PlotFest: Plotting a Function

\(Off[General::spell]; 
Off[General::spell1]; 

    Off[ParametricPlot3D::ppcom]; 
Off[ParametricPlot::ppcom]; 

    Off[Plot::plnr]; 
SetOptions[Limit, Analytic -> True]; \)

CMView = {2.7, 1.6, 1.2};

Vector Drawer

(* :Title:  NewVectorDraw  *)
(* :Author: Bruce Carpenter *)
(* :Summary:  Construct 2 and 3 dimensional vectors,
     arrows, and arrowheads as graphics 
     objects.
*)
(* :Date:  Copyright 1997, Math Everywhere, Inc. 
*)
(*  :Drawbacks:  Since the vectors are drawn out of 
     context, the shapes of the arrowheads are only 
     correct with AspectRatio set to Automatic.
*)
BeginPackage["Vector3D`","Graphics`Colors`"];

Axes3D::usage = "Axes3D[a, b] creates a Graphics3D object of cartesian axes with x, y, and z running from -a/3 to a, and with axes labels b units beyond the tips of the axes.  Axes3D[a] is Axes3D[a, a/8].";

Perpend::usage = "Perpend[a] returns a unit vector which is perpendicular to the vector a.";

Arrow::usage = "Arrow[a] produces a 2 or 3 dimensional vector from the origin to a. Arrow[a, Tail -> tail] gives a vector from tail to a.";
	
Vector::usage = "Vector[a, b] gives a 2D or 3D vector from a to b.";

ArrowHead::usage = "ArrowHead[a, vec] produces an arrowhead with its tip placed at the point a, pointing in the direction of the vector vec.";

Tail::usage = "Tail -> point puts the tail of the vector at point.";

Norm::usage = "Norm[vec] returns the Euclidean length of vec.";

Normalize::usage = "Normalize[vec] returns a unit vector in the direction of vec, provided vec is nonzero.";

Aperture::usage = "Aperture is the ratio of the radius of the base to the length of the head of a vector.";

TipSize::usage = "TipSize specifies an absolute size for the inner tip length of the head of a vector.";

TipRatio::usage = "TipRatio is the ratio of the inner length of the arrowhead to the length of the shaft of a vector.";

HeadRatio::usage = "HeadRatio is the ratio of the outer length of the arrowhead to the length of the shaft of a vector.";

HeadSize::usage = "HeadSize specifies an absolute size for the head of a vector.";

ScaleFactor::usage = "ScaleFactor specifies the amount to scale a vector in length and may be either a number or a pure function.";

ZeroVectorPointSize::usage = "ZeroVectorPointSize is the size of the point used to represent a zero vector.";

Options[Vector] = Options[Arrow] = Options[ArrowHead] =
{HeadRatio -> 0.18, TipRatio -> 0.14, Aperture -> 0.3, ScaleFactor -> 1, HeadType -> Polygon, ShaftQ -> True, EdgesQ -> True, VectorColor -> RGBColor[0,0,1], ShaftWidth -> 0.005, ZeroVectorPointSize -> 0.01};
		
Begin["`Private`"];

SetOptions[ParametricPlot,AspectRatio -> Automatic];
SetOptions[Plot,AspectRatio -> Automatic];
SetOptions[Graphics,AspectRatio -> Automatic];

Axes3D[u_,v_] := 
      Graphics3D[{
            {Blue,Line[{{-u/3,0,0},{u,0,0}}]},
                    Text["x",{u + v,0,0}],
            {Blue,Line[{{0,-u/3,0},{0,u,0}}]},
                    Text["y",{0, u + v,0}],
            {Blue,Line[{{0,0,-u/3},{0,0,u}}]},
                    Text["z",{0,0,u + v}]}]; 
Axes3D[u_] := Axes3D[u,u/8];
				
Norm[a_?VectorQ] := N[Sqrt[a.a]];

Normalize[a_?VectorQ] := If[a.a == 0, a, a/Norm[a]];

Perpend[a_?VectorQ] := Normalize[{a[[2]],-a[[1]]}]/;Length[a]==2
Perpend[a_?VectorQ] := Normalize[If[a[[1]]==0,{1,0,0},{0,a[[3]],-a[[2]]}]]/;Length[a]==3

