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GenerativeComponents PrinciplesValentines Day 2012

Volker Mueller

Research Director

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GenerativeComponents Principles A quick introduction to the most important things you need to know about GC

Status:GC 8.11.9.93

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Originally inspired by Ben Dohertys

generative componentstheoretical frameworksthe stuff youneed to know

www.notionparallax.co.uk

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A Starting Point andGrounding in SpaceGive me a lever long enough and a fulcrum [fixed point in space] on which to place it, and I shall move the world. Archimedes

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GenerativeComponents is a change propagation system.Changes are propagated through a dependency graph (the Graph).

Graph is directed and acyclic = directed acyclic graph = DAG.

Graph needs a (logical) starting point =baseCS .

A Starting Point

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Theoretical geometrygives us an infiniteuniverse as a blankcanvas. There is noinherent concept of up, or of where weare in this space.

We add a grid toprovide someorientation.

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Theoretical geometrygives us an infiniteuniverse as a blankcanvas. There is noinherent concept of up, or of where weare in this space.

We add a grid toprovide someorientation.

We add a coordinate

system baseCS to provideA Starting Point for the geometry, too.

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The most commonway of describingspace is theCartesian grid with{X, Y, Z} coordinatetriples.

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The positive direction

of theZ axis isconsidered up ininfrastructure design.

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A CoordinateSystemin GC determines

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(a) A location (X, Y, Z),often at X, Y, and Z =0 for the baseCS.

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(b) Three axes, normalto each other.

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(b) Three axes, normalto each other.

(c) Three planes,spanned by pairs of the X, Y, Z directionvectors for 3 planes(XY, XZ, YZ),therefore,perpendicular to eachother (b).

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Planes and their normals have thesame color-coding.

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Planes and their normals have thesame color-coding.

The yellow lineindicates thecurrently activeplane.

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This is a GCCoordinateSystem.

It is the Starting Pointfor the geometricmodel.

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and for the Graph.

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and for the Graph.

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Populating the Model View &Populating the GraphGraph & Model View show the same Model

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direct modelingpoint,plane,coordinate systemplacement.

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old fashionedplacement throughNew Node task.

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old fashionedplacement throughNew Node task.

Provide inputsthrough graphicselection of Nodes or Geometry.

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Drag and drop fromNew Node pane toGraph.

(future Node palette.)

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Drag and drop fromNew Node pane toGraph.

(future Node palette.)

No dropping intomodel

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drag

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drop

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wire

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wire

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generate result

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generate result

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3 ModesModel, Graph, GC-Script

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1: Model

2: Graph

12

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1: Model

2: Graph

3: Editable inputproperties /expressions.

12 3

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1: Model

2: Graph


12 3

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12 333

1: Model

2: Graph


Editable Transaction

Script.

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3 MethodsLeft to right, right to left, hybrid

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Left to right

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Left to right

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Left to right

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Left to right

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Left to right

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Left to right

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Right to left

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Right to left

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Right to left

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Right to left

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Right to left

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Right to left

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Right to left

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Right to left

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Any which way

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I n c o r p o r a

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Any which way

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I n c o r p o r a

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Any which way

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I n c o r p o r a

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Any which way

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I n c o r p o r a

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Any which way

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Any which way

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Any which way

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Any which way

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Any which way

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Any which way

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Any which way

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Relationships/DependenciesDirected Acyclic Graph (DAG)Propagation System

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Dependency Graph.

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Dependency Graph.

We are interested inrelationships, not justwhere and how bigthings are.

d h

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Dependency Graph.


Move point01 andcone01 as well asline01 move, too.

D d G h

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Dependency Graph.


Move point01 andcone01 as well asline01 move, too.

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Input?Graphic or typing?

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Lik d h t

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Like a spreadsheet.

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Lik d h t

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5

5+2

Sin(5)

Value

Simpleexpression

Function

Like a spreadsheet.

Expression go intothis box.

Single values.

Simple expressions.

Functions drag & drop

Like a spreadsheet

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5

5+2

Sin(5)

Value

Simpleexpression

Function

Like a spreadsheet.


Single values.

Simple expressions.


Like a spreadsheet

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Like a spreadsheet.


