atex graphics with pstricks · 3. pstricks to use pstricks you must include the following lines in...

$: ATEX Graphics with PSTricks · 3. PSTricks To use PSTricks you must include the following lines in the preamble of your document. 1 \usepackage{pst-eucl} 2 \usepackage{calc} 3 \usepackage{pst-3dplot}%$
LATEX Graphics with PSTricks

This presentation is also available online. Please visit my home page and follow the links.

http://www.math.msu.edu/∼hensh

1. Resources

(a) ImageMagick is a collection of (free) image manipulation tools. You can find out more byvisiting

http://www.imagemagick.com

(b) The LATEX Graphics Companion.

- Paperback: 608 pages

- Publisher: Addison-Wesley Pub Co; 1st edition (April 15, 1997)

- ISBN: 0201854694

(c) The LATEX Graphics Companion (2nd Edition).


- Publisher: Addison-Wesley Professional; 2 edition (August 12, 2007)

- ISBN: 0321508920

(d) PSTricks: Graphics and PostScript for TEX and LATEX.


- Publisher: UIT Cambridge Ltd. (September 1, 2011)

- ISBN: 1906860130

(e) The PSTricks web site.

https://www.tug.org/PSTricks

(f) PostScript(R) Language Tutorial and Cookbook (also called the “The Blue Book”)


- Publisher: Addison-Wesley Professional (January 1, 1985)

- ISBN: 0201101793

2. PSTricks

1

(a) We start with some examples.

−5 −4 −3 −2 −1 1 2 3 4 5

5

10

15

20

25

30

y = x2

Figure 1: Graphing simple functions.

1 2 3 4 5 6

−1

1

2

3

r = 1 + 2 sin θ

r-θ Coordinate System

Figure 2: Graphing with some fancy effects.

2

−2 −1 1 2

1

2

3

r = 1 + 2 sin θx-y Coordinate System

Figure 3: Polar Graphs

b

1

P (2, 6)

Figure 4: Area between two curves.

3

~u

~v

Figure 5: A pair of vectors

~u

~v

~v

~u + ~v

Figure 6: Vector addition with a grid.

4

π

12

1

bb

b

(a)(b)

(c)

Figure 7: Polar Grid

xy

z

Figure 8: Sketching a cylinder.

5

x

y

z

b

b

Figure 9: Position Vectors

Figure 10: Level Curves

6

θ = α

θ = β

r = g1(θ)

r = g2(θ)

r = a

Figure 11: Polar Area

xy

z

b

Figure 12: A Tangent Line

7

−1 1

−1

1

c = 0.01

c = 0.025

c = 0.05

c = 0.075

c = 0.1

c = 0.11

c = 0.125

b

xy = c

P0

Figure 13: Lagrange Multipliers

8

x

y

z

Figure 14: Exposed Solid

9

x

y

z

Figure 15: Surface Integrals

x

y

z

Figure 16: Distorted Surface

10

Figure 17: Fractals

11

Isn’tPSTricks

lots of fun

2013 !

Isn’tPSTricks

lots of fun

2013

Isn’tPSTricks

lots of fun

2013 !Figure 18: Lens Effects

−1 1 2 3

−1

1

2

C

Increasing Vector Magnitude

Figure 19: Vector Field

12

3. PSTricks

To use PSTricks you must include the following lines in the preamble of your document.

1 \usepackage{pst-eucl}

2 \usepackage{calc}

3 \usepackage{pst-3dplot}%

4 \usepackage{pst-grad}

5 \usepackage{pst-plot,pst-math,pstricks-add}%

6 \usepackage{pst-all}

7 %\RequirePackage{pst-xkey}

We should mention that there have been some incompatibilities between the pstcol package(used by PSTricks) and the graphics packages mentioned above.

Using colors with PSTricks is similar to what has already been discussed. The real power ofthe PSTricks package is the ability to create graphics using LATEX-like syntax.

13

(a) Preliminaries

PSTricks provides users with the capability to draw using the familiar syntax of LATEX.

