PH36010
Numerical Methods
Advanced Plotting and Animation
Advanced Plotting in
MathCAD
• Polar Plots
• 3D Plots
–
–
–
–
Surface
Contour
Scatter
Bar
• Vector Field Plots
Polar Plots #1
• X-Y Plot wrapped around a
circle
r=cos2()
90
1 20
1
60
0 .8
0 .6
1 50
30
0 .4
0 .2
F(  )
1 80
0
0
2 10
3 30
2 40
3 00
2 70

 0 33
3 .7 491
Polar Plots #2
k
•r=e
k  0.2
r     e
k 
  0  0.1  6 
90
1 20
40
60
30
1 50
30
20
10
r  
1 80
0
0
2 10
3 30
2 40
3 00
2 70

Uses for Polar Plots
• Rotating Machines
– Stirling engine
– Vibration
• Arctic & Antarctic systems
3D surface plots
Simple #1
• Plot of data stored in array
• Indices of array represent x-y
plane
• Value of matrix is z value
–
–
–
–
Define size of matrix & indices
Determine values for x & y
Fill matrix
Plot
3D surface plots
Simple #2
N
100
xi
0  N
5 0.1i
f( x  y)
Mi j
i
e
2
x
f xi  yj
yj
y
2
j
0  N
5 0.1j
• Define size &
indices
• Define Range
for x & y
• 2D-Gaussian
• Fill matrix
3D surface plots
Simple #3
M
3D surface plots
Simple #4 – Lighting
M
Contour Plot
- from 3d plot data
• Format plot type -> Contour plot
M
Simple 3D plots
Limitations
• X & Y axes scaled in indices
• Suited to continuous single
valued functions
• More advanced 3D plots
available
3D Scatter plot
• Plot of individual points
• Supply with 3 vectors of equal
length (X,Y,Z)
• Points plotted:
– X0,Y0,Z0
– X1,Y1,Z1
–…
3D scatter plot
Photon Scatter in 3D
( XX  YY  ZZ)
3D surface plots
Parametric Plot
• Plot complex surfaces
• From Resource Centre Guide to 3D
plotting
( X  Y  Z)
3D parametric plot
Example - 1002 for H
X- & YAxes
scaled
in m
( X  Y  Z100 )
Parametric Plots
How to… #1
•
•
•
•
•
•
Rubber sheet N x M
Wrap sheet around surface
XXa,b is x-coord of point a,b
YYa,b is y-coord of point a,b
ZZa,b is z-coord of point a,b
(XX,YY,ZZ) will plot surface
Parametric Plots
Example 1 - Gaussian
N
100
xmin
5
xmax
5
x
xmax xmin
ymin
5
ymax 5
y
ymax ymin
i
0  N
j
xmin x i
YYi j
ymin y j
f( x  y)
ZZi j
e
N
0  N
XXi j
2
x
N
• Ranges
defined as
before
y
2
f XXi j  YYi j
• Fill XX & YY
matrices
• Fill ZZ matrix
Parametric Plots
Example 1 - Gaussian
• Note:
– Axes
scaled
( XX  YY  ZZ)
The SeriesVector()
program
SeriesVector( Start  Finish  nSteps )  

Finish  Start
nSteps
for i  0  nSteps
Resulti  Start  i
Result
 (increment) and i (loop count) are
‘local’ to function.
Function builds Result vector
Returned on last line of program
Parametric Plots
Polar 3D #1
nr
r
n

Xi j
100
i
0  nr
SeriesVector 0 m 10 a 0  nr
30
j
0  n
SeriesVector
(     n)
ricos j
Yi j
risin j
• Create r vector
0-10 radii, 100
teps
• Create  vector
-180..180 deg,
30 steps
• Fill X & Y
vectors
Paramteric Plots
Polar 3D #2
r
100 ( r )
 a0
Define 100 function
a
e 0
1
3
2
Z100i j
3
 a 0 100 ri
Fill Z matrix with
100 function
NB Only depends on r
3D parametric plot
Example - 1002 for H
X- & YAxes
scaled
in m
( X  Y  Z100 )
Plot of 2102 for H
• X & Y matrices as before
• Z now depends on r,q
r
210 ( r   )
1
r e
2 a 0
cos (  )
32  a 0
5
2
Z210i j
3
 a 0 210 ri  j
3D parametric plot
Example - 2102 for H
( X  Y  Z210 )
Pretty Pictures
From Resource Centre
Plot multiple Surfaces
Pretty Pictures #3
Planetary Orbit
• 3 objects
– Planet : (xe,ye,ze)
– Star : (xs,ys,zs)
– Orbit: (xp,yp,zp)
( xe  ye  ze)  ( xs  ys  zs)  ( xp  yp  zp)
Animation
Principles
• Make output depend on FRAME
• Tools|Animation|Record…
to display dialog
–
–
–
–
–
Fill in first & last frames
Drag cursor around area to animate
Select ‘Animate’ to go
View with AVI player
Save to disk
• Help|Quicksheets|Graphing and
Visualization
Animation
Hints and Tips
• Fix scale of graphs
• Watch out for:
– Size of AVI files
– Long calculation times
• Can test by using
FRAME:=
– Delete afterwards
• Put indicator in movie
Animation Example #1
Cannon Ball with drag
Animation Example #2
LCR at resonance
Animation Example #3
Heat Flow in Plate
Animation Example #4
Planetary Orbits
3D with lighting &
shadows