Applications, Examples and Libraries

Circumscribed ellipse

by: one man 1355 Product: Maple 17

July 31 2020

0 2

One way to find the equation of an ellipse circumscribed around a triangle. In this case, we solve a linear system of equations, which is obtained after fixing the values of two variables ( t1 and t2). These are five equations: three equations of the second-order curve at three vertices of the triangle and two equations of a linear combination of the coordinates of the gradient of the curve equation.
The solving of system takes place in the ELS procedure. When solving, hyperboles appear, so the program has a filter. The filter passes the equations of ellipses based on by checking the values of the invariants of the second-order curves.
FOR_ELL_ТR_OUT_PROCE_F.mw ( Fixed comments in the text 01, 08, 2020)

Doing plots "a la XKCD"

by: mmcdara 7896 Product: Maple

July 25 2020

6 0

A lot of scientific software propose packages enabling drawing figures in XKCD style/
Up to now I thought this was restricted to open products (R, Python, ...) but I recently discovered Matlab and even Mathematica were doing same.

Layton S (2012). “XKCDIFY! Adding flair to boring Matlab Axes one plot at a time.” Last accessed on December 08, 2014, URL https://github.com/slayton/matlab-xkcdify.

Woods S (2012). “xkcd-style graphs.” Last accessed on December 08, 2014, URL http://mathematica.stackexchange.com/questions/11350/xkcd-style-graphs/ 11355#11355.

So why not Maple?

As a regular user of R, I could have visualize the body of the corresponding procedures to see how these drawings were made and just translate theminto Maple.
But copying for the sake of copying is not of much interest.
So I started to develop some primitives for "XKCD-drawing" lines, polygons, circles and even histograms.
My goal is not to write an XKCD package (I don't have the skills for that) but just to arouse the interest of (maybe) a few people here who could continue this preliminary work

A main problem is the one of the XKCD fonts: no question to redefine them in Maple and I guess using them in a commercial code is not legal (?). So no XKCD font in this first work, nor even the funny guy who appears recurently on the drawings (but it could be easily constructed in Maple).

In a recent post (Plot styling - experimenting with Maple's plotting...) Samir Khan proposed a few styles made of several plotting options, some of which he named "Excel style" or "Oscilloscope style"... maybe a future "XKCD xtyle" in Maple ?

This work has been done with Maple 2015 and reuses an old version of a 1D-Kriging procedure

>

restart:

>	with(LinearAlgebra): with(plots): with(Statistics):

The principle is always the same:
    1/   Let L a straight line which is either defined by its two ending points (xkvd_hline) or taken as the default [0, 0], [1, 0] line.
          For xkvd_hline the given line L is firstly rotate to be aligned with the horizontal axis.

    2/   Let P1, ..., PN N points on L. Each Pn writes [xn, yn]

    3/   A random perturbation rn is added yo the values y1, ..., yN

    4/   A stationnary random process RP, with gaussian correlation function is used to build a smooth curve passing through the points
          (x1, y1+r1), ..., (xN, yN+rN) (procedure KG where "KG" stands for "Kriging")

    5/   The result is drawn or mapped to some predefined shape :
                  xkcd_hist,
                  xkcd_polyline,
                  xkcd_circle

    6/   A procedure xkcd_func is also provided to draw functions defined by an explicit relation.

>

KG := proc(X, Y, psi, sigma)
  local NX, DX, K, mu, k, y:
  NX := numelems(X);
  DX := < seq(Vector[row](NX, X), k=1..NX) >^+:
  K := (sigma, psi) -> evalf( sigma^2 *~ exp~(-((DX - DX^+) /~ psi)^~2) ):
  mu := add(Y) / NX;
  k := (x, sigma, psi) -> evalf( convert(sigma^2 *~ exp~(-((x -~ X ) /~ psi)^~2), Vector[row]) ):
  y := mu + k(x, sigma, psi) . (K(sigma, psi))^(-1) . convert((Y -~ mu), Vector[column]):
  return y
end proc:

xkcd_hline := proc(p1::list, p2::list, a::nonnegative, lc::positive, col)
  # p1 : first ending point
  # p2 : second ending point
  # a : amplitude of the random perturbations
  # lc : correlation length
  # col: color
  local roll, NX, LX, X, Z:
  roll := rand(-1.0 .. 1.0):
  NX   := 10:
  LX   := p2[1]-p1[1]:
  X    := [seq(p1[1]..p2[1], LX/(NX-1))]:
  Z    := [p1[2], seq(p1[2]+a*roll(), k=1..NX-1)]:
  return plot(KG(X, Z, lc*LX, 1), x=min(X)..max(X), color=col, scaling=constrained):
end proc:

xkcd_line := proc(L::list, a::nonnegative, lc::positive, col, {lsty::integer:=1})
  # L : list which contains the two ending point
  # a : amplitude of the random perturbations
  # lc : correlation length
  # col: color
  local T, roll, NX, DX, DY, LX, A, m, M, X, Z, P:
  T    := (a, x0, y0, l) ->
             plottools:-transform(
               (x,y) -> [ x0 + l * (x*cos(a)-y*sin(a)), y0 + l * (x*sin(a)+y*cos(a)) ]
             ):
  roll := rand(-1.0 .. 1.0):
  NX   := 5:
  DX   := L[2][1]-L[1][1]:
  DY   := L[2][2]-L[1][2]:
  LX := sqrt(DX^2+DY^2):
  if DX <> 0 then
     A := arcsin(DY/LX):
  else
     A:= Pi/2:
  end if:
  X := [seq(0..1, 1/(NX-1))]:
  Z := [ seq(a*roll(), k=1..NX)]:
  P := plot(KG(X, Z, lc, 1), x=0..1, color=col, scaling=constrained, linestyle=lsty):
  return T(A, op(L[1]), LX)(P)
end proc:

xkcd_func := proc(f, r::list, NX::posint, a::positive, lc::positive, col)
  # f : function to draw
  # r : plot range
  # NX : number of equidistant "nodes" in the range r (boundaries included)
  # a : amplitude of the random perturbations
  # lc : correlation length
  # col: color
  local roll, F, LX, Pf, Xf, Zf:
  roll := rand(-1.0 .. 1.0):
  F    := unapply(f, indets(f, name)[1]);
  LX   := r[2]-r[1]:
  Pf   := [seq(r[1]..r[2], LX/(NX-1))]:
  Xf   := Pf +~ [seq(a*roll(), k=1..numelems(Pf))]:
  Zf   := F~(Pf) +~ [seq(a*roll(), k=1..numelems(Pf))]:
  return plot(KG(Xf, Zf, lc*LX, 1), x=min(Xf)..max(Xf), color=col):
end proc:

xkcd_hist := proc(H, ah, av, ax, ay, lch, lcv, lcx, lcy, colh, colxy)
  # H   : Histogram
  # ah : amplitude of the random perturbations on the horizontal boundaries of the bins
  # av : amplitude of the random perturbations on the vertical boundaries of the bins
  # ax : amplitude of the random perturbations on the horizontal axis
  # ay : amplitude of the random perturbations on the vertical axis
  # lch : correlation length on the horizontal boundaries of the bins
  # lcv : correlation length on the vertical boundaries of the bins
  # lcx : correlation length on the horizontal axis
  # lcy : correlation length on the vertical axis
  # colh: color of the histogram
  # col : color of the axes
  local data, horiz, verti, horizontal_lines, vertical_lines, po, rpo, p1, p2:
  data := op(1..-2, op(1, H)):
  verti := sort( [seq(data[n][3..4][], n=1..numelems([data]))] , key=(x->x[1]) );
  verti := verti[1],
           map(
                n -> if verti[n][2] > verti[n+1][2] then
                        verti[n]
                      else
                        verti[n+1]
                      end if,
                [seq(2..numelems(verti)-2,2)]
           )[],
           verti[-1];
  horiz := seq(data[n][[4, 3]], n=1..numelems([data])):

  horizontal_lines := NULL:
  for po in horiz do
    horizontal_lines := horizontal_lines, xkcd_hline(po[1], po[2], ah, lch, colh):
  end do:

  vertical_lines := NULL:
  for po in [verti] do
    rpo := po[[2, 1]]:
    vertical_lines := vertical_lines, xkcd_hline([0, rpo[2]], rpo, av, lcv, colh):
  end do:

