Adventures in 3D Modeling Using VPython ICTCM 2009

Russell HermanGabriel Lugo

UNC WilmingtonWilmington, NC

http://www.enzim.hu/~szia/emanim/emanim.htm

3D – Fast and Furious!Visual Python

Open SourcePython BasedMultiple OS

Summer Ventures in Science and Mathematics

State-wide Program14 Talented HS StudentsMath Modeling CourseFour Week Culminating in

Presentation

http://www.phy.syr.edu/~salgado/vpython/dillard/

http://bio.physics.drexel.edu/VPython.html

Course Web Site

http://people.uncw.edu/hermanr/SVSM.htm

First Two weeks of course:1. Modeling2. Examples3. Software Excel, Maple, Vpython, and more

the process of representing a real-world phenomenon as a set of mathematical equations

FieldsNatural sciences, engineering, social sciencesMathematics, economics, and more

ClassificationsLinear vs nonlinearDeterministic vs stochasticContinuous vs discrete

What is Mathematical Modeling?

TypesObservationExperimentComputationTheory

Processing SimulationVisualizationData Analysis

End ProductsPredictionVerificationNew Models

Using Computers to do science … Shodor

Methods in Science

What is Needed?SoftwareNumerical Computation 3*5+16 =Symbolic Analysis (x+y)^2 = Visualization

Mathematics Algebra Trigonometry Graphing Functions Calculus Linear Algebra Differential Equations … and more

Examples of Models

Given a roll of paper find (without unrolling it) the total length of the paper on the roll. A single sheet may be provided for measuring.

Projectile Motion

http://ffden-2.phys.uaf.edu/211_fall2004.web.dir/Brendan_Karchere/Body%20Pages%20(2-5)/Page%203.html

Gravitation and Planetary Motion

http://www.bnsc.gov.uk/lzcontent.aspx?nid=4709 OscillationsMass-SpringPendulaNonlinear Systems

gravity.py

springmass.py

incline.py

http://www.andrewclem.com/Chaos.html

http://en.wikipedia.org/wiki/Asteroid_belt

Fractal CoastlinesFractal MusicIterated Function SystemsXaos ExplorationMandelbrot and Julia Sets

http://home.pacbell.net/bulens/SelfSimi.htm

http://www.evl.uic.edu/aej/488/diagrams/fractal.gif

http://www.sustainablescale.org/AreasofConcern/Population/PopulationandScale/QuickFacts.aspx

http://council.wisconsinforestry.org/deer/deerpop.php

transmission dynamics of communicable diseaseshttp://www.scholarpedia.org/article/Image:Animated_SIR_Epidemic.gif

sir.py

Rate of Change of a Quantity = Rate in – Rate outRate of Change = (Population Change)/TimeRate in = Birth rate = bPRate out = Mortality rate = dP

Rate of Change of a Quantity = Rate in – Rate out

P bP dP kPt

∆ = − ≡∆

1 00 1 0 0/

P P ktkP P P Pt N N

− = ⇒ = +

Let Time = [0, t] in N steps: ∆t = t/N Pn=n∆t , n=0,1,2,…,N

2

2 1 01 1kt ktP P PN N

= + = +

01N

NktP PN

= +

1 01 1N

j j Nkt ktP P P PN N−

= + ⇒ = +

As N gets large, PN is what?

0( ) ktP t P e=

Plot Data -MS ExcelFit given function – Regression CurveDetermine Parameters (like k)Modifying Model

