Algo Geom / NCTM 08 Reno Algorithmic Geometry Pierre Bierre AlgoGeom.org Pleasanton CA NCTM 2008 Reno NV copyright Spatial Thoughtware 2008

Algo Geom / NCTM 08 Reno

Algorithmic Geometry

Pierre Bierre

AlgoGeom.orgPleasanton CA

NCTM 2008 Reno NV

copyright Spatial Thoughtware 2008


What is Algorithmic Geometry??

MATHEMATICS COMPUTER SCIENCE

Advanced geometry problem-solving

paper & pencil sketching Java programming

vector features interactive graphics

SCIENCE ENGINEERING

new formal concepts automate your math know-how


Pierre Bierre

BS Theoretical Physics MS Computer Science Neuropsychology

motion picture artificial intelligence immunology

data analysis

robotics2003: algorithmic geometry crystallized

2004 book: Flexing the Power of Algorithmic Geometry

2005: pilot course (proof-of-concept)


Today’s outline

Core 2D representations

Getting started with Java 2D graphics

Group exercise: problem from week 6

Lab problem exercises

Core 3D representations

3D graphics workspace

How can I get involved?

2D

3DResults from 2005 pilot


Getting started with Java…


Core 2D mathematical representations


How to represent 2D direction?

unitcircle

[ 1 1 ]

[ -1 -1 ]

y

x

In the computer,

is just a number

/4

.785397.06858-5.49778…

0 <= < 2

0.0000016.283184

discontinuity at 2 !

not a 1:1 representation!

A single real number doesn’t have the information capacity to do the job



unitcircle

[ 1 1 ]

[ -1 -1 ]

y

x

d1[ .707 .707 ]

[ .894 -.447 ]d2



y

x

unitcircle

d1

d2

[ 1 1 ]

[ -1 -1 ]

[ .707 .707 ]

[ .894 -.447 ]

direction vector

unit vector used to represent spatial direction

Advantages (computational and mental):

1:1 representation continuity extends naturally to 3D

[ .708, .706 ]


How do you compute direction vectors?y

x2

[ 1 1 ]

[ 2 -1 ]

unitcircle

d1

d2

d2 = [ 2 -1 ]norm = = [ .894 -.447 ]

d1 = [ 1 1 ]norm = len ([ 1 1 ]) sqrt ( 12 + 12 )

= = [ .707 .707 ][ 1 1 ]

1.414=

[ 2 -1 ]

len ([ 2 -1 ]) sqrt ( 22 + -12 )=

2.236=

[ 2 -1 ]

[ 1 1 ] [ 1 1 ]

[ 2 -1 ]

from points!


You can compute direction from p1 to p2…

d

y

x

p1

p2

d = normalize (p2 – p1)


Line “hit-testing” problem….


How to represent infinite 2D lines?

y

x

L

p • o ==

l

What is invariant (numerically) about all points on line L?

l

l = signed distance from origin

o

o = orientation (direction perpendicular to line)

y = mx + b

(m = dy/dx)

fails for vertical lines!

[ o l ] representation of 2D line

algos must handle all cases


How to do coordinate rotation?

(without angles)


Coordinate rotation…

x

y

newYaxisp1

newXaxis

What are new coordinates of p1 if axes are rotated as shown?

Instead of angle , just say the newXaxis direction you want to use


Learn how to visualize dot product of two vectors

y

xd

p2

p3

p1

d •p gives new coordinate of p along new axis d

dotProd (d, p) = d.x * p.x + d.y * p.y


p • newYaxis --> new y coord…

x

y

p1

p2

newYaxis

p3

newXaxis

Write a general purpose Rotate (point) function:

p • newXaxis --> new x coord…

p’ = [ p • newXaxis p • newYaxis ]


New point values after rotation…

p1_prime = rotate ( newXaxis, p1 );

y/

x/

p1/

p2/

p3/

p1 = unrotate ( newXaxis, p1_prime );


Editable tilted rectangle problem…


2D object intersection problems:

i

L2L1

i1

i2

C1

C2

i1

i2

CIR

L


Problem-solving methodology

Problem statement

Sketch out a mental solution

Write pseudocode

0: dist (c1, c2) > r1 + r2

Translate algo into Java

if (Vec2.distance(C1.c, C2.c)

Test algo graphically


Solve: Intersection of 2 circles

i1

i2

C1

C2

Given: C1 = [ c1 r1 ]C2 = [ c2 r2 ](any possible)

Solve for: numIntersectionPts (0, 1, 2, 3 =∞i1 i2


Subproblem: How many intersection points?