  p1 := [2*verti[1][1]-verti[2][1], 0]:
  p2 := [2*verti[-1][1]-verti[-2][1], 0]:

  return
    display(
      horizontal_lines,
      T~([vertical_lines]),
      xkcd_hline(p1, p2, ax, lcx, colxy),
      T(xkcd_hline([0, 0], [1.2*max(op~(2, [verti])), 0], ay, lcy, colxy)),
      axes=none,
      scaling=unconstrained
    );
end proc:

xkcd_polyline := proc(L::list, a::nonnegative, lc::positive, col)
  # xkcd_polyline reduces to xkcd_line whebn L has 2 elements
  # L : List of points
  # a : amplitude of the random perturbations
  # lc : correlation length
  # col: color
  local T, roll, NX, n, DX, DY, LX, A, m, M, X, Z, P:
  T    := (a, x0, y0, l) ->
             plottools:-transform(
               (x,y) -> [ x0 + l * (x*cos(a)-y*sin(a)), y0 + l * (x*sin(a)+y*cos(a)) ]
             ):
  roll := rand(-1.0 .. 1.0):
  NX   := 5:
  for n from 1 to numelems(L)-1 do
    DX   := L[n+1][1]-L[n][1]:
    DY   := L[n+1][2]-L[n][2]:
    LX := sqrt(DX^2+DY^2):
    if DX <> 0 then
      A := evalf(arcsin(abs(DY)/LX)):
      if DX >= 0 and DY <= 0 then A := -A end if:
      if DX <= 0 and DY > 0 then A := Pi-A end if:
      if DX <= 0 and DY <= 0 then A := Pi+A end if:
    else
      A:= Pi/2:
      if DY < 0 then A := 3*Pi/2 end if:
    end if:
    if n=1 then
      X := [seq(0..1, 1/(NX-1))]:
      Z := [seq(a*roll(), k=1..NX)]:
    else
      X := [0    , seq(1/(NX-1)..1, 1/(NX-1))]:
      Z := [Z[NX], seq(a*roll(), k=1..NX-1)]:
    end if:
    P    := plot(KG(X, Z, lc, 1), x=0..1, color=col, scaling=constrained):
    P||n := T(A, op(L[n]), LX)(P):
  end do;
  return seq(P||n, n=1..numelems(L)-1)
end proc:

xkcd_circle := proc(a::nonnegative, lc::positive, r::positive, cent::list, col)
  # a   : amplitude of the random perturbations
  # lc : correlation length
  # r   : redius of the circle
  # cent: center of the circle
  # col : color
  local roll, NX, LX, X, Z, xkg, A:
  roll := rand(-1.0 .. 1.0):
  NX   := 10:
  X    := [seq(0..1, 1/(NX-1))]:
  Z    := [0, seq(a*roll(), k=1..NX-1)]:
  xkg := KG(X, Z, lc, 1):
  A    := Pi*roll():
  return plot([cent[1]+r*(1+xkg)*cos(2*Pi*x+A), cent[2]+r*(1+xkg)*sin(2*Pi*x+A), x=0..1], color=col)
end proc:

T := plottools:-transform((x,y) -> [y, x]):

>	# Axes plot x_axis := xkcd_hline([0, 0], [10, 0], 0.03, 0.5, black): y_axis := xkcd_hline([0, 0], [10, 0], 0.03, 0.5, black): display( x_axis, T(y_axis), axes=none, scaling=constrained )

>	# A simple function f := 1+10*(x/5-1)^2: F := xkcd_func(f, [0.5, 9.5], 6, 0.3, 0.4, red): display( x_axis, T(y_axis), F, axes=none, scaling=constrained )

>	# An histogram S := Sample(Normal(0,1),100): H := Histogram(S, maxbins=6): xkcd_hist(H, 0, 0.02, 0.001, 0.01, 1, 0.1, 0.01, 1, red, black)

>

# Axes plus grid with two red straight lines

r := rand(-0.1 .. 0.1):

x_axis := xkcd_line([[-2, 0], [10, 0]], 0.01, 0.2, black):
y_axis := xkcd_line([[0, -2], [0, 10]], 0.01, 0.2, black):
d1     := xkcd_line([[-1, 1], [9, 9]] , 0.01, 0.2, red):
d2     := xkcd_line([[-1, 9], [9, -1]], 0.01, 0.2, red):
display(
  x_axis, y_axis,
  seq( xkcd_line([[-2+0.3*r(), u+0.3*r()], [10+0.3*r(), u+0.3*r()]], 0.005, 0.5, gray), u in [seq(-1..9, 2)]),
  seq( xkcd_line([[u+0.3*r(), -2+0.3*r()], [u+0.3*r(), 10+0.3*r()]], 0.005, 0.5, gray), u in [seq(-1..9, 2)]),
  d1, d2,
  axes=none,
  scaling=constrained
)

>	# Axes and a couple of polygonal lines d1 := xkcd_polyline([[0, 0], [1, 3], [3, 5], [7, 1], [9, 7]], 0.01, 1, red): d2 := xkcd_polyline([[0, 9], [2, 8], [5, 2], [8, 3], [10, -1]], 0.01, 1, blue): display( x_axis, y_axis, d1, d2, axes=none, scaling=constrained )

>

# A few polygonal shapes

display(
  xkcd_polyline([[0, 0], [1, 0], [1, 1], [0, 1], [0, 0]], 0.01, 1, red),
  xkcd_polyline([[1/3, 1/3], [2/3, 1/3], [2/3, 4/3], [-1, 4/3], [1/3, 1/3]], 0.01, 1, blue),
  xkcd_polyline([[-1/3, -1/3], [4/3, 1/2], [1/2, 1/2], [1/2,-1], [-1/3, -1/3]], 0.01, 1, green),
  axes=none,
  scaling=constrained
)

>

# A few circles

cols := [red, green, blue, gold, black]:                                # colors
cents := convert( Statistics:-Sample(Uniform(-1, 3), [5, 2]), listlist): # centers
radii := Statistics:-Sample(Uniform(1/2, 2), 5):                         # radii
lcs   := Statistics:-Sample(Uniform(0.2, 0.7), 5):                       # correlations lengths

display(
  seq(
    xkcd_circle(0.02, lcs[n], radii[n], cents[n], cols[n]),
    n=1..5
  ),
  axes=none,
  scaling=constrained
)

>

# A 3D drawing

x_axis := xkcd_line([[0, 0], [5, 0]], 0.01, 0.2, black):
y_axis := xkcd_line([[0, 0], [4, 2]], 0.01, 0.2, black):
z_axis := xkcd_line([[0, 0], [0, 5]], 0.01, 0.2, black):

f1 := 4*cos(x/6)-1:
F1 := xkcd_func(f1, [0.5, 5], 6, 0.001, 0.8, red):
F2 := xkcd_line([[0.5, eval(f1, x=0.5)], [3, 4]], 0.01, 0.2, red):
f3 := 4*cos((x-2)/6):
F3 := xkcd_func(f3, [3, 7], 6, 0.001, 0.8, red):
F4 := xkcd_line([[5, eval(f1, x=5)], [7, eval(f3, x=7)]], 0.01, 0.2, red):

dx := xkcd_line([[2, 1], [4, 1]], 0.01, 0.2, gray, lsty=3):
dy := xkcd_line([[2, 0], [4, 1]], 0.01, 0.2, gray, lsty=3):
dz := xkcd_line([[4, 1], [4, 3]], 0.01, 0.2, gray, lsty=3):

po := xkcd_circle(0.02, 0.3, 0.1, [4, 3], blue):

# Numerical value come from "probe info + copy/paste"

nvect   := xkcd_polyline([[4, 3], [4.57, 4.26], [4.35, 4.1], [4.57, 4.26], [4.58, 4.02]], 0.01, 1, blue):
tg1vect := xkcd_polyline([[4, 3], [4.78, 2.59], [4.49, 2.87], [4.78, 2.59], [4.46, 2.57]], 0.01, 1, blue):
tg2vect := xkcd_polyline([[4, 3], [4.79, 3.35], [4.70, 3.13], [4.79, 3.35], [4.46, 3.35]], 0.01, 1, blue):
rec1    := xkcd_polyline([[4.118, 3.286], [4.365, 3.396], [4.257, 3.108]], 0.01, 1, blue):
rec2    := xkcd_polyline([[4.257, 3.108], [4.476, 2.985], [4.259, 2.876]], 0.01, 1, blue):

display(
  x_axis, y_axis, z_axis,
  F1, F2, F3, F4,
  dx, dy, dz,
  po,
  nvect, tg1vect, tg2vect, rec1, rec2,
  axes=none,
  scaling=constrained
)

>

# Arrow

d1 := xkcd_polyline([[0, 0], [1, 0], [0.9, 0.05], [1, 0], [0.9, -0.05]], 0.01, 1, red):

T := (a, x0, y0, l) ->
             plottools:-transform(
               (x,y) -> [ x0 + l * (x*cos(a)-y*sin(a)), y0 + l * (x*sin(a)+y*cos(a)) ]
             ):

display(
  seq( T(2*Pi*n/10, 0.5, 0, 1/2)(
           display(
              xkcd_polyline(
                  [[0, 0], [1, 0], [0.9, 0.05], [1, 0], [0.9, -0.05]],
                  0.01,
                  1,
                  ColorTools:-Color([rand()/10^12, rand()/10^12, rand()/10^12])
               )
           )
        ),
       n=1..10
  ),
  axes=none,
  scaling=constrained
)

>

Download XKCD.mw

An ellipse inscribed in a given triangle

by: one man 1355 Product: Maple 17

July 24 2020

3 11

An attempt to find the equation of an ellipse inscribed in a given triangle.
The program works on the basis of the ELS procedure. After the procedure works, the solutions are filtered.
ELS procedure solves the system of equations f1, f2, f3, f4, f5 for the coefficients of the second-order curve.
The equation f1 corresponds to the condition that the side of the triangle intersects t a curve of the second order at one point.
The equation f2 corresponds to the condition that the point x1,x2 belongs to a curve of the second order.
Equation f3 corresponds to the condition that the side of the triangle is tangent to the second order curve at the point x1,x2.
The equation f4 is similar to the equation f2, and the equation f5 is similar to the equation f3.
FOR_ELL_ТR_PROCE.mw
For example

Plot styling - experimenting with Maple's plotting...

by: Samir Khan 2151 Product: Maple

July 13 2020

11 4

I like tweaking plots to get the look and feel I want, and luckily Maple has many plotting options that I often play with. Here, I visualize the same data several times, but each time with different styling.

First, some data.

restart:
data_1 := [[0,0],[1,2],[2,1.3],[3,6]]:
data_2 := [[0.5,3],[1,1],[2,5],[3,2]]:
data_3 := [[-0.5,3],[1.3,1],[2.5,5],[4.5,2]]:

This is the default look.

plot([data_1, data_2, data_3])

I think the darker background on this plot makes it easier to look at.

gray_grid :=
 background      = "LightGrey"
,color           = [ ColorTools:-Color("RGB",[150/255, 40 /255, 27 /255])
                    ,ColorTools:-Color("RGB",[0  /255, 0  /255, 0  /255])
                    ,ColorTools:-Color("RGB",[68 /255, 108/255, 179/255]) ]
,axes            = frame
,axis[2]         = [color = black, gridlines = [10, thickness = 1, color = ColorTools:-Color("RGB", [1, 1, 1])]]
,axis[1]         = [color = black, gridlines = [10, thickness = 1, color = ColorTools:-Color("RGB", [1, 1, 1])]]
,axesfont        = [Arial]
,labelfont       = [Arial]
,size            = [400*1.78, 400]
,labeldirections = [horizontal, vertical]
,filled          = false
,transparency    = 0
,thickness       = 5
,style           = line:

plot([data_1, data_2, data_3], gray_grid);

I call the next style Excel, for obvious reasons.

excel :=
 background      = white
,color           = [ ColorTools:-Color("RGB",[79/255,  129/255, 189/255])
                    ,ColorTools:-Color("RGB",[192/255, 80/255,   77/255])
                    ,ColorTools:-Color("RGB",[155/255, 187/255,  89/255])]
,axes            = frame
,axis[2]         = [gridlines = [10, thickness = 0, color = ColorTools:-Color("RGB",[134/255,134/255,134/255])]]
,font            = [Calibri]
,labelfont       = [Calibri]
,size            = [400*1.78, 400]
,labeldirections = [horizontal, vertical]
,transparency    = 0
,thickness       = 3
,style           = point
,symbol          = [soliddiamond, solidbox, solidcircle]
,symbolsize      = 15:

plot([data_1, data_2, data_3], excel)

This style makes the plot look a bit like the oscilloscope I have in my garage.

dark_gridlines :=
 background      = ColorTools:-Color("RGB",[0,0,0])
,color           = white
,axes            = frame
,linestyle       = [solid, dash, dashdot]
,axis            = [gridlines = [10, linestyle = dot, color = ColorTools:-Color("RGB",[0.5, 0.5, 0.5])]]
,font            = [Arial]
,size            = [400*1.78, 400]:

plot([data_1, data_2, data_3], dark_gridlines);

The colors in the next style remind me of an Autumn morning.

autumnal :=
 background      =  ColorTools:-Color("RGB",[236/255, 240/255, 241/255])
,color           = [  ColorTools:-Color("RGB",[144/255, 54/255, 24/255])
                     ,ColorTools:-Color("RGB",[105/255, 108/255, 51/255])
                     ,ColorTools:-Color("RGB",[131/255, 112/255, 82/255]) ]
,axes            = frame
,font            = [Arial]
,size            = [400*1.78, 400]
,filled          = true
,axis[2]         = [gridlines = [10, thickness = 1, color = white]]
,axis[1]         = [gridlines = [10, thickness = 1, color = white]]
,symbol          = solidcircle
,style           = point
,transparency    = [0.6, 0.4, 0.2]:

plot([data_1, data_2, data_3], autumnal);

In honor of a friend and ex-colleague, I call this style "The Swedish".

swedish :=
 background      = ColorTools:-Color("RGB", [0/255, 107/255, 168/255])
,color           = [ ColorTools:-Color("RGB",[169/255, 158/255, 112/255])
                    ,ColorTools:-Color("RGB",[126/255,  24/255,   9/255])
                    ,ColorTools:-Color("RGB",[254/255, 205/255,   0/255])]
,axes            = frame
,axis            = [gridlines = [10, color = ColorTools:-Color("RGB",[134/255,134/255,134/255])]]
,font            = [Arial]
,size            = [400*1.78, 400]
,labeldirections = [horizontal, vertical]
,filled          = false
,thickness       = 10:

plot([data_1, data_2, data_3], swedish);

This looks like a plot from a journal article.

experimental_data_mono :=

background       = white
,color           = black
,axes            = box
,axis            = [gridlines = [linestyle = dot, color = ColorTools:-Color("RGB",[0.5, 0.5, 0.5])]]
,font            = [Arial, 11]
,legendstyle     = [font = [Arial, 11]]
,size            = [400, 400]
,labeldirections = [horizontal, vertical]
,style           = point
,symbol          = [solidcircle, solidbox, soliddiamond]
,symbolsize      = [15,15,20]:

plot([data_1, data_2, data_3], experimental_data_mono, legend = ["Annihilation", "Authority", "Acceptance"]);

If you're willing to tinker a little bit, you can add some real character and personality to your visualizations. Try it!

I'd also be very interested to learn what you find attractive in a plot - please do let me know.

Realistic 3D rendering of a complex system

by: mmcdara 7896 Product: Maple 2015

July 02 2020

5 1

Hi,

I would like to share this work I've done.
No big math here, just a demonstrator of Maple's capabilities in 3D visualization.

All the plots in the file have been discarded to reduce the size of this post. Here is a screen capture to give you an idea of what is inside the file.

Download 3D_Visualization.mw

Numerical evaluation of integrals over polygons...

by: mmcdara 7896

June 28 2020

2 2

Hi,

In a recent post (Monte Carlo Integration) Radaar shared its work about the numerical integration, with the Monte Carlo method, of a function defined in polar coordinates.
Radaar used a raw strategy based on a sampling in cartesian coordinates plus an ad hoc transformation.
Radaar obtained reasonably good results, but I posted a comment to show how Monte Carlo summation in polar coordinates can be done in a much simpler way. Behind this is the choice of a "good" sampling distribution which makes the integration problem as simple as Monte Carlo integration over a 2D rectangle with sides parallel to the co-ordinate axis.