1 \psline[linecolor=blue,linewidth=1.25pt](-3,1)(2,2)

The previous example might be easier to understand if we include more detail in thesketch. Thus

1 \showgrid

2 \psline[linecolor=blue,linewidth=1.25pt,arrowscale=2]{->}(-3,1)(2,2)

1 \newpsobject{showgrid}{psgrid}{%

2 gridlabels=0pt%

3 ,griddots=0%

4 ,gridwidth=0.5pt%

5 ,gridcolor=gray%

6 ,subgriddiv=0%

7 ,subgridwidth=0.25pt%

8 ,subgridcolor=red}

14

(b) Basic Graphics Objects

Here’s a curve. Notice that the points used can be turned on (as shown) or off.

1 %\begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)

2 \showgrid

3 \pscurve[linecolor=red,linewidth=1.5pt,showpoints=true]%

4 (\xmin,1)(0,2)(3,1)(\xmax,\ymax)

5 %\end{pspicture}

bb

b

b

where the values \xmin, \ymin, etc. have been defined previously as

1 \def\xmin{-6}\def\xmax{6}

2 \def\ymin{-6}\def\ymax{6}

15

We begin by setting the default unit(s) in PSTricks using the command\psset{unit=1cm} . This is actually the default value.


2 \showgrid

3 \pscurve[linecolor=red,linewidth=1.5pt,showpoints=true]

4 (\xmin,1)(0,2)(3,1)(\xmax,\ymax)

5 \psbezier[style=myCurveStyle,linecolor=green]{-}%

6 (-4,1)(-2,3)(1,-4)(5,5)

7 %\end{pspicture}

bb

b

b

There are built-in shapes


2 \showgrid

3 \psellipse[linecolor=blue,linewidth=1.5pt]

4 (1,0)(1,1.5)

5 \psdots[linecolor=red,linewidth=1.25pt](0,0)

6 \SpecialCoor

7 \uput{6pt}[180](0,0){$(0,0)$}

8 \NormalCoor

9 %\end{pspicture}

b(0, 0)

16

Here is a circle centered at (−2, −1) of radius 2.


2 \showgrid

3 \pscircle[linecolor=red,linewidth=1.5pt]

4 (-2,-1){2}

5 %\end{pspicture}

Here is the same object filled-in and clipped.


2 \showgrid

3 \psclip{\myframe(\xmin,\ymin)(\xmax,\ymax)}

4 \pscircle*[linecolor=red,linewidth=1.5pt]

5 (-2,-1){2}

6 \endpsclip

7 \pswedge*[linecolor=white](-2,-1){1}{15}{105}

8 %\end{pspicture}

17

Finally, we plot some functions. To do this we’ll use some custom macros that give theuser better control over the coordinate system.

1 \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)

2 \showgrid

3 \pstVerb{%

4 /f@ {dup mul} def % x^2

5 }

6 %%%%%%%%%%%%%%%%%%%%

7 %% Axes and Ticks %%

8 %%%%%%%%%%%%%%%%%%%%

9 \myaxes{<->}(0,0)(\xmin,\ymin)(\xmax,\ymax)

10 \xTickMarks{\xmin}{\xmax}{1}

11 \yTickMarks{\ymin}{\ymax}{5}

12 %%%%%%%%%%%%%%%%%%%%%%%

13 %% Graphical Objects %%

14 %%%%%%%%%%%%%%%%%%%%%%%


16 \psplot[style=myPlotStyle]

17 {\xmin}{\xmax}{x f@}

18 \endpsclip

19 %%%%%%%%%%%%

20 %% Labels %%

21 %%%%%%%%%%%%

22 \SpecialCoor

23 \uput{6pt}[0](!3 dup f@){$y=x^2$}

24 \NormalCoor

25 \end{pspicture}

18

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6−5

0

5

10

15

20

25

30

35

40

y = x2

And again, using better grid controls.

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6−5

0

5

10

15

20

25

30

35

40y = x2

19

Here’s something useful for integration theory. Use the sketch to estimate the integralbelow.

