What?

• procedural animation
• direct manipulation

• shapes express mathematical relationships
• before-and-after diagrams describe changes of state
• "parallel" execution of rules

• geometry is a useful way to explain a computation

Vector Projection

u = (w · v) · v / ||v||

Reflections

R = 2 · (L · N) · N - L

Bézier Subdivision

• geometry is a useful way to explain a computation to people
• why not computers, too?

Visual Language

• visual problems
• most important data type is position
• most important operation is movement
• the simplest movement is translation (vector addition)

Vectors

• frequently used
• frequently drawn
• too abstract: no base, no position

Snapping is Assignment

Triangles

• triangles can represent rotation and scaling
• similar triangles are used in many geometric proofs

Rotating and Scaling

Special Triangles

Cams

Summary

• geometry is a useful way to explain a computation to people
• geometry can be used to explain an expression to a computer
• what about control flow?

Rule Rewriting

• Pictorial Janus
  
• Cocoa / KidSim
  
• 3D-Visulan
  

State

Demonstrating a Step

• geometry is a useful way to explain a computation to people
• geometry can be used to explain an expression to a computer
• you can describe changes in state this way
• what about conditionals?

Events / Messages:

• clock ticks
• user input
• collisions
• communication

Conditions

• presence of a condition indicates "X must be true"
• point in circle:
  
• stick intersection
  

• geometry is a useful way to explain a computation to people
• geometry can be used to explain an expression to a computer
• you can describe changes in state this way
• you can describe conditions that must be met to trigger a rule
• programs - algorithms = data structures

Links

• critters can contain links
  
• links can be assigned to by connecting them to a critter
  

Data Structures

• critters are used for composing state
• binary tree node:
  two links and no behavior
  

• geometry is a useful way to explain a computation to people
• geometry can be used to explain an expression to a computer
• you can describe changes in state this way
• you can describe conditions that must be met to trigger a rule
• geometry is also good for describing data structures
• hasn't somebody done this before?

• inspiration for geometric computation
• limited to compass/ruler constructions
• not programming languages

• visual equivalents of concurrent constraint languages
• Pictorial Janus has accidental layout
• ToonTalk can manipulate sprites numerically
• powerful data structures and communication

• grid-based rules - no smooth motion
• uses a calculator for arithmetic

• similar application area, but for end users
• sophisticated multiple-example inference algorithm:
  often unclear what it has learned
• deck of cards metaphor for choices

• commercial video game construction kit
• lots of built-in knowledge, little flexibility

• more like a keyframe animation system
• built-in limited knowledge of directional movement
• extensive use of numerals

Contributions

• framework for geometric computation
• set of useful shapes
• integration of geometric expressions with programming constructs

Status

• have prototype, written in STk
• stick manipulation and biases work
• rule construction and user input events work
• can build:
  • straight-line motion
  • parabolic trajectory
  • spawning new critters

Evaluation

• examples
  • asteroids
  • particle system
  • bresenham
• user experience

Conclusions

• geometry is a useful way to explain a computation to people
• geometry can be used to explain an expression to a computer
• you can describe changes in state this way
• you can describe conditions that must be met to trigger a rule
• geometry is also good for describing data structures
• this work combines the above in a new and powerful way