Approaches to Infinity

Computer Graphics

Presentation

Sabih Ahmed Khan 07-0053

Imran Jawaid 07-0037

Approaches to Infinity

Introduction

Computers are particularly good at repetition.

The high precesion with which the modern

computers can do calculations allows an alogorithm

to get closer and closer looks.

Now we are jumping to infinitely, however it has the

finite resolution and finite size and it must be made

in finite amount of time.

Thus we make the approximation to the creatures

being studied.

Fractals and self Similarity

We want methods that takes us to infinity.

This will feature Recursion.

Self similar here means the same at every scale

some curves are Exactly self similar and others are

Statistically similar.

Self similar curves were called “fractal”.

Successive Refinement in

Curves

Successive genration of KOCH curve are denoteds

as Ko, K1, K2 ….

The zeroth is just a horizontal line.

To create the K1 divide the line K0 in to three equal

parts and replace the middle section with a

triangular bump. Sides of length 1/3.

The total length of one line is 4/3.

Successive Refinement in

Curves

To form Kn+1 from Kn

Sub divide each segment of kn into three equal

parts and replace the middle part with a bump in the

shape of an equilateral Triangle.

Ki has total length of (4/3)I

As I tend to infinity, the length of the curve become

infinite

Drawing Koch Curve and

Snow Flake

Drawing Koch Curve and

Snow Flake

String Production and Peano

Curves

A large number of curves can be generated by

Refining the line Segments. (to generate these

curves, L-systems)

Turtle Graphics Reads the String and interprets

each character a command to perform some

operation.

„F‟ means forward(1,1) (Go forward 1 m the current direction)

„+‟ means turn(A) (turn right through angle A degrees)

„-‟ means turn(A) (turn left through angle A degrees)


Curves

“F-F++F-F “ with an angle of 60 the turtle would

draw an the first genration Koch curve.


Curves

How to generate a simple String in to the Longer

one that will generate Rich curve?

This is based on the String Production Rules .

„F‟->”F-F++F-F”

In the first stage the initial string is called the atom.

S1=“F-F++F-F”(1st curve)

S2=“F-F++F-F-F-F++F-F++F-F++F-F-F-F++F-F”(2nd

curve)


Curves

Call Produce String Function and read a string from

the Input file which contains the atoms and finally

writes it in the Output file the Desired Curve.


Curves

Where in each string is repeatedly fed back into same function and

produce the next higher order object.

Extending the Language

A richer set of curves can be generated by adding more rules to

string production process

The rule is that any X and Y characters are ignored where as F , +

and – are interpreted as before.

The atom is FX here.

For Example X and Y in FX+YF is interpreted as F+F+

Extending the Language

For the second order Dragon the turtle Responds to F+F++-F-F+

instead of S2=FX+YF++-FX-YF+.

The Resultant Curve is shown Below.

Producing String Recursively

and Drawing in a program.

To do this we store the variable called Order, the number of times to

apply the rule to the String.

For + and – the turtle turn accordingly but if the order is zero the turtle

moves forward for F. else it calls the Function it self to produce the

desired string

Curves based on String

Production

The five key intergradient's for each curve (atom, F-string, X-String,

Y-String, angle in degrees)

Curves based on String

Production

Allowing Branching

We need a character that commands the turtle to save its

current state

State of turtle = [cp,cd].

A turtle Stack is maintained.

Fractal Trees

The bush is based on the atom F an angle of 22 and the F

string.

F-> “FF-[-F+F+F]+[+F-F-F]”

Tiling the Plane

Another way to move towards the infinity is to repeat a shape

again and again, covering the Entire Plane

The notion is to take many copy of some shape .

Tiling the Plane

Another way to move towards the infinity is to repeat a shape

again and again, covering the Entire Plane

The notion is to take many copy of some shape .

Monohedral Tiling

Those based on a single Polygon

The polygon in question is called the prototile of the

tesselation

Only three possible regular tiling can be shown. A square ,

Hexagon and triangle.

Monohedral Tiling

Famous polyominoes formed by connecting unit squares Edge

to edge

Polyimanoids are formed by connecting congruent triangles

together..

Dihedral Tiling

It permits the use of two prototiles and therefore offer many

more possiblities.

Dihedral Tiling

The most famous dihedral tiling are the Archmedian tilings.

Reptiles

Reptiles are class of non periodic tilings that are most Easily

described Recursively.

Different replicas of a reptile fit together to form a large reptile

of the same shapes

Reptiles

Reptiles are class of non periodic tilings that are most Easily

described Recursively.

Different replicas of a reptile fit together to form a large reptile

of the same shapes

Approaches to Infinity

Documents