..2\mathrm{π},q=0..2\mathrm{π},\mathrm{vectorfield}=\mathrm{VectorField}\left(⟨r,0,0⟩,{\mathrm{spherical}}_{r,\mathrm{φ},\mathrm{θ}}\right),\mathrm{surfaceoptions}=\left[\mathrm{color}=\mathrm{red}\right],\mathrm{vectorgrid}=\left[3,3\right]\right)$