Base3D = Table[{Cos[2 k Pi/8.], Sin[2 k Pi/8.]},{k,0,8}]; 

Vector[from_?VectorQ, to_?VectorQ, opts___] := Module[
   {sq, sw, ht, hs, hr, tr, ap, sf, vc, zvps, diff = to - from, tip},
   {sq, sw, ht, hs, hr, ts, tr, ap, sf, vc, zvps} = {ShaftQ, ShaftWidth, HeadType, HeadSize, HeadRatio, TipSize, TipRatio, Aperture, ScaleFactor, VectorColor, ZeroVectorPointSize} /. {opts} /. Options[Vector];
   If[from==to, Return[Graphics[{PointSize[zvps],vc,Point[from]}]]];
   diff = If[NumberQ[N[sf]], sf diff, sf[diff]];
   tip = from + diff;
   If[NumberQ[hs], hr = hs/Norm[diff]; tr = 0.75 hr];
   If[NumberQ[ts], tr = ts/Norm[diff]];
   Graphics[{If[sq,{vc,Thickness[sw],Line[{from,tip - tr diff}]},{}],
      With[{trans = ap hr Norm[diff] Perpend[diff]},
      {vc, Thickness[sw], ht[{tip,
                   (tip - hr diff + trans),
                   (tip - tr diff),
                   (tip - hr diff - trans),tip}]}]}]
] /; Length[from]==Length[to]==2

Vector[from_?VectorQ, to_?VectorQ, opts___] := Module[
   {sq, eq, sw, hs, hr, tr,ap, sf, vc, zvps, diff = to - from, tip},
   {sq, eq, sw, hs, hr, ap, sf, vc, zvps} = {ShaftQ, EdgesQ, ShaftWidth, HeadSize, HeadRatio, Aperture, ScaleFactor, VectorColor, ZeroVectorPointSize} /. {opts} /. Options[Vector];
   If[from==to, Return[Graphics3D[{PointSize[zvps],vc,Point[from]}]]];
   diff = If[NumberQ[N[sf]], sf diff, sf[diff]];
   tip = from + diff;
   If[NumberQ[hs], hr = hs/Norm[diff]];
   tr = hr/2;
   Graphics3D[{If[sq,{vc,Thickness[sw],Line[{from,tip-tr diff}]},{}], If[eq, {}, EdgeForm[]], {SurfaceColor[vc],
      With[{temp = Perpend[diff]},
      Polygon[Append[#,tip]]& /@ Partition[(tip - hr diff + #)&/@((ap hr Norm[diff] Base3D) . {temp,Normalize[Cross[temp,diff]]}),2,1]]}}]
] /; Length[from]==Length[to]==3


Arrow[a_, Tail->b_, opts___] := Vector[b,a+b,opts]

Arrow[a_, opts___] := Vector[Table[0,{Length[a]}],a,opts]

ArrowHead[a_, b_, opts___] := Vector[a-b,a,ShaftQ->False,opts];

End[];

EndPackage[];
Null

Graphics`Colors`GosiaGreen = RGBColor[0,  0.392187,  0];

Spherical Setup

\(Clear[polar]; 

    polar[{x_, y_, z_}] := {\@\(x\^2 + y\^2 + z\^2\), ArcTan[y\/x], 
        ArcCos[z\/\@\(x\^2 + y\^2 + z\^2\)]}; 
Clear[X, length]; 

    length[X_] := \@\(X . X\); 
\(rr = 3; \)\)R = 5; scale = 0.8; red = RGBColor[1, 0, 0]; blue = RGBColor[0, 0, 1]; color1 = RGBColor[0.8, 0.2, 0.8]; color2 = RGBColor[0.2, 0.8, 0.2]; black = RGBColor[0, 0, 0]; \(Clear[axes]; 

    axes := Graphics3D[{Line[{X, {1.5\ rr, 0, 0}}], 
          Line[{X, {0, 1.5\ rr, 0}}], Line[{X, {0, 0, 2\ rr}}]}]; 