LOGISTIC GROWTH

2 2P bP dP mP kP mPt

∆ = − − ≡ −∆

Spinning Tennis BallVibrations of Hollow Core BatsDriving Cars with High Center of GravityProjectiles with Air DragSky DivingAnd more

pinpongtable.py

pool.py

bungee.pybasketball.py

Separate out harmonic frequencies from signals

http://www.thayerbirding.com/dnn/v39Features/Spectrogram/tabid/756/Default.aspx

http://physics.ucsc.edu/lecturedemonstrations/waves/sound.html

Brownian Motion is the random movement of particles suspended in a liquid or gas or the mathematical model used to describe such random movements

plotrand.py

the behavior of connected clusters in a random graphExamples

Forest FiresGalaxy FormationOil FieldsElectrical Resistance

http://ccl.northwestern.edu/netlogo/models/Percolation

percolation.py

BabyloniansClassic FormulaBinary vs HexadecimalMonte Carlo

http://zoonek2.free.fr/UNIX/48_R/16.html

montecarlo.py

Happy π Day: 3-14-2009

A cellular automaton is an array of identically programmed automata, or “cells”, which interact with one another.

http://vlab.infotech.monash.edu.au/resources/tutorials/cellular-automata/

http://people.uncw.edu/hermanr/mat367/FCABook/Book/ComplexI.pdf

Modeling in VPython

“3D Programming for Ordinary Mortals”http://vpython.org/

- based on Python http://www.python.org/ - David Scherer, David Andersen, Jonathan

Brandmeyer, Ruth Chabay, Ari Heitner, Ian Peters, Bruce Sherwood.

- for physics instructionEasy to learn, 3D Interactive modelingRecently updated to Version 5

Install Python and then VPythonRun IDLEFirst line

from visual import *Next line

sphere()Save – CTRL-S

Save as test.pyRun – F5

Colorsphere(color=color.red)

Radius sphere(radius=0.5,color=color.red)

Nameball = sphere(radius=0.5,color=color.red)

Positionball = sphere(pos=(0,2,0),radius=0.5,color=color.red)

Change positionball.pos = (1,2,3)

... And navigate Zoom: Hold middle button and move mouse

Or, hold both left and right buttons Rotate: Hold right button

from visual import *

#Create a sphereball = sphere(pos=(0,2,0),color=color.yellow,radius=1)

#Create a boxfloor = box(length=10, height=2, width=4,color=color.blue)

Cartesian CoordinatesLocate points with (x,y,z)

VectorsDraw arrow from originVectors have magnitude and

direction(1,2,3)

General VectorDraw from one point to another

ExamplesPosition, velocity, force

Draw arrow from origin to center of ballarrow(pos=(0,0,0), axis=ball.pos)Begins at posEnds at axis

Arbitrary arrowarrow(pos=(1,2,3), axis=(0,2,-1))

Arrow attributesarrow(pos=(1,2,3), axis=(0,2,-1),

shaftwidth=1,headwidth=2,headlength=3)

Moving Ball – Constant Velocity

Goal: Make ball move and bounce off wallCreate new program

File – New WindowSave as motion.py

Create scene

Run Example

VelocityVelocity = displacement/elapsed timev = ∆x/∆t

Rewrite∆x=new position – old position∆x=v∆tnew position = old position + v∆t

In Vpythonball.pos = ball.pos + ball.velocity*dt

Need several time stepsUse while loop

Keep executing commands over and overAdditional Information

Specify dt and velocityNeed rate – while loop executed 100 times/sec

Run Example

Logical TestsIf the ball moves to far to right, reverse its direction.

If statementIf ball.x > wallR.x:Then what?

Reverse directionball.velocity = -ball.velocity

Run Example

Assignment 1Add wallL on left at (-6,00)Add test to have ball bounce off this wall too.

Assignment 2Change ball velocity to move ball at an angle

with no z-component

Assignment 3Add a ceiling and a floor that touch vertical wallsAdd back wallAdd invisible front wall (i.e., only use if statement)Run program with ball bouncing off all walls.

Run Example

Uniform Motion – Adding Acceleration

Goal: Drop a ball that bounces on the floor.Free fall

All objects near the surface of the Earth fall with an acceleration -9.8 m/s2, or -32 ft/s2 (or a = -g).