This comment I sent pushed me to share the present work on Monte Carlo integration over simple polygons ("simple" means that two sides do not intersect).
Here again one can use raw Monte Carlo integration on the rectangle this polygon is inscribed in. But as in Radaar's post, a specific sampling distribution can be used that makes the summation method more elegant.

This work relies on three main ingredients:

The Dirichlet distribution, whose one form enables sampling the 2D simplex in a uniform way.
The construction of a 1-to-1 mapping from this simplex into any non degenerated triangle (a mapping whose jacobian is a constant equal to the ratio of the areas of the two triangles).
A tesselation into triangles of the polygon to integrate over.

This work has been carried out in Maple 2015, which required the development of a module to do the tesselation. Maybe more recent Maple's versions contain internal procedures to do that.

Monte_Carlo_Integration.mw

Why Staying at Home is Good to Avoid the Spread...

by: Roanes-Lozano 70 Product: Maple

June 16 2020

13 0

Hi. My name is Eugenio and I’m a Professor at the Departamento de Didáctica de las Ciencias Experimentales, Sociales y Matemáticas at the Facultad de Educación of the Universidad Complutense de Madrid (UCM) and a member of the Instituto de Matemática Interdisciplinar (IMI) of the UCM.

I have a 14-year-old son. In the beginning of the pandemic, a confinement was ordered in Spain. It is not easy to make a kid understand that we shouldn't meet our friends and relatives for some time and that we should all stay at home in the first stage. So, I developed a simplified explanation of virus propagation for kids, firstly in Scratch and later in Maple, the latter using an implementation of turtle geometry that we developed long ago and has a much better graphic resolution (E. Roanes-Lozano and E. Roanes-Macías. An Implementation of “Turtle Graphics” in Maple V. MapleTech. Special Issue, 1994, 82-85). A video (in Spanish) of the Scratch version is available from the Instituto de Matemática Interdisciplinar (IMI) web page: https://www.ucm.es/imi/other-activities

Introduction

Surely you are uncomfortable being locked up at home, so I will try to justify that, although we are all looking forward going out, it is good not to meet your friends and family with whom you do not live.

I firstly need to mention a fractal is. A fractal is a geometric object whose structure is repeated at any scale. An example in nature is Romanesco broccoli, that you perhaps have eaten (you can search for images on the Internet). You can find a simple fractal in the following image (drawn with Maple):

Notice that each branch is divided into two branches, always forming the same angle and decreasing in size in the same proportion.

We can say that the tree in the previous image is of “depth 7” because there are 7 levels of branches.

It is quite easy to create this kind of drawing with the so called “turtle geometry” (with a recursive procedure, that is, a procedure that calls itself). Perhaps you have used Scratch programming language at school (or Logo, if you are older), which graphics are based in turtle geometry.

All drawings along these pages have been created with Maple. We can easily reform the code that generated the previous tree so that it has three, four, five,… branches at each level, instead of two.

But let’s begin with a tale that explains in a much simplified way how the spread of a disease works.

- o O o -

Let's suppose that a cat returns sick to Catland suffering from a very contagious disease and he meets his friends and family, since he has missed them so much.

We do not know very well how many cats each sick cat infects in average (before the order to STAY AT HOME arrives, as cats in Catland are very obedient and obey right away). Therefore, we’ll analyze different scenarios:

Each sick cat infects two other cats.
Each sick cat infects three other cats.
Each sick cat infects five other cats

1. Each Sick Cat Infects Two Cats

In all the figures that follow, the cat initially sick is in the center of the image. The infected cats are represented by a red square.

· Before everyone gets confined at home, it only takes time for that first sick cat to see his friends, but then confinement is ordered (depth 1)

As you can see, with the cat meeting his friends and family, we already have 3 sick cats.

· Before all cats confine themselves at home, the first cat meets his friends, and these in turn have time to meet their friends (depth 2)

In this case, the number of sick cats is 7.

· Before every cat is confined at home, there is time for the initially sick cat to meet his friends, for these to meet their friends, and for the latter (friends of the friends of the first sick cat) to meet their friends (depth 3).

There are already 15 sick cats...

· Depth 4: 31 sick cats.

· Depth 5: 63 sick cats.

Next we’ll see what would happen if each sick cat infected three cats, instead of two.

2. Every Sick Cat Infects Three Cats

· Now we speed up, as you’ve got the idea.

The first sick cat has infected three friends or family before confining himself at home. So there are 4 infected cats.

· If each of the recently infected cats in the previous figure have in turn contact with their friends and family, we move on to the following situation, with 13 sick cats:

· And if each of these 13 infected cats lives a normal life, the disease spreads even more, and we already have 40!

· At the next step we have 121 sick cats:

· And, if they keep seeing friends and family, there will be 364 sick cats (the image reminds of what is called a Sierpinski triangle):

4. Every Sick Cat Infects Five Cats

· In this case already at depth 2 we already have 31 sick cats.

5. Conclusion

This is an example of exponential growth. And the higher the number of cats infected by each sick cat, the worse the situation is.

Therefore, avoiding meeting friends and relatives that do not live with you is hard, but good for stopping the infection. So, it is hard, but I stay at home at the first stage too!

Monte Carlo Integration

by: radaar 202 Product: Maple

June 15 2020

1 1

Monte Carlo integration uses random sampling unlike classical techniques like the trapezoidal or Simpson's rule in evaluating the integration numerically.

Download MONTE_CARLO_INTEGRATION1.mw

Minimal Road Radius - solving a physics problem...

by: Scot Gould 1039 Product: Maple 2020

June 14 2020

3 2

The purpose of this document is:

a) to correct the physics that was used in the document "Minimal Road Radius for Highway Superelevation" recently submitted to the Maple Applications Center;

b) to confirm the values found in the manual for the American Association of State Highway and Transportation Officials (AASHTO) that engineers use to design and build these banked curves are physically sound.

c) to highlight the pedagogical value inherent in the Maple language to distinguish between assignment ( := ) and equivalence ( = );

d) but most importantly, to demonstrate the pedagogical value Maple has in thinking about solving a problem involving a physical process. Given Maple's symbolic mathematics capabilities, one can implement a top-down approach to the physics and the mathematics, working from the general principle to the specific example. This allows one to avoid the types of errors that occur when translating the problem into a bottom up approach, from specific values of the example to the general principle, an approach that is required by most other computational systems.

I hope that others are willing to continue to engage in discussions related to the pedagogical value of Maple beyond mathematics.

I was asked to post this document to both here and the Maple Applications Center

[Document edited for typos.]

Minimum_Road_Radius.mw

Solving multi-span Euler beams with the Method...

by: Rouben Rostamian 8566 Product: Maple 2020

June 08 2020

14 15

Maple's pdsolve() is quite capable of solving the PDE that describes the motion of a single-span Euler beam. As far as I have been able to ascertain, there is no obvious way of applying pdsolve() to solve multi-span beams. The worksheet attached to this post provides tools for solving multi-span Euler beams. Shown below are a few demos. The worksheet contains more demos.

A module for solving the Euler beam with the method of lines

beamsolve

The beamsolve proc solves a (possibly multi-span) Euler beam equation:

subject to initial and boundary conditions. The solution is the

transverse deflection of the beam at point at time , subject to the load
density (i.e., load per unit length) given by . The coefficient

is the beam's mass density (mass per unit length), is the Young's modulus of

the beam's material, and is the beam's cross-sectional moment of inertia

about the neutral axis. The figure below illustrates a 3-span beam (drawn in green)
supported on four supports, and loaded by a variable density load (drawn in gray)
which may vary in time. The objective is to determine the deformed shape of the
beam as a function of time.

The number of spans, their lengths, and the nature of the supports may be specified

as arguments to beamsolve.

In this worksheet we assume that are constants for simplicity. Since only
the product of the coefficients and enters the calculations, we lump the two

together into a single variable which we indicate with the two-letter symbol .

Commonly, is referred to as the beam's rigidity.

The PDE needs to be supplied with boundary conditions, two at each end, each

condition prescribing a value (possibly time-dependent) for one of
(that's 36 possible combinations!) where I have used subscripts to indicate

derivatives. Thus, for a single-span beam of length , the following is an admissible

set of boundary conditions:
. (Oops, coorection, that last
condition was meant to be u_xxx(L,t) = sin t.)
Additionally, the PDE needs to be supplied with initial conditions which express

the initial displacement and the initial velocity:

The PDE is solved through the Method of Lines. Thus, each span is subdivided into

subintervals and the PDE's spatial derivatives are approximated through finite differences.