∫

5

2

1

xdx

y =1

x

1 2 3 4 5 6

1

20

Here is the code

1 \def\xmin{0}\def\xmax{6}

2 \def\ymin{0}\def\ymax{1}

3 \def\dommin{\xmin}\def\dommax{\xmax}

4 \VR{3in}{2.5in}

5

6 \newpsobject{newgrid}{psgrid}{%

7 gridlabels=0pt%

8 ,griddots=0%

9 ,gridwidth=0.5pt%

10 ,gridcolor=gray%

11 ,subgriddiv=4%

12 ,subgridwidth=0.25pt%

13 ,subgridcolor=red}

14 \begin{pspicture}(\xmin,\ymin)(\xmax,\ymax)

15 \newgrid

16 %%%%%%%%%%%%%%%%%%%%


18 %%%%%%%%%%%%%%%%%%%%

19 \SpecialCoor

20 %% Labels go here

21 \rput[lr](!\xmax\space\xmax\space\xmin\space sub 15 div sub

22 \ymax\space\ymax\space\ymin\space sub 25 div sub){$y=\dfrac{1}{x}$}

23 \NormalCoor


25 %% Change these as needed. #1 - start, #2 - end, #3 - increment

26 \xTickMarks{\xmin}{\xmax}{1}

27 \yTickMarks{\ymin}{\ymax}{1}

28 \psclip{\psframe(\xmin,\ymin)(\xmax,\ymax)}

29 %% Graphing directives go here, e.g.,

30 \psplot[style=myPlotStyle]

31 {0.1}{\xmax}{1 x div}

32 \endpsclip

33 \SpecialCoor

34 \psline[fillstyle=crosshatch]{-}(3,0)(3.5,0)(!3.5 1 3

35 div)(!3 1 3 div)(3,0)

36 \psline[fillstyle=crosshatch]{-}(3.5,0)(4,0)(!4 1 3.5

37 div)(!3.5 1 3.5 div)(3.5,0)

38 \psline[fillstyle=crosshatch]{-}(4,0)(4.5,0)(!4.5 1 4

39 div)(!4 1 4 div)(4,0)

40 \psline[fillstyle=crosshatch]{-}(4.5,0)(5,0)(!5 1 4.5

41 div)(!4.5 1 4.5 div)(4.5,0)

42 \NormalCoor

43 \end{pspicture}

21

(c) Plotting Data from a File

Suppose that you wish to plot the following data.

0, 00.0628, 0.062790.1256, 0.12533

... ...

The following code does the trick.


2 \showgrid



5 %% Graphical Objects %%


7 \fileplot{plotData.txt}

8 \endpsclip

9 \SpecialCoor

10 \rput[lt](!\xmax\space\xmin\space sub 15 div

11 \ymax\space\ymax\space\ymin\space sub 25 div sub){$y=\sin x$}

12 \NormalCoor

13 %\end{pspicture}

y = sin x

22

4. Several examples from geometry.

D

C

A B

E

80◦

30◦

A80◦

B

C

77◦

40◦

DE

53◦50◦60◦

A

B

C

P

T V

R

E

23

5. A few exotic tricks.

(a) A vector field.

−2 −1 1 2

−2

−1

1

2

24

(b) An ice-cream cone.

x

y

z

x

y

z

x y

z

25

(c) A level surface.

R

S

f(x, y, z) = c

b

b

∆σk

∆Pk

∆Ak

p

p∇f (xk, yk, zk)

ukvk

26

Figure 20: Fractals - Sierpinski Triangle

x

y

z

Figure 21: Sphere

The equation of a sphere with radius r centered at the origin is

x2 + y2 + z2 = r2

27

0

80

160

−80

−160

20 · 103 40 · 103 60 · 103 80 · 103 100 · 103

x

y

Figure 22: Brownian Motion

1

2

3

4

−1

−2

1 2 3 4−1−2−3−4−5

Figure 23: Not Sure What to Call This

28

Figure 24: Tessellations?