Single values.

Simple expressions.


Complex expressionsfollow BODMAS

5

5+2

Sin(5)

Value

Simpleexpression

Function

Complexexpression(1/Sin(5)) + 90

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B ( ) Brackets firstO Orders (Powers, Roots)

DM / * Division and Multiplication (left to right)

AS + - Addition and Subtraction (left to right)

Power: xy Pow(x,y) expl. x Pow(x,0.5)

Root: x Sqrt(x)

x Pow(x,(1/3))

2

3

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15 / ( 3 + 2) = ?

15 / 3 + 2 = ?

( 15 / 3) + 2 = ?

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15 / ( 3 + 2) = 3

15 / 3 + 2 = 7

( 15 / 3) + 2 = 7

IN COMPUTING, THEREARE NEVER TOO MANYROUND BRACKETS

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5

5+2

Sin(5)

(1/Sin(5)) + 90

dave

dave*2

Value

Simpleexpression

Function

Complexexpression

Named variable

Expression withnamed variable

dave = 8

Once a variable isdefined (named) itcan take on a valueand be used in placeof a value.

Dave < > dave

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ObjectsEverything is an object

Almost everything

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Almost everythingshown so far hasbeen an object.

Including variables.

In computing, objectsare not equal.

They have Type.

dave = 8

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TypesWhat objects are

Data come in

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Data come indifferent kinds, likefreight on a train.

Each kind of freightrequires a matchingkind of freight car.

Computers aresimilarly picky, theyonly deal with whatthey have beenprepared to handle.

In some computer languages, type declares the whatof a data object.

GC uses two

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GC uses twocategories of type:

simple variabletypes and

classes = GCfeature types

l bl ( l )

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bool (Boolean) true, falseint (integer) whole numbers (0, 5, -4, 1000, -503)double real or decimal numbers (0.5, -7.8, 15.0, 158.543679789)string some text (hello world, 450, dave)

Simple variable types (selection)

Cl

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Object-oriented computing implements the idea that data and their methods arebundled in packages, as classes.

A Class is a data type that includes the methods that construct (create and initialize)instances, manipulate the data contained in them, and destruct them at end of their lifecycle.

Data contained in a class instance may be exposed as instance properties.Methods also determine how class instancesbehave, i.e. how they react to changes of their properties.

In GC classes are called features; methods are calledtechniques.

Classes

GC f ( l i )

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Point GCsfeature type pointPlaneCoordinateSystem GCs combination of point, directions, planes Direction GCs ray Line

Curve one-dimensional feature type with 0 to 1 parameterizationBSplineSurfaceSolidGenerated Feature Type user defined feature type composed from other features

GC feature types (selection)

GC d

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GC node types

GC d

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GC node types

GC d t

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now alsonon-featurenode types

GC node types

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PropertiesHow objects are

P ti

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Features have properties.

They can be simple variable types or sub-features.

The dot operator (viadotNet, Microsoft) accesses feature.properties.

Property Type

myCoordinateSystem.Name =point01 stringmyCoordinateSystem.XYPlane.X = 5.35 doublemyCoordinateSystem.XYPlane.IsReplicated = false bool(ean)

Properties

feature dot operator propertysub-feature

Properties GC features have

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Properties input properties andoutput properties.


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They are accessedthrough the Inputsand Outputs porteditors on a featuresnode.


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They are accessedthrough theInputs and Outputs porteditors on a featuresnode.


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They are accessedthrough theInputs and Outputs porteditors on a featuresnode.


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Properties Ports Default input

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Properties Ports properties for theactive technique are

visible as ports towhich connectionscan be made.


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visible asports towhich connectionscan be made.


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They are pinned bydefault.


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Additional input portsmay be pinned to thenode.


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Properties Ports properties for theactive technique arevisible as ports towhich connectionscan be made.



dProperties Ports Default inputf h

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Properties Ports properties for theactive technique arevisible as ports towhich connectionscan be made.



dProperties Ports Default output portish f i lf

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Properties Ports the feature itself.

dProperties Ports Default output port ish f i lf

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Properties Ports the feature itself. Additional outputproperties may beexposed as portspinned to the node,too.

dProperties Ports Default output port ish f i lf

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Properties Ports the feature itself. Additional outputproperties may beexposed as portspinned to the node,too.

d

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Casting and InterfacesFlexibility in Relationships

dCasting and Interfaces

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Object-oriented computing is very strict about membership in aclass or feature type.