The time derivatives, however, are not discretized. This reduces the PDE into a set of

ODEs which are solved with Maple's dsolve().

Calling sequence:

beamsolve(, options)

Parameters:

: List of span lengths, in order from left to right, as in .

The number of subintervals in the shortest span (for the finite difference approximation)

Notes:

•	It is assumed that the spans are laid back-to-back along the axis, with the left end of the overall beam at .

•

The interior supports, that is, those supports where any two spans meet, are assumed
to be of the so-called simple type. A simple support is immobile and it doesn't exert
a bending moment on the beam. Supports at the far left and far right of the beam can
be of general type; see the BC_left and BC_right options below.

•	If the beam consists of a single span, then the argument may be entered as a number rather than as a list. That is, is equivalent to .

Options:

All options are of the form option_name=value, and have built-in default values.

Only options that are other than the defaults need be specified.

rho: the beam's (constant) mass density per unit length (default: rho = 1)

EI: the beam's (constant) rigidity (default: EI = 1)

T: solve the PDE over the time interval (default: T = 1)

        F: an expression in and that describes the applied force   (default: F = 0)
        IC: the list of the initial displacement and velocity, as
                expressions in (default: IC = [0,0])

        BC_left: a list consisting of a pair of boundary conditions at the left end of
                the overall (possibly multi-span beam. These can be any two of
                . The right-hand sides of these equations

can be any expression in . The left-hand sides should be entered literally as indicated.

If a right-hand side is omitted, it is taken to be zero. (default: BC_left = which

corresponds to a simple support).

BC_right: like BC_left, but for the right end of the overall beam (default: BC_right =

The returned module:

A call to beamsolve returns a module which presents three methods. The methods are:

plot (t, refine=val, options)

plots the solution at time . If the discretization in the direction

is too coarse and the graph looks non-smooth, the refine option

(default: refine=1) may be specified to smooth out the graph by introducing

val number of intermediate points within each discretized subinterval.

All other options are assumed to be plot options and are passed to plots:-display.

plot3d (m=val, options)

plots the surface . The optional specification requests

a grid consisting of subintervals in the time direction (default:

Note that this grid is for plotting purposes only; the solution is computed

as a continuous (not discrete) function of time. All other options are assumed

to be plot3d options and are passed to plots:-display.

animate (frames=val, refine=val, options)

produces an animation of the beam's motion. The frames option (default = 50)

                specifies the number of animation frames. The refine option is passed to plot
                (see the description above. All other options are assumed to be plot options and
                are passed to plots:-display.

Note:

In specifying the boundary conditions, the following reminder can be helpful. If the beam

is considered to be horizontal, then is the vertical displacement, is the slope,

is the bending moment, and is the transverse shear force.

>

A single-span simply-supported beam with initial velocity

The function is an exact solution of a simply supported beam with

The solution is periodic in time with period .

>	sol := beamsolve(1, 25, 'T'=2/Pi, 'IC'=[0, Pi^2sin(Pix)]): sol:-animate(size=[600,250]);

The initial condition does not lead to a separable form, and

therefore the motion is more complex.

>	sol := beamsolve(1, 25, 'T'=2/Pi, 'IC'=[0, 1]): sol:-animate(frames=200, size=[600,250]);

>

A single-span cantilever beam

A cantilever beam with initial condition where is the
first eigenmode of its free vibration (calculated in another spreadsheet). The motion is
periodic in time, with period

>	g := 0.5cos(1.875104069x) - 0.5cosh(1.875104069x) - 0.3670477570sin(1.875104069x) + 0.3670477570sinh(1.875104069x): sol := beamsolve(1, 25, 'T'=1.787018777, 'BC_left'=[u,u_x], 'BC_right'=[u_xx,u_xxx], 'IC'=[g, 0]): sol:-animate(size=[600,250]);

If the initial condition is not an eigenmode, then the solution is rather chaotic.

>	sol := beamsolve(1, 25, 'T'=3.57, 'BC_left'=[u,u_x], 'BC_right'=[u_xx,u_xxx], 'IC'=[-x^2, 0]): sol:-animate(size=[600,250], frames=100);

>

A single-span cantilever beam with a weight hanging from its free end

>	sol := beamsolve(1, 25, 'T'=3.57, 'BC_left'=[u,u_x], 'BC_right'=[u_xx,u_xxx=1]): sol:-animate(size=[600,250], frames=100);

>

A single-span cantilever beam with oscillating support

>	sol := beamsolve(1, 25, 'T'=Pi, 'BC_left'=[u=0.1sin(10t),u_x], 'BC_right'=[u_xx,u_xxx]): sol:-animate(size=[600,250], frames=100);

>

A dual-span simply-supported beam with moving load

Load moves across a dual-span beam.

The beam continues oscillating after the load leaves.

>

d := 0.4: T := 4: nframes := 100:
myload := - max(0, -6*(x - t)*(d + x - t)/d^3):
sol := beamsolve([1,1], 20, 'T'=T, 'F'=myload):
sol:-animate(frames=nframes):
plots:-animate(plot, [2e-3*myload(x,t), x=0..2, thickness=1, filled=[color="Green"]], t=0..T, frames=nframes):
plots:-display([%%,%], size=[600,250]);

>

A triple-span simply-supported beam with moving load

Load moves across a triple-span beam.

The beam continues oscillating after the load leaves.

>

d := 0.4: T := 6: nframes := 100:
myload := - max(0, -6*(x - t)*(d + x - t)/d^3):
sol := beamsolve([1,1,1], 20, 'T'=T, 'F'=myload):
sol:-plot3d(m=50);
sol:-animate(frames=nframes):
plots:-animate(plot, [2e-3*myload(x,t), x=0..3, thickness=1, filled=[color="Green"]], t=0..T, frames=nframes):
plots:-display([%%,%], size=[600,250]);

>

z3d;

>

A triple-span beam, moving load falling off the cantilever end

In this demo the load move across a multi-span beam with a cantilever section at the right.

As it skips past the cantilever end, the beam snaps back violently.

>

d := 0.4: T := 8: nframes := 200:
myload := - max(0, -6*(x - t/2)*(d + x - t/2)/d^3):
sol := beamsolve([1,1,1/2], 10, 'T'=T, 'F'=myload, BC_right=[u_xx, u_xxx]):
sol:-animate(frames=nframes):
plots:-animate(plot, [1e-2*myload(x,t), x=0..3, thickness=1, filled=[color="Green"]], t=0..T, frames=nframes):
plots:-display([%%,%], size=[600,250]);

>

Download worksheet: euler-beam-with-method-of-lines.mw

Scalable Subplots within Figure

by: Bland3 30 Product: Maple 2019

May 29 2020

0 3

This is my attempt to produce a subplot within a larger plot for magnifying data in a small region, and putting that subplot into the white space of the figure.
Based on the questions: How to insert a plot into another plot? and Inset figure in Maple, I wrote a couple of procedures that create sub-plots and allow the user to place the subplot window as he/she chooses. This avoids the graininess issues mentioned by acer in the second link (and experienced by me).

So far, I only have this completed for point plots, but using acer's method of piecewise functions posted in the plotin2b.mw of the second article, with the subplot function being defined only if it satisfies your conditions, would allow the subplot generating procedure to be generalized easily enough. But the data I'm working with all point plots, so that's the example here.

The basic idea is to use one procedure to create boxes, make tickmarks on the expanded region, and make tickmark labels, combine all of those into one graph. Then create scaled and shifted versions of the data series, then make graphs of those. Lastly, combine them all into one picture.

Hope this helps someone who has to do the same.

Mapleprimes isn't inserting the contents, but here is the download of the file: SubPlotBoxesandVectorDataSeries.mw

Partitioning a cube into pyramids: An animation

by: Rouben Rostamian 8566 Product: Maple

May 21 2020

7 3

Here is a little cute demo that shows how a cube may be paritioned into three congruent pyramids. This was inspired by a Mathematica demo that I found in the web but I think this one's better :-)

A Cube as a union of three right pyramids

Here is an animated demo of the well-known fact that a cube may be partitioned

into three congruent right pyramids.

2020-05-21

>

restart;

>	with(plots):

>	with(plottools):

A proc to plot a general polyhedron.
list of vertices
list of faces

An entry in describes a face made of the vertices etc.