    Clear[xyplane]; 

    xyplane := 
      Graphics3D[
        Polygon[{{\(-2\)\ rr, \(-2\)\ rr, 0}, {\(-2\)\ rr, 2\ rr, 0}, {2\ rr, 
              2\ rr, 0}, {2\ rr, \(-2\)\ rr, 0}}]]; 
\(X = {0, 0, 0}; \)\)origin = Graphics3D[{black, Point[X]}]; Z = {2, 3, 5}; \(Clear[point]; 

    point := Graphics3D[{red, PointSize[0.03], Point[scale\ Z]}]; 

    \(Z0 = {0, 0, length[0.8\ Z]}; \)\)\(Clear[zpoint]; 
zpoint = Graphics3D[{black, Point[Z0]}]; 

    \(XY = scale\ Z\ {1, 1, 0}; \)\)\(Clear[xypoint]; 
xypoint := Graphics3D[{black, Point[XY]}]; 

    \(X0 = {length[scale\ Z\ {1, 1, 0}], 0, 0}; \)\)xpoint = Graphics3D[{black, Point[X0]}]; Y0 = {0, length[scale Z {1, 1, 0}], 0}; ypoint = Graphics3D[{black, Point[Y0]}]; \(Clear[ray]; 
ray := Graphics3D[{blue, Line[{X, scale\ Z}]}]; 

    Clear[ray1]; 
ray1 := Graphics3D[{color1, Line[{scale\ Z, Z}]}]; 

    Clear[xyray]; 
xyray := Graphics3D[{red, Line[{X, Z\ {1, 1, 0}}]}]; 

    \(rays := {ray, xyray}; \)\)xyaux = 
      ((length[0.8 Z] (0.8 Z {1, 1, 0}))/length)[0.8 Z {1, 1, 0}]; 
    \(Clear[zarc]; 

    zarc = Graphics3D[{blue, 
          Line[Table[
              \(Cos[t]\ \) {0, 0, length[0.8\ Z]} + Sin[t]\ xyaux, {t, 0, 
                \(polar[0.8\ Z]\)[[3
                  ]],  .1}]]}]; 
Clear[zarc1]; 

    zarc1 = Graphics3D[{color1, 
          Line[Table[
              \(Cos[t]\ \) {0, 0, length[0.8\ Z]} + Sin[t]\ xyaux, {t, 
                \(polar[0.8\ Z]\)[[3]], 
                1,  .1}]]}]; 
Clear[zarc2]; 

    zarc2 = Graphics3D[{color1, 
          Line[Table[
              \(Cos[t]\ \) {0, 0, length[0.8\ Z]} + Sin[t]\ xyaux, {t, 
                \(-0.8\), 0,  .1}]]}]; 
Clear[xyarc]; 

    xyarc = Graphics3D[{blue, 
          Line[Table[
              Cos[t]\ X0 + Sin[t]\ Y0, {t, 0, 
                \(polar[XY]\)[[2
                  ]],  .1}]]}]; 
Clear[xyarc1]; 

    xyarc1 = Graphics3D[{color1, 
          Line[Table[Cos[t]\ X0 + Sin[t]\ Y0, {t, \(- .5\), 0,  .1}]]}]; 

    Clear[xyarc2]; 

    xyarc2 = Graphics3D[{color1, 
          Line[Table[
              Cos[t]\ X0 + Sin[t]\ Y0, {t, 
                \(polar[XY]\)[[2]], 
                \(polar[XY]\)[[2]] + 
                  0.3,  .1}]]}]; 
Clear[stick]; 

    stick = Graphics3D[{red, Line[{scale\ Z, XY}]}]; 