However, you can sometimes stuff one data type into another slot(casting) but the type is generally a good hint as to what is required.

There are some specificcasting functions.In GC, casting works only for simple variable types, not for featuretypes.

Casting and Interfaces

d

Interfaces draw oni il ti d

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similar properties andmethods of featuretypes to let us useone feature type inplace of another one.

This softens theclass boundaries for more flexiblemodeling options.

Interfaces areprefixed with an I.

d

Plane is like a Pointb f X YZ

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because of X,Y,Zlocation.

CoordinateSystem islike a Point becauseof X,Y,Z location.

Plane is like a

Direction because of Planes normalvector.

Etc.

e d

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ListsCollecting stuff, retrieving it, and leveraging its power

te dLists TypeCurly Brackets{ }

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{ , , , , , , }

[0] [1] [2] [3] [4]

Curly Brackets{ } to define a list.

Things in a list areindexed from 0.

Indexing usesSquare Brackets [ ]

List lengthcounts

how many items arein the list, startingwith 1 (here it is 5).

te dLists Lists can have emptycontainers (null)

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[0] [1] [2] [3] [4]

containers (null ).

Lists can be of differenttypes.

t e dLists If wedeclare avariable called dave

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variable called daveas a list having thecontents{A,B,C,D,E,F,G}

Then we can refer toany item of that listindividually by itsindex.

Remember to countindices from 0.

dave = {A,B,C,D,E,F,G}[0] [1] [2] [3] [4] [5] [6]

dave[4] = ?

dave [4] = E

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t e dDimensions

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0

123

Geometry Reduction through Data

intersections

t e dMulti-Dimensional Lists

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Lists use nested Curly Brackets { } to indicate dimensionality of their nesting(property: List.Rank).

0d a1

1d {a1,a2,a3}

2d {{a1,a2,a3},{b1,b2},{c1,c2,c3,c4}}

3d {{{a1,a2,a3,a4,a5},{b1,b2}},{{{c1,c2,c3},{d1,d2,d3,d4,d5,d6}},{{{e1},{f 1,f 2},{g1,g2,g2}},{{{h1,h2,h3,h4},{i1,i2,i3},{j1,j2}}}

N d {{{{{a,a,a},{b,b}},{c,c,c}}},{d,d,d},{{{e}}}, etc }

a t e d

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ReplicationThe Power of Lists in GenerativeComponents

a t e dREPLICATION This circles radius isdefined using a

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defined using asingle value.

That is how youdexpect it to work fromexperience.

5

a t e dREPLICATION This circles radius isdefined using alist

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defined using alist .

Lists are really wherethe power of GCkicks in.

{3,4,4.5,5}

ra t e dREPLICATION This circles radius isdefined using alist

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{3,4,4.5,5}

r a t e dREPLICATION This circles radius isdefined using alist

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{3,4,4.5,5}

r a t e d

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y S y s t e m s ,

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(repl.)=

replicatable

r a t e d

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2

2

XTranslation: double (repl.)

YTranslation: double (repl.)

r a t e d

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{2,4,6,8}

2



[0] [1] [2] [3] [0] [1] [2] [3]

r a t e d

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{2,4,6,8}

{2,4,6,8}



[0] [1] [2] [3]

[0] [1] [2] [3]

[0] [1] [2] [3]

o r a

t e d

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{2,4,6,8,10,12}

{2,4,6,8}



[0] [1] [2] [3] [4] [5] [0] [1] [2] [3]

[0] [1] [2] [3]

ReplicationOption.CorrespondingIndexingToggle Replication

o r a

t e d

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{2,4,6,8,10,12}

{2,4,6,8}



[0] [1] [2] [3] [4] [5] [0][0] [1][0] [2][0] [3][0] [4][0] [5][0]

[0] [1] [2] [3]

ReplicationOption.AllCombinations

[0][1] [1][1] [2][1] [3][1] [4][1] [5][1]

[0][2] [1][2] [2][2] [3][2] [4][2] [5][2]

[0][3] [1][3] [2][3] [3][3] [4][3] [5][3]

Toggle Replication

o r a

t e d

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Observations: As in the geometry itself, there is dimensionality in the data, too.

