PH36010 PH36010 Numerical MethodsNumerical Methods

Advanced Plotting and AnimationAdvanced Plotting and Animation

Advanced Plotting in Advanced Plotting in MathCADMathCAD

• Polar PlotsPolar Plots• 3D Plots3D Plots

– SurfaceSurface– ContourContour– ScatterScatter– BarBar

• Vector Field PlotsVector Field Plots

Polar Plots #1Polar Plots #1

• X-Y Plot wrapped around a circleX-Y Plot wrapped around a circle

0

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0.8

0.6

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0

1

3.7491033F ( )

r=cosr=cos22(())

Polar Plots #2Polar Plots #2k 0.2

r ek 0 0.1 6

0

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0r

•r=er=ekk

Uses for Polar PlotsUses for Polar Plots

• Rotating MachinesRotating Machines– Stirling engineStirling engine– VibrationVibration

• Arctic & Antarctic systemsArctic & Antarctic systems

3D surface plots3D surface plotsSimple #1Simple #1

• Plot of data stored in arrayPlot of data stored in array• Indices of array represent x-y planeIndices of array represent x-y plane• Value of matrix is z valueValue of matrix is z value

– Define size of matrix & indicesDefine size of matrix & indices– Determine values for x & yDetermine values for x & y– Fill matrixFill matrix– PlotPlot

3D surface plots3D surface plotsSimple #2Simple #2

• Define size & Define size & indicesindices

• Define Range Define Range for x & yfor x & y

• 2D-Gaussian2D-Gaussian

• Fill matrixFill matrix

N 100 i 0 N j 0 N

xi 5 0.1 i yj 5 0.1 j

f x y( ) ex2

y2

Mi j f xi yj

3D surface plots3D surface plotsSimple #3Simple #3

M

3D surface plots3D surface plotsSimple #4 – LightingSimple #4 – Lighting

M

Contour PlotContour Plot- from 3d plot data- from 3d plot data

• Format plot type -> Contour plotFormat plot type -> Contour plot

M

Simple 3D plotsSimple 3D plotsLimitationsLimitations

• X & Y axes scaled in indicesX & Y axes scaled in indices

• Suited to continuous single valued Suited to continuous single valued functionsfunctions

• More advanced 3D plots availableMore advanced 3D plots available

3D Scatter plot3D Scatter plot

• Plot of individual pointsPlot of individual points• Supply with 3 vectors of equal Supply with 3 vectors of equal

length (X,Y,Z)length (X,Y,Z)• Points plotted:Points plotted:

– XX00,Y,Y00,Z,Z00

– XX11,Y,Y11,Z,Z11

– ……

3D scatter plot3D scatter plotPhoton Scatter in 3DPhoton Scatter in 3D

XX YY ZZ( )

3D surface plots3D surface plotsParametric PlotParametric Plot

• Plot complex surfacesPlot complex surfaces• From Resource Centre Guide to 3D From Resource Centre Guide to 3D

plottingplotting

X Y Z( )

3D parametric plot3D parametric plotExample - Example - 100100

22 for H for H

X Y Z100( )

X- & Y-X- & Y-Axes Axes scaled scaled in min m

Parametric PlotsParametric PlotsHow to… #1How to… #1

• Rubber sheet N x MRubber sheet N x M• Wrap sheet around surfaceWrap sheet around surface

• XXXXa,ba,b is x-coord of point a,b is x-coord of point a,b

• YYYYa,ba,b is y-coord of point a,b is y-coord of point a,b

• ZZZZa,ba,b is z-coord of point a,b is z-coord of point a,b

• (XX,YY,ZZ) will plot surface(XX,YY,ZZ) will plot surface

Parametric PlotsParametric PlotsExample 1 - GaussianExample 1 - Gaussian

• Ranges defined Ranges defined as beforeas before

• Fill XX & YY Fill XX & YY matricesmatrices

• Fill ZZ matrixFill ZZ matrix

N 100

xmin 5 xmax 5 xxmax xmin

N

ymin 5 ymax 5 yymax ymin

N

i 0 N j 0 N

XXi j xmin x i

YYi j ymin y j

f x y( ) ex2

y2

ZZi j f XXi j YYi j

Parametric PlotsParametric PlotsExample 1 - GaussianExample 1 - Gaussian

• Note:Note:– Axes Axes

scaledscaled

XX YY ZZ( )