>	polyhedron := proc(V::list, F::list) seq(plottools:-polygon([seq( V[F[i][j]], j=1..nops(F[i]))]), i=1..nops(F)); plots:-display(%); end proc:

Define the vertices and faces of a pyramid:

>	v := [[0,0,0],[1,0,0],[1,1,0],[0,1,0],[0,0,1]]; f := [ [1,2,3,4], [5,2,3], [5,3,4], [1,5,4], [1,2,5] ];

Build three such pyramids:

>	P1 := polyhedron(v, f): P2 := reflect(P1, [[1,0,0],[1,1,0],[1,0,1]]): P3 := reflect(P1, [[0,1,0],[1,1,0],[0,1,1]]):

This is what we have so far:

>	display(P1,P2,P3, scaling=constrained);

Define an animation frame. The parameter goes from 0 to 1.

Any extra options are assumed to be plot3d options and are

passed to plots:-display.

>	frame := proc(t) plots:-display( P1, rotate(P2, Pi/2t, [[1,1,0],[1,0,0]]), rotate(P3, Pi/2t, [[0,1,0],[1,1,0]]), color=["Red", "Green", "Blue"], _rest); end proc:

Animate:

>	display(frame(0) $40, seq(frame(t), t=0..1, 0.01), frame(1) $40, insequence, scaling=constrained, axes=none, orientation=[45,0,120], viewpoint=circleleft);

Download square-partitioned-into-pyramids.mw

The equations of motion in curvilinear coordinates,...

by: ecterrab 14630 Product: Maple

May 21 2020

5 0

The equations of motion in curvilinear coordinates, tensor notation and Coriolis force

The formulation of the equations of motion of a particle is simple in Cartesian coordinates using vector notation. However, depending on the problem, for example when describing the motion of a particle as seen from a non-inertial system of references (e.g. a rotating planet, like earth), there is advantage in using curvilinear coordinates and also tensor notation. When the particle's movement is observed from such a rotating referential, we also see "acceleration" that is not due to any force but to the fact that the referential itself is accelerated. The material below discusses and formulates these topics, and derives the expression for the Coriolis and centripetal force in cylindrical coordinates as seen from a rotating system of references.

The computation below is reproducible in Maple 2020 using the Maplesoft Physics Updates v.681 or newer.

Vector notation

Generally speaking the equations of motion of a particle are easy to formulate: the position vector is a function of time, the velocity is its first derivative and the acceleration is its second derivative. For instance, in Cartesian coordinates

>

(1)

>

(2)

>

(3)

Newton's 2^nd law, that in an inertial system of references when there is force there is acceleration and viceversa, is then given by

>

(4)

where represents the total force acting on the particle. This vectorial equation represents three second order differential equations which, for given initial conditions, can be integrated to arrive at a closed form expression for as a function of .

Tensor notation

In Cartesian coordinates, the tensorial form of the equations (4) is also straightforward. In a flat spacetime - Galilean system of references - the three space coordinates form a 3D tensor, and so does its first derivate and the second one. Set the spacetime to be 3-dimensional and Euclidean and use lowercaselatin indices for the corresponding tensors

>

Physics:-g_[mu, nu] = Matrix(%id = 18446744078329083054)

(5)

The position, velocity and acceleration vectors are expressed in tensor notation as done in (1), (2) and (3)

>

(6)

>

(7)

>

(8)

Setting a tensor to represent the three Cartesian components of the force

>

${Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-d_[a], Physics:-g_[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(9)

Newton's 2^nd law (4), now expressed in tensorial notation, is given by

>

(10)

The three differential equations behind this tensorial form of (4) are as expected

>

${F__x(t) = m*(diff(diff(x(t), t), t)), F__y(t) = m*(diff(diff(y(t), t), t)), F__z(t) = m*(diff(diff(z(t), t), t))}$

(11)

Things are straightforward in Cartesian coordinates because the components of the line element are exactly the components of the tensor

>

(12)

and so, the form factors (see related Mapleprimes post) are all equal to 1.

In the case of no external forces, and the equations of motion, whose solution are the trajectory, can be formulated as the path of minimal length between two points, a geodesic. In the case under consideration, because the spacetime is flat (Galilean) those two points lie on a plane, we are talking about Euclidean geometry, that information is encoded in the metric (the 3x3 identity matrix (5)), and the geodesic is a straight line. The differential equations of this geodesic are thus the equations of motion (11) with , and can be computed using Geodesics

>

(13)

Curvilinear coordinates

Vector notation

The form of these equations in the case of curvilinear coordinates, for example in cylindrical or spherical variables, is obtained performing a change of coordinates.

>

(14)

This change keeps the z axis unchanged, so the corresponding unit vector remains unchanged.

Changing the basis and coordinates used to represent the position vector , it becomes

>

(15)

where since in (1) the coordinates () depend on t, in (15), not just and but also the unit vector depends on t. The velocity is computed as usual, differentiating

>

(16)

The second term is due to the dependency of on the coordinate together with the chain rule diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`(t), t) = (diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) and (diff(`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`, phi))*(diff(phi(t), t)) = `#mover(mi("φ",fontstyle = "normal"),mo("&and;"))`(t)*(diff(phi(t), t)) . The dependency of curvilinear unit vectors on the coordinates is automatically taken into account when differentiating due to the Setup setting geometricdifferentiation = true.

For the acceleration,

>

(17)

where the term involving comes from differentiating in (16) taking into account the dependency of on the coordinate This result can also be obtained by directly changing variables in the acceleration , in equation (3)

>

(18)

Newton's 2^nd law becomes

>

(19)

In the absence of external forces, equating to 0 the vector components (coefficients of the unit vectors) of the acceleration we get the system of differential equations in cylindrical coordinates whose solution is the trajectory of the particle expressed in the (

>

$`~`[`=`]({coeffs(rhs(diff(diff(r_(t), t), t) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k), [`#mover(mi("ρ",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("φ",fontstyle = "normal"),mo("&and;"))`(t), `#mover(mi("k"),mo("&and;"))`])}, 0)$

${2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}$

(20)

>	$solve({2(diff(rho(t), t))(diff(phi(t), t))+rho(t)(diff(diff(phi(t), t), t)) = 0, diff(diff(rho(t), t), t)-rho(t)(diff(phi(t), t))^2 = 0, diff(diff(z(t), t), t) = 0}, {diff(phi(t), t, t), diff(rho(t), t, t), diff(z(t), t, t)})$

${diff(diff(phi(t), t), t) = -2*(diff(rho(t), t))*(diff(phi(t), t))/rho(t), diff(diff(rho(t), t), t) = rho(t)*(diff(phi(t), t))^2, diff(diff(z(t), t), t) = 0}$

(21)

In this formulation (21) with , although , no acceleration in the direction, is naturally expected, the same cannot be said about the other two equations for and . Those two equations are discussed below under Coriolis and Centripetal forces. The key observation at this point, however, is that the right-hand sides of both unexpected equations involve , rotation around the z axis.

Tensor notation

The same equations (19) and (21) result when using tensor notation. For that purpose, one can transform the position, velocity and acceleration tensors (6), (7), (8), but since they are expressed as functions of a parameter (the time), it is simpler to transform only the underlying metric using TransformCoordinates. That has the advantage that all the geometrical subtleties of curvilinear coordinates, like scale factors and dependency of unit vectors on curvilinear coordinates, get automatically, very succinctly, encoded in the metric:

>

Physics:-g_[a, b] = Matrix(%id = 18446744078263848958)

(22)

The computation of geodesics assumes that the coordinates () depend on a parameter. That parameter is passed as the first argument to Geodesics

>

(23)

These equations of motion (23) are the same as the equations (21) computed using standard vector notation, differentiating and taking into account the dependency of curvilinear unit vectors on the curvilinear coordinates in (16) and (17). One of the interesting features of computing with tensors is, as said, that all those geometrical algebraic subtleties of curvilinear coordinates are automatically encoded in the metric (22).

To understand how are the geodesic equations computed in one go in (23), one can perform the calculation in steps:

1.	Make be a function of directly in the metric

2.	Compute - not the final form of the equations (23) - but the intermediate form expressing the geodesic equation using tensor notation, in terms of Christoffel symbols

3.	Compute the components of that tensorial equation for the geodesic (using TensorArray)

For step 1, we have

>

Physics:-g_[a, b] = Matrix(%id = 18446744078354237910)

(24)

Set this metric where

>

Physics:-g_[a, b] = Matrix(%id = 18446744078342604430)

(25)

Step 2, the geodesic equations in tensor notation with the coordinates depending on the time t are computed passing the optional argument tensornotation

>

(26)

Step 3: compute the components of this tensorial equation

>

(27)

These are the same equations (23).