1 singleton Point 0d

1 repl. parameter 1d2 repl. parameters AND CorrespondingIndexing 1d

2 repl. parameters AND AllCombinations 2d

3 repl. parameters AND AllCombinations 3dn repl. parameters AND AllCombinations nd

o r a

t e d

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Observation:There is exponential growth in created features based on 10:

1 singleton Point 0d 1

1 repl. parameter 1d 102 repl. parameters AND CorrespondingIndexing 1d 10

2 repl. parameters AND AllCombinations 2d 100

3 repl. parameters AND AllCombinations 3d 10004 repl. parameters AND AllCombinations 4d 10000

n repl. parameters AND AllCombinations nd 10n

o r a

t e d

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Observation:There is exponential growth in created features based on 30:

1 singleton Point 0d 1

1 repl. parameter 1d 302 repl. parameters AND CorrespondingIndexing 1d 30

2 repl. parameters AND AllCombinations 2d 900

3 repl. parameters AND AllCombinations 3d 270004 repl. parameters AND AllCombinations 4d 810000

n repl. parameters AND AllCombinations nd 30n

p o r a

t e d

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SpaceEuclidean Space, non-Euclidean Space,Coordinate Space (Cartesian, Cylindrical, Spherical),Parametric Space,

p o r a

t e dCartesian space

U l

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Unless you are aquantum physicist or a theoreticalmathematician,3dimensionalCartesian space is all

you will ever need(almost).Descartes added theorthogonal coordinate system tothe principledEuclidian space.

p o r a

t e dIn Euclidian space

( i i ll d fi d i

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(originally defined in2d space, i.e. on aplane, eg. a sheet of paper) there arestraight lines that canbe parallel.

In non-Euclidianspace (eg.thesurface of a sphere),Euclidian rules do notapply, for example,there are no straightlines (but arcs,circles, curves).

rp o r a

t e dOn the surface,

E lidi (t ) d

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Euclidian (top) andnon-Euclidian(bottom)2d spaces.

In 3d, these are

curved surfaces,which wouldpeacefully coexist or intersect with anystraight lines throughspace

r p o r a

t e d

Parameter SpaceThi i b dd d

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Line.StartPointT = 0.0

Line.EndPointT = 1.0

T = 0.35

This is anembedded space.From within the linethe universe onlyextends as far as theend of the line.

This space is definedas 1 unit (of self),regardless of itssize externally.The parameter is theT value

r p o r a

t e d

While on a Line theT parameter space is

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T = 0.25 T = 0.5

T = 0.75

T = 0.0

T = 1.0

T-parameter space ishomogenous, this isnot the case for BSplineCurves.T = 0.5 is notnecessarily the

geometric centre, it isthe parametric centre.Parametric distancesbetween controlpoints are equal, if they are equallyweighted

r p o r a

t e dThe analog is true for

surfaces with the 2d

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surfaces, with the 2d parameter space being a 1 by 1square.Instead of theCartesian 3d XYZ

coordinates they are2d UV coordinates.

r p o r a

t e d

A UV coordinate tuple{0 2 0 7} can be

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UV = {0.2,0.7}XYZ = {70,30,3.969}

{0.2, 0.7} can besampled into an XYZcoordinate triplet{x,y,z}, for example byplacing aPoint.ByParametersO

nSurface().

r p o r a

t e d

One way to thinkabout how UV

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UV = {0.2,0.7}XYZ = {70,30,3.969}

about how UVparametric spacedeals with distortion isto draw a grid on aballoon and then blowit up & squeeze it

about a bit. The gridchanges shape, butthe relationships(topology) stay thesame.

o r p o r a

t e dThere are also update

methods using

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methods usingcylindrical or sphericalcoordinate systems.These are handy for cylindrical and sphericalthings, but also for

survey data.

o r p o r a

t e d

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The End

2012 02 14 GC Principles

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