# numIntPts Draw examples Decision logic

∞identical (C1, C2)

dist(c1 , c2) == r1 + r2

dist(c1 , c2) == abs(r1 - r2)

dist(c1 , c2) > r1 + r2

dist(c1 , c2) < abs(r1 - r2)

none of the above


x’

y’

C1’

C2’

Subproblem: Locations of intersection points?

i2

C1i1

C2x

y

newOrigin

translatecoords i2’

i1’

newXaxis

rotatecoords

C1’’ x’’

y’’

C2’’

C2’ = translate (newOrigin, C2)C2’ = translate (C1.c, C2)

C2’’ = rotate (newXaxis, C2’) C2’’ = rotate (C2’.cnorm, C2’)

i1’’

i2’’


a2 + h2 = r12

b2 + h2 = r22

a2 - b2 = r12 - r2

2

(a + b) (a - b) = r12 - r2

2

c (a - b) = r12 - r2

2

a - b = (r12 - r2

2) / c

a + b = c

2a = c + (r12 - r2

2) / c

c = C2’’.c.x

i1’’ = [ a h ]

a = (c2 + r12 - r2

2) / 2c

i2’’ = [ a -h ]

h = sqrt(r12 - a2)

2a = c2/c + (r12 - r2

2) / c

a = (c2 + r12 - r2

2) / 2c

x’’

y’’

C2’’i1’’

i2’’

a b

h

c = a + b

c

r1 r2

C1’’


x’

y’

C1’

C2’

Subproblem: Locations of intersection points?

i2

C1i1

C2x

y

newOrigin

translatecoords i2’

i1’

newXaxis

rotatecoords

C1’’ x’’

y’’

C2’’

C2’ = translate (C1.c, C2)

C2’’ = rotate (C2’.cnorm, C2’)

i1’’

i2’’

i1’ = unrotate (C2’.cnorm, i1’’) i2’ = unrotate (C2’.cnorm, i2’’)

i1 = untranslate (C1.c, i1’)i2 = untranslate (C1.c, i2’)


Java coding…


Exploring Problem Challenges…..

a

60 socket rotating caddy

x

y

wt

c


Exploring Problem Challenges…..

“I’m crashed in the Nevada desert, and this will be my last call. I’ll give you the info you need to place my location on a curved search path…

Mt. Whitney Mt. Ritter

30 30

36 19/128


3D algoGeom


Develop formal representations for these 3D objects:

• planes

• lines

• coordinate rotations (rotators)

• spheres

• circles

Object intersections:

plane-plane plane-line 3-plane

sphere-sphere sphere-circle 3-sphere (GPS)

x

z

y

o

l

• 3D direction vectorz

x

y

[ 1 1 1] [-1 -2 2]

d1

d2

unitsphere

[-.33 -.66 .66][.58 .58 .58]

x

z

y

newZaxisnewXaxis

newYaxis

x

z

y

L

newZaxis

z/

L/

x/

y/

[ x/ y

/ ]

L

PL1 PL2LPL

i

SPH1

SPH2

CIR

SPHCIR

i1

i2


3D graphicsworkspace

QuickTime™ and a decompressor

are needed to see this picture.


Results from 2005 pilot course

student feedback:

getting my computer to do all the gruntwork - way cool!

Java programming picked up in 2 weeks

gave me advantage later in college math/engineering

elegant compared to traditional geometry / trig

course “proof of concept”:

syllabus fit into 72-hours classtime (no homework)

labs easy to grade

learning results impressive - student programs sophisticated seek funding for feasibility


Example of student work (from final exam)


Summary of algorithmic geometry

spatial problem-solving as human-computer partnership

representations modernized for ease-of-algorithm-writing

concepts scale up intuitively from 2D 3D

surge in problem-solving confidence

learn Java programming by osmosis

fast-track for students considering STEM


How can I get involved?

Participate in Algorithmic Geometry summer workshop

Pick a student TA w/ Java - we’ll train you as a team

Team-teach a pilot course on your home turf

Get paid as educational researchers

Help leapfrog U.S. math education

Contact Algo Geom Project:

[email protected]

925-200-1558

Pierre Bierre

(contingent on funding)


Documents

Algo Geom / NCTM 08 Reno Algorithmic Geometry Pierre Bierre AlgoGeom.org Pleasanton CA NCTM 2008 Reno NV copyright Spatial Thoughtware 2008