Accelerationa = ∆v/∆t

Rewrite for VPythonnew velocity = old velocity + a ∆tball.velocity = ball.velocity + ball.acceleration*dt

Free fall with constant horizontal velocity.

bounce.py

floor = box(length=300, height=0.5, width=4, color=color.blue)ball = sphere(pos=(-90,100,0),radius=2, color=color.red)ball.velocity = vector(30,0,0)ball.trail = curve(color=color.yellow)

ball2 = sphere(pos=(-90,100,0),radius=2, color=color.red)ball2.velocity = vector(30,0,0)ball2.trail = curve(color=color.green)

ball3 = sphere(pos=(-90,100,0),radius=2, color=color.red)v = vector(30,0,0)ball3.trail = curve(color=color.blue)a = vector(0,-9.8,0)

dt = 0.01while 1: rate(100) ball.pos = ball.pos + ball.velocity*dt if ball.y < 1: ball.velocity.y = -ball.velocity.y else: ball.velocity.y = ball.velocity.y - 9.8*dt ball.trail.append(pos=ball.pos)

ball.pos = ball.pos + ball.velocity*dt vel0=ball2.velocity if ball2.y < 1: ball2.velocity.y = -ball2.velocity.y else: ball2.velocity.y = ball2.velocity.y - 9.8*dt ball2.pos = ball2.pos + (vel0+ball2.velocity)*dt/2 ball2.trail.append(pos=ball2.pos)

if ball3.y < 1: v.y= -v.y else: v += a * dt ball3.pos += v* dt + .5 * a * dt**2 ball3.trail.append(pos=ball3.pos)bounces.py

Mass-Spring System

Hooke’s Law for Springs F = -kx

Newton’s 2nd Law of Motion F = ma Or, a = F/m

from visual import *

scene.range = (1,1,1)scene.center = (0,.1,0)scene.width = 800scene.height = 275scene.background = (1,1,1)

x0 = vector(.65,0,0)Surface = box(size=(2,.02,.5),pos=(0,-.1,0))wall = box(size=(.04,.5,.3),pos=(-.77,.15,0))spring = helix(pos=(-.75,0,0),axis=x0,

radius=.08,coils=6,thickness=.01,color=color.blue)block = box(pos=(0,0,0),size=(.2,.2,.2),color=color.red)

k = 10m = 10b=2v = vector(0,0,0)a = vector(0,0,0)F = vector(0,0,0)

block.pos=(0.25,0,0) # Set Initial position of block and connect spring x = block.posspring.axis = x0 + x

finished = False dt = .01 while not finished: rate(100) F = -k*x - b*v a = F/m v += a * dt x += v + .5 * a * dt**2

block.pos = x spring.axis = x0 + x

spring.py springmass.py

What happens when the mass is changed?What happens when k is changed?Change the initial block position

i.e., block.pos=(0.25,0,0)Do you need to change other settings?

NOW – EXPLORE AND MODIFY

Adding More ...

LotkaVolterra.py

Rubik's Cube

Rubik5.exeStand Alone Application

Stochastic Spatial Dynamics of Epidemic Models

EpidemicLooperWallsQuad.py.py

More Examples?

bounce.pygravity.py

springmass.pyspring.py

bounces.pymotion.py

motioninbox.pycouposc.py

balls.pypool.py

basket.pybasketball.py

More Python

Scientific Computing with Python ...NumPySciPyMatplotlibSymPyhttp://www.pythonxy.com

...

Thank You!Further information can be found at

http://people.uncw.edu/hermanr/SVSM.htm or hermanr@uncw.edu

ABSTRACT

We demonstrate the use of Vpython for developing and analyzing mathematical models of real world continuous and discrete processes with minimal training and graphics programming. VPython lets students focus on computations without having to write explicit graphics statements while creating their own 3D visualizations. We will provide examples, taken mostly from a summer course for high school students, of how easy it is to simulate projectile motion, nonlinear dynamics, the spread of epidemics, percolation and cellular automata. Such simulations can be used well into applied courses in the undergraduate mathematics and science curriculum.