The SeriesVector()The SeriesVector()programprogram

SeriesVector Start Finish nSteps( ) Finish Start

nSteps

Resulti Start i

i 0 nStepsfor

Result

(increment) and (increment) and ii (loop count) are (loop count) are ‘local’ to function.‘local’ to function.Function builds Result vectorFunction builds Result vectorReturned on last line of programReturned on last line of program

Parametric PlotsParametric PlotsPolar 3D #1Polar 3D #1

• Create r vectorCreate r vector0-10 radii, 100 0-10 radii, 100 tepsteps

• Create Create vector vector-180..180 deg, -180..180 deg, 30 steps30 steps

• Fill X & Y vectorsFill X & Y vectors

nr 100 i 0 nr

r SeriesVector 0 m 10 a 0 nr

n 30 j 0 n

SeriesVector n( )

Xi j ri cos j Yi j ri sin j

Paramteric PlotsParamteric PlotsPolar 3D #2Polar 3D #2

100 r( )1

a 03

e

r

a 0

Z100i j a 03 100 ri

2

Define Define 100 function100 function

Fill Z matrix withFill Z matrix with 100 function100 function

NB Only depends on NB Only depends on rr

3D parametric plot3D parametric plotExample - Example - 100100

22 for H for H

X Y Z100( )

X- & Y-X- & Y-Axes Axes scaled scaled in min m

Plot of Plot of 21021022 for H for H

• X & Y matrices as beforeX & Y matrices as before• Z now depends on r,qZ now depends on r,q

210 r ( )1

32 a 05

r e

r

2 a 0

cos ( )

Z210i j a 03 210 ri j

2

3D parametric plot3D parametric plotExample - Example - 210210

22 for H for H

X Y Z210( )

Pretty PicturesPretty Pictures From Resource Centre From Resource Centre

Plot multiple SurfacesPlot multiple Surfaces

Pretty Pictures #3Pretty Pictures #3 Planetary Orbit Planetary Orbit

• 3 objects3 objects– Planet : (xe,ye,ze)Planet : (xe,ye,ze)– Star : (xs,ys,zs)Star : (xs,ys,zs)– Orbit: (xp,yp,zp)Orbit: (xp,yp,zp)

xe ye ze( ) xs ys zs( ) xp yp zp( )

AnimationAnimation Principles Principles

• Make output depend on FRAMEMake output depend on FRAME• Tools|Animation|Record…Tools|Animation|Record…

to display dialog to display dialog– Fill in first & last framesFill in first & last frames– Drag cursor around area to animateDrag cursor around area to animate– Select ‘Animate’ to goSelect ‘Animate’ to go– View with AVI playerView with AVI player– Save to diskSave to disk

• Help|Quicksheets|Graphing and Help|Quicksheets|Graphing and VisualizationVisualization

AnimationAnimationHints and TipsHints and Tips

• Fix scale of graphsFix scale of graphs• Watch out for:Watch out for:

– Size of AVI filesSize of AVI files– Long calculation timesLong calculation times

• Can test by usingCan test by usingFRAME:=FRAME:=– Delete afterwardsDelete afterwards

• Put indicator in moviePut indicator in movie

Animation Example #1Animation Example #1 Cannon Ball with drag Cannon Ball with drag

Animation Example #2Animation Example #2LCR at resonanceLCR at resonance

Animation Example #3Animation Example #3Heat Flow in PlateHeat Flow in Plate

Animation Example #4Animation Example #4Planetary OrbitsPlanetary Orbits

3D with lighting & 3D with lighting & shadowsshadows