Function Repository Resource:

DirectionParametricPlot3D

Create a parametric plot of a curve in space with direction indicated by arrowheads and color

Contributed by: Dennis M Schneider

ResourceFunction["DirectionParametricPlot3D"][{f_x,f_y,f_z},{t,t_min,t_max}]

generates a parametric plot of a curve {f_x,f_y,f_z} in space whose coordinate functions are functions of t with the direction of the curve indicated by color and arrowheads.

Details and Options

The following options are supported:

"ArrowNumber"

number of arrowheads to include in the plot

"ArrowSize"

Large

size of the arrowheads

ColorFunction

Automatic

specify the color function

"DrawArrowheads"

True

whether to include the arrowheads in the plot

PlotStyle

Automatic

style to be applied to the curve

Examples

Basic Examples (1)

A conical helix climbing up the z-axis:

In[1]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{t Cos[4 \[Pi] t], t Sin[4 \[Pi] t], t}, {t, 0, 2}]$

Out[1]=

Options (11)

ArrowSize (2)

The option "ArrowSize"→1 corresponds (roughly) to the default "ArrowSize"→Large:

In[2]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{t Cos[4 \[Pi] t], t Sin[4 \[Pi] t], t}, {t, 0, 2}, "ArrowSize" -> 1]$

Out[2]=

Make the arrowheads smaller:

In[3]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}, {t, 0, \[Pi]}, "ArrowSize" -> Medium]$

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ArrowNumber (2)

Plot more arrowheads:

In[4]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}, {t, 0, \[Pi]}, "ArrowNumber" -> 25]$

Out[4]=

Suppress the arrowheads by setting "ArrowNumber" to 0:

In[5]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}, {t, 0, \[Pi]}, "ArrowNumber" -> 0]$

Out[5]=

DrawArrowheads (1)

Suppress the arrowheads by setting "DrawArrowheads"→False:

In[6]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}, {t, 0, \[Pi]}, "DrawArrowheads" -> False]$

Out[6]=

PlotStyle (4)

Setting PlotStyle to a color plots the curve in that color:

In[7]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}, {t, 0, \[Pi]}, PlotStyle -> Purple]$

Out[7]=

Plot two curves with a different style as well as a different size arrowhead for each curve:

In[8]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{{2 Cos[5 t], 2 Sin[5 t], t}, {Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}}, {t, 0, \[Pi]}, PlotStyle -> {Thickness[.007], Thickness[.013]}, "ArrowSize" -> {Medium, Large}]$

Out[8]=

Non-color-related values for PlotStyle will not affect the automatic coloring:

In[9]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}, {t, 0, \[Pi]}, PlotStyle -> Thickness[.013]]$

Out[9]=

For multiple curves, if just one style is specified, it is applied to both curves:

In[10]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{{2 Cos[5 t], 2 Sin[5 t], t}, {Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}}, {t, 0, \[Pi]}, PlotStyle -> Thickness[.013]]$

Out[10]=

ColorFunction (1)

Change the color function:

In[11]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{Cos[t] Cos[10 t], Cos[t] Sin[10 t], t}, {t, 0, \[Pi]}, ColorFunction -> Function[{x, y, z, t}, GrayLevel[.8 t]]]$

Out[11]=

PlotLegends (1)

Since PlotLegends depends on the color of the curve, they will not work unless ColorFunction is set to None or the specifications for PlotStyle contain a color:

In[12]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{{3/2 Cos[3 \[Pi] t], 3/2 Sin[3 \[Pi] t], 2 t}, {2 t Cos[3 \[Pi] t], 2 t Sin[3 \[Pi] t], 2 t}}, {t, 0, 2}, ColorFunction -> None, PlotLegends -> "Expressions"]$

Out[12]=

In[13]:=

$ResourceFunction[ "DirectionParametricPlot3D"][{{3/2 Cos[3 \[Pi] t], 3/2 Sin[3 \[Pi] t], 2 t}, {2 t Cos[3 \[Pi] t], 2 t Sin[3 \[Pi] t], 2 t}}, {t, 0, 2}, PlotStyle -> {Thickness[.01], {Magenta, Thickness[.013]}}, PlotLegends -> "Expressions"]$