Having the tensorial equation (26) is also useful to formulate the equations of motion in tensorial form in the presence of force. For that purpose, redefine the contravariant tensor to represent the force in the cylindrical basis

>

${Physics:-D_[a], Physics:-Dgamma[a], F[j], Physics:-Psigma[a], Physics:-Ricci[a, b], Physics:-Riemann[a, b, c, d], Physics:-Weyl[a, b, c, d], Physics:-d_[a], Physics:-g_[a, b], Physics:-Christoffel[a, b, c], Physics:-Einstein[a, b], Physics:-LeviCivita[a, b, c], Physics:-SpaceTimeVector[a](X)}$

(28)

Newton's 2^nd law (19)

>

(29)

now using tensorial notation, becomes

>

(30)

>

Array(%id = 18446744078329063774)

(31)

where we recall (see related Mapleprimes post) that to obtain the vector components entering from these tensor components we need to multiply the latter by the scale factors (), the component of in the direction of is given by .

Coriolis force and centripetal force

After changing variables the position vector of the particle got expressed in (15) as

A distinction needs to be made here, according to whether the unit vector depends or not on the time , the former being the general case. When is a constant, the value of the coordinate - the angle between and the x axis - does not change, there is no rotation around the z axis. On the other hand, when , the orientation of and so the coordinate changes with time, so either the force acting on the particle has a component in the direction that produces rotation around the z axis, or the system of references - itself - is rotating around the z axis.

Likewise, the expression (15) can represent the position vector measured in the original Galilean (inertial) system of references, where a force is producing rotation around the z axis, or it can represent the position of the particle measured in a rotating, non-inertial system references. Hence the transformation (14) can also be interpreted in two different ways, as representing a choice of different functions (generalized coordinates) to represent the position of the particle in the original inertial system of references, or it can represent a transformation from an inertial to another rotating, non-inertial, system of references.

This equivalence between the trajectory of a particle subject to an external force, as observed in an inertial system of references, and the same trajectory observed from a non-inertial accelerated system of references where there is no external force, actually at the root of the formulation of general relativity, is also well known in classical mechanics. The (apparent) forces observed in the rotating non-inertial system of references, due only to its acceleration, are called Coriolis and centripetal forces.

To see that the equations

diff(rho(t), t, t) = (diff(phi(t), t))^2*rho(t), diff(phi(t), t, t) = -2*(diff(phi(t), t))*(diff(rho(t), t))/rho(t)

that appeared in (27) when in the inertial system of references , are related to the Coriolis and centripetal forces in the non-inertial referencial, following [1] introduce a vector representing the rotation of that referencial around the z axis (when, in the inertial system of references, the non-inertial system rotates clockwise, in the non-inertial system increases in value in the anti-clockwise direction)

>

(32)

According to [1], (39.7), the acceleration in the inertial system is expressed in terms of the quantities in the non-inertial rotating system by the sum of the following three vectorial terms.

First, the components of the acceleration measured in the non-inertial system are given by the second derivatives of the coordinates () multiplied by the scale factors, which in this case are given by () (see this post in Mapleprimes)

>

(33)

Second, the Coriolis force divided by the mass, by definition given by

>

(34)

Third the centripetal force divided by the mass, defined by

>

(35)

Adding these three terms,

>

a_(t)+2*Physics:-Vectors:-`&x`(diff(r_(t), t), omega_)+Physics:-Vectors:-`&x`(omega_, Physics:-Vectors:-`&x`(r_(t), omega_)) = _rho(t)*(diff(diff(rho(t), t), t)-rho(t)*(diff(phi(t), t))^2)+_phi(t)*(2*(diff(rho(t), t))*(diff(phi(t), t))+rho(t)*(diff(diff(phi(t), t), t)))+(diff(diff(z(t), t), t))*_k

(36)

So that

>

(37)

and where the right-hand side of (36) is, actually, the result computed lines above in (18)

>

(38)

>

(39)

From (37), in the absence of external forces and so the acceleration measured in the rotating system is given by the sum of the Coriolis and centripetal accelerations

>

(40)

In other words: in the absence of external forces, the acceleration of a particle observed in a rotating (non-inertial) system of references is not zero.

Expressing this equation (40) in terms of () we get

>

(41)

resulting in the three equations

>

(42)

>

(43)

>

(44)

which are the equations returned by Geodesics in (23)

>

(45)

>

References

[1] L.D. Landau, E.M. Lifchitz, Mechanics, Course of Theoretical Physics, Volume 1, third edition, Elsevier.

Download The_equations_of_motion_in_curvilinear_coordinates.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

The fractal structure’s researching. Modeling...

by: Alsu 445 Product: Maple

May 16 2020

1 0

Research work

The fractal structure’s researching.

Modeling of the fractal sets in the Maple program.

Municipal Budget Educational Establishment “School # 57” of Kirov district of Kazan

Author: Ibragimova Evelina

Scientific advisor: Alsu Gibadullina - mathematics teacher

Translator: Aigul Gibadullina

In Russian

ИбрагимоваЭ_Фракталы.docx

In English

Fractals_researching.doc

( Images - in attached files )

Table of contents:

Introduction

I. Studying of principles of fractals construction

II. Applied meaning of fractals

III. Researching of computer programs of fractals construction

Conclusion

Introduction

We don’t usually think about main point of things, which we have to do with every day. Environmental systems are many-sided, ever–changing and complicated, but they are formed by a little number of rules. Fractals are apt example of this – they are complicated, but based on simple regulations. Self – similarity is the main attribute of them. Just one fractal element contains genetically information about all system. This information have a forming role for all system. But sometimes self – similarity is partial.

Hypothesis of the research. Fractals and various elements of the Universe have general principles of structural organization. It is a reason why the theory of fractals is instrument for cognition of the world.

Purpose of the research. Studying of genetic analogy between fractals and alive and non-living Universe systems with computer-based mathematical modeling in the Mapel’s computer space.

Problems of the research.

studying of principles of fractal’s construction;
Detection of general fractal content of physical, biological and artificial systems;
Researching of applied meaning of fractals;
Searching of computer programs which can generate all of known fractals;
Researching of fractals witch was assigned by complex variables;
Formation of innovative ideas of using of fractals in different spaces;

The object of research. Fractal structures, nature and society objects.

The subject of research. Manifestation of fractality in different objects of the Universe.

Methods of the researching.

Studying and analysis of literature of research’s problem;
Searching of computer programs which can generate fractals and experimentation with them;
Comparative analysis of principles of generating of fractals and structural organizations of physical, biological and artificial systems;
Generation and formulation of innovation ways to applied significance of fractals.

Applied significance.

Researching of universality of fractals gives general academic way of cognition of nature and society.

I. Studying of principles of fractal construction

We can see fractal constructions everywhere – from crystals and different accumulations (clouds, rivers, mountains, stars etc.) to complex ecosystems and biological objects like fern leafs or human brain. Actually, the idea that fractal principles are genetic code of our Universe has been discussed for about fifteen years. The first attempt of modeling of the process of the Universe construction was done by A.D. Linde. We also know that young Andrey Saharov had solved “fractal” calculation problem – it was already half a century ago.

Now therefore, fractal picture of the world was intuitively anticipated by human genius and it inevitably manifested in its activity.

Fractals are divided into four groups in the traditional way: geometric (constructive), algebraical (dynamical), stochastical and natural.

The first group of fractals is geometric. It is the most demonstrative type of fractals, because we can instantly observe the self-similarity in it. This type of fractals is constructed in the basis of original figure by her fragmentation and realizing of different transformations. Geometrical fractals ensue on repeating of this procedure. They are using in computer-generated graphics for generating the pictures of leafs, bush, dimensional structures, etc.

The second large group of fractals - algebraical. This fractals are constructed by iteration of nonlinear displays, which set by simple formulas. There are two types of algebraical fractals – linear and nonlinear. The first of them are determined by first order equates (linear equates), and the second by nonlinear equates, their nature significantly brighter, richer and more diverse than first order equates.