    \(xaxislabel = Text["x", {0.363, 0.26}]; \)\)yaxislabel = Text["y", {0.73, 0.3}]; zaxislabel = Text["z", { .5, 0.67}]; stickplot = 
      Show[xyplane, axes, ray, point, Boxed -> False, 
        ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; rho = Text["r", {0.53, 0.45}]; rlabel = 
      Show[xyplane, axes, ray, point, Boxed -> False, 
        ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {rho, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; phi = Text["s", {0.55, 0.57}]; slabel = 
      Show[xyplane, zpoint, axes, ray, point, zarc, zarc1, zarc2, 
        Boxed -> False, ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {phi, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; theta = Text["t", { .5, 0.3}]; tlabel = 
      Show[xyplane, axes, xypoint, xpoint, ray, stick, point, xyarc, xyarc1, 
        xyarc2, Boxed -> False, ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {theta, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity]; all = 
      Show[xyplane, axes, ray, stick, point, zpoint, xpoint, xypoint, xyarc, 
        xyarc1, xyarc2, zarc, zarc1, zarc2, Boxed -> False, 
        ViewPoint -> {2.747, 1.594, 1.168}, 
        Epilog -> {theta, phi, rho, xaxislabel, yaxislabel, zaxislabel}, 
        DisplayFunction -> Identity];

Crisscross

\(Clear[x]; 

    crisscross = 
      Plot[{Cos[x], Sin[x]}, {x, π\/8, \(3\ π\)\/8}, 
        PlotStyle -> {{RGBColor[1, 0, 0], Thickness[0.02]}}, 
        Axes -> None, AspectRatio -> 1\/3, 
        DisplayFunction -> Identity]; \)

Plotting a function

Here's how you program the function 
        f[x] = 5.0 (E^(-0.1 x)) Sin[ x]
into Mathematica:

\(Clear[f, x]; 
f[x_] = 5.0\ E\^\(\(-0.1\)\ x\)\ Sin[x]\)

Here is how you get a table of the values
       {x, f[x]} 
for x = 0, 0.5, 1, 1.5, 2, 2.5, 3,  ... , 9.5, 10:

\(xjump\  = \ 0.5; 
Table[{x, f[x]}, {x, 1, 10, xjump}]\)

See these points:

\(dataplot\  = \ Table[Graphics[Point[{x, f[x]}]], {x, 1, 10, xjump}]; 

    Show[dataplot, Axes -> True]; \)

You can make the dots bigger:

dataplot = Table[Graphics[{PointSize[0.03],Point[{x, f[x]}]}],{x, 1, 10, xjump}];
Show[dataplot, Axes -> True];

You can label the x and f[x] axes:

Show[dataplot,Axes -> True,AxesLabel -> {"x", "f[x]"}];

You can put the points in color:

dataplot = Table[Graphics[{Blue,PointSize[0.03],Point[{x, f[x]}]}],{x, 1, 10, xjump}];
colorplot  = Show[dataplot, Axes -> True];

Nice graphics, Mathematica!

But why settle for point plots when you can see the whole curve for 0 ≤ x ≤ 10:

curveplot = 
      Plot[f[x], {x, 0, 10}, 
        PlotStyle -> {{Thickness[0.01], CadmiumOrange}}, 
        AxesLabel -> {"x", "f[x]"}];

See the curve and the data:

Show[curveplot, colorplot];

If you want to spot a certain point on the curve, you can do it.  For instance, here is the curve shown with a data point at {x, f[x]} for x = 5.9:

\(x = 5.9; 

    datapoint = Graphics[{Blue, PointSize[0.03], Point[{x, f[x]}]}]; 

    \(Show[curveplot, datapoint]; \)\)

Here's how to spot the data point {x, f[x]} for x = 2.9:

\(x = 2.9; 

    datapoint = Graphics[{Blue, PointSize[0.03], Point[{x, f[x]}]}]; 

    \(Show[curveplot, datapoint]; \)\)

See {x, f[x]} dance along the curve as x runs from 0 to 10.

\(Clear[datapoint, x]; 

    datapoint[x_]\  := \ 
      Graphics[{Blue, PointSize[0.03], Point[{x, f[x]}]}]\)\(xjump\  = \ 1; 

    Table[Show[curveplot, datapoint[x], PlotLabel -> x\ "= x"], {x, 0.5, 
        9.5, xjump}]; \)

To see a movie of {x, f[x]} dancing along the curve,
-> select all the plots immediately above and then 
-> go to the Cell menu and select :  Animate Selected Graphics.