3

.141

59 26535

897

9323846264

3383279502884

19716

9399

375105820974944592

3078164062862089986

2803

482534

2117

0679821480865132823066

470

9384460955058223172535940

8128

4811

174

502

841

0270

193852110555964462294895493

038

1964

428

81097566593344612847564823378

6783

1652

7120190

9145

6485

66

92346034861045432664821339360726024

9141

2737

24587006606315588174881520920962829

25

4091

7153

643

678

9259

03600

1133

0530

54882046652138414695194151160943305727

03657

59591

953092186117381932611793105118548074462379

96274

9567

3518

8575

2724891

227

938

1830

1194

9129

83367336244065664308602139494639522473719070

217986

09437

02770539217176293176752384674818467669405132000

56812

7145

2635

60827

785

771

342

757

789

609

1736

3717

87214

6844090122495343014654958537105079227968925892354201

995611212

90219608640344181598136297747713099605187072113499999983

7297

8049

9510

59731

732

8160963

185

9502

445

9455

3469

0830

2642

522308253344685035261931188171010003137838752886587533208

38142061717

76691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959

Figure 25: π Spiral

30

Transparency Effects

The sketch below shows the infinite plane x = 0.

xy

z

xy

z

x = 0

In a similar manner one can sketch the graphs of the equations y = 0 and z = 0.

x

y

z

x

y

z

x = 0y =0

z =0

Notice that these three planes break up three-space into eight octants. The first octant coincides withpositive x, y and z-coordinates and is three-dimensional analogue of quadrant I in the plane.

31

A Saddle Point

x

y

z

x

y

z

32

Example 1. A Familiar Curve

Find the areas of the shaded regions.

r1 = 2 cos θ − sin θ

r2 = cos θ

1 2

−1

1

Pb

We first need to find the polar coordinates of pointof intersection, P . Setting r1 = r2 and solving we seethat θ = π/4. It follows that the area of green portionof the shaded region is given by

AG =1

2

∫ π/4

0

(

r21 − r2

2

)

dθ

It follows that

AG =1

2

∫ π/4

0

sin2 θ + 3 cos2 θ − 4 sin θ cos θ dθ

=1

2

∫ π/4

0

1 + 2 cos2 θ − 2 sin 2θ dθ

=1

2

∫ π/4

0

2 + cos 2θ − 2 sin 2θ dθ

=1

4(4θ + sin 2θ + 2 cos 2θ)

π/4

0

=1

4{(π + 1 + 0) − (0 + 0 + 2)} =

π − 1

4

Notice that

AG + AB =1

2

∫

arctan 2

0

r21 dθ

(Compare this last equation to the gas tank problem.)

Finally, can you find AY?

33

Definition. The Gamma Function

Γ(x) =

∫

∞

0

tx−1e−t dt (1)

Here the (improper) integral converges absolutely for all x ∈ R except for the non-positive integers. Infact, the Gamma function can be extended throughout the complex plane (again, except for thenon-positive integers).

y = Γ(x)

b b b

b

b

b

−4 −3 −2 −1 1 2 3 4 5

2

6

24

Observe that

Γ(1) =

∫

∞

0

t0e−t dt

=−1

et

∞

0

= 0 − (−1) = 1

and for positive integers n, integration by parts yields the recursive relation

Γ(n + 1) =

∫

∞

0

tne−t dt

= −tne−t

∞

0

+ n

∫

∞

0

tn−1e−t dt

= 0 + nΓ(n)

and Euler had found his extension. That is, for each nonnegative integer n, he could now define thefactorial by

n! = Γ(n + 1) (2)

34

Sketch the curve given by the equation below in polar coordinates.

r1 = f(θ) = 2 cos θ − sin θ, 0 ≤ θ ≤ π (3)

−2 −1 1 2

−1

1

m = tan(arctan 2) = 2

r1 = 2 cos θ − sin θ, 0 ≤ θ ≤ π

b

rθ-Coordinate System

1 2 3 4 5 6

−2

−1

1

2 r = f(θ)

This busy sketch requires some explanation. Recall that the given (polar) equationdefines a circle of radius

√5/2 centered at (1, −1/2). The blue part of circle identifies

that portion of polar equation r1 = 2 cos θ−sin θ, 0 ≤ θ ≤ arctan 2, i.e., when r1 ≥ 0.The red part identifies the part of the curve for arctan 2 ≤ θ ≤ π, i.e., when r1 < 0.

The yellow portion is actually the sketch of the curve

r2 = |f(θ)|, arctan 2 ≤ θ ≤ π (4)

The vectors (arrows) trace out the curve r2 for 0 ≤ θ ≤ 2π. Notice that the equationin (4) yields two circles which are symmetric about the line y = 2x (shown in green).

35

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