The third known group of fractals – stochastical. It is generated by method of random modification of options in iterative process. Therefore, we get an objects which is similar to nature fractals – asymmetrical trees, rugged coasts, mountain scenery etc. Such fractals are useful in modeling of land topography, sea–surface and electrolysis process etc.

The fourth group of fractals is nature, they are dominate in our life. The main difference of such fractals is that they can’t demonstrate infinite self-similarity. There is “physical fractals” term in the classification concept for nature fractals, this term notes their naturalness. These fractals are created with two simple operations: copy and scaling. We can indefinitely list examples of nature fractals: human’s circulatory system, crowns and leafs of trees, lungs, etc. It is impossible to show all diversity of nature fractals.

II. Applied meaning of fractals

Fractals are having incredibly widespread application nowadays.

In the medicine. Human’s organism is consists from fractal structures: circulatory system, bronchus, muscle, neuron system, etc. So it’s naturally that fractal algorithms are useful in the medicine. For example, assessment of rhythm of fractal dimension while electric diagrams analyzing allows to make more informative and accurate view on the beginning of specific illnesses. Also fractals are using for high–quality processing of X–ray images (in the experimental way). There are designing of new methods in the gastroenterology which allows to explore gastrointestinal tract organs qualitative and painlessly. Actually, there are discoveries of application of fractal methods for diagnosis and treatment of cancer.

In the science. There are no scientific and technical areas without fractal calculations nowadays. It happens due to the fact that majority of nature objects have fractal structures and dimension: coasts of the continents; natural resources allocation; magnetic field anomaly; dissemination of surges and vibrations in an elastic environments; porous, solid and fungal bodies; crystals; turbulence; dynamic of complicated systems in general, etc. Fractals are useful in geology, geophysics, in the oil sciences… It’s impossible to list all the spaces of adaptability.

Modeling of chaotic processes, particularly, in description of population models.

In telecommunications. It’s naturally that fractals are popular in this area too. Natan Coen is person, who had started to use fractal antennas. Fractal antenna has very compact form which provides high productivity. Due to this, such antennas are used in marine and air transport, in personal devises. The theory of fractal antennas has become an independent, well-developed apparatus of synthesis and analysis of electric small antenna (ESA) nowadays. There are developments of possibility of fractal compression of the traffic which is transmitted through the web. The goal of this is more effective transfer of information.

In the visual effects. The theory of fractals has penetrated area of formation of different kinds of visualizations and creation of special effects in the computer graphics soon. This theory are very useful in modeling of nature landscapes in computer games. The film industry also has not been without fractal geometry. All the special effects are based in fractal structure: mountain landscape, lava, flame, fog, large flows and the same. In the modern level of the cinema creation of the special effects is impossible without modeling of fractals.

In the economics. The Veirshtrass’s function is famous example of stochastic fractals. Analysis of graph of the function in interactive mathematic environment Maple allows to make sure in fractal structure of function by way of entry of different ranges of graphic visualization. In any indefinitely small area of the part graph of the function absolutely looks like area of this part in the all . The property of function is used in analysis of stock markets.

In the architect. Notably, fractal structures have become useful in the architect more earlier than B. Mandelbrode had discovered them. S.B. Pomorov, Doctor of Architecture, Professor, member of Russian Architect Union, talks about application of fractal theory in the architect in his article. Let’s see on the part of this article:

“Fractal structures were found in configuration of African tribal villages, in ancient Vavilon’s ziggurats, in iconic buildings of ancient India and China, in gothic temples of ancient Russia .

We can see the high fractal level in Malevich’s Architectons. But they were created long before emergence of the notion of fractals in the architect. People started to use fractal algorithms on the architect morphogenesis consciously after Mandelbrot’s publications. It was made possible to use fractal geometry for analyzing of architectural forms.

Fractals had become available to the majority of specialists due to the computerization. They had been incredibly attractive for architectors, designers and town planners in aesthetic, philosophical and psychological way. Fractal theory was perceived on emotional, sensual level in the first phase. The constant repression leading to loss of sensuality.

Application of fractal structures is effective on the microenvironment designing level: interior, household items and their elements. Fractal structures introduction allows creating a new surroundings for people with fractal properties on all levels. It corresponds to nesting spaces.

Fractal formations are not a panacea or a new era in the architect history. But it’s a new way to design architect forms which enriches the architectural theory and practice language. The understanding of nature fractal impacts on architectural view of urban environment. An attempt to develop the method of architectural designing which will base in an in-depth fractal forms is especially interesting. Will this method base only on mathematics? Will it be different methods and features symbiosis? The practice experiments and researches will show us. It’s safe to say that modern fractal approach can be useful not only for analysis, but also for harmonic order and nature’s chaos, architect which may be semantic dominant in nature and historic context.”

Computer systems. Fractal data compression is the most useful fractal application in the computer science. This kind of compression is based on the fact that it’s easy to describe the real world by fractal geometry. Nevertheless, pictures are compressed better than by other methods (like jpeg or gif). Another one advantage is that picture isn’t pixelateing while compressing. Often picture looks better after increase in fractal compressing.

Basic concept for fractal computer graphics is “Fractal triangle”. Also there are “Fractal figure”, “Fractal object”, “Fractal line”, “Fractal composition”, “Parent object” and “Heir object”. However, it should be noted that fractal computer graphics has recently received as a kind of computer graphics of 21th century.

The opportunities of fractal computer graphics cannot be overemphasized. It allows creating abstract compositions where we can realize a lot of moves: horizontal and vertical, diagonal directions, symmetry and asymmetry etc. Only a few programmers from all over the world know about fractal graphics today. To what can we compare fractal picture? For example, with complex structure of crystal or with snowflake, the elements of which line up in the one complex composition. This property of fractal object can be useful in ornament creating or designing of decorative composition. Algorithms of synthesis of fractal rates which allows to reproduce copy of any picture too close to the original are developed today.

III. Researching of computer programs of fractal construction

Strict algorithms of fractals are really good for programming. There are a lot of computer programs which introduce fractals nowadays. Computer mathematic systems are stand out from over programs, especially, Maple. Computer mathematics is mathematic modeling tool. So programming represents genetic structure of fractal in these systems and we can see precise submission of fractal structure in the picture while we enter a number of iterations . This is the reason why mathematic fractals should be studied with computer mathematics. The last discovery in fractal geometry has been made possible by powerful, modern computers. Fractal property researching is almost completely based on computer calculations. It allows making computer experiments which reproduce processes and phenomenon which we can’t experiment in the real world with.

Our school has been worked with computer mathematics Maple package more than 10 years. So we have unique opportunity to experiment with mathematic fractals, thanks to that we can understand how initial values impact on outcome (it is stochastic fractal). For example, we have understood the meaning of the fact that color is the fourth dimension: color changing leads to changing of physical characteristics. That is what astrophysics mean talking about “multicolored” of the Universe. While fractal constructing in interactive mathematic environment we received graphic models which was like A. D. Linde’s model of the Universe. Perhaps, it demonstrates that Universe has fractal structure.

Conclusion

Scientists and philosophers argue, can we talk about universality of fractals in recent years. There are two groups of two opposite positions. We agree with the fact that fractals are universal. Due to the fact that movement is inherent property of material also we always have fractals wherever we have movement.

We are convinced that fractal is genetic property of the Universe, but it is not mean that all the Universe elements to the one fractal organization. In deployment process fractal structure is undergoing a lot of fluctuations (deviations) and a lot of points of bifurcation (branching) lead grate number of fractal development varieties.

Therefore, we think that fractals are general academic method of real world researching. Fractals give the methodology of nature and community researching.

In transitional, chaotic period of society development social life become harder. Different social systems clash. Ancient values are exchanged for new values literally in all spaces. So it’s vitally important for science to develop behavior strategies which allow to avoid tragic mistakes. We think that fractals play important role in developing of such technologies. And – synergy is theory of evolving systems self- organization. But evolution happens on fractal principles, as we know now.

P.S. Images - in attached files

Examining Maxwell's Equations

by: Steve Roper 180 Product: Maple

May 13 2020

6 4

This is still a work in progress, but might be of use to anybody interested in Maxwell's Equations :-)

Examining_Maxwells_Equations.mw

E-Mail Address:
Password:
Remember Me:	Automatically sign in on future visits

E-Mail Address:
Password:
Remember Me:	Automatically sign in on future visits

Ask a Question

Create a Post