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Unit-6: Curves and Fractals: Year & Branch: S.E. Computer Engineering Subject Name: Computer Graphics (210251)

The document discusses Hilbert curves, which are continuous space-filling curves named after German mathematician David Hilbert. Hilbert curves are fractals that are self-similar - zooming in on a section of the curve reveals the same pattern. The curves are generated through successive approximations by subdividing squares and drawing connecting lines between the subdivision centers in a particular order. Hilbert curves have a topological dimension of 1 but a fractal dimension of 2, so they appear to fill a 2D space despite being a 1D line. The document provides C++ code to generate a Hilbert curve using recursion.

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270 views13 pages

Unit-6: Curves and Fractals: Year & Branch: S.E. Computer Engineering Subject Name: Computer Graphics (210251)

The document discusses Hilbert curves, which are continuous space-filling curves named after German mathematician David Hilbert. Hilbert curves are fractals that are self-similar - zooming in on a section of the curve reveals the same pattern. The curves are generated through successive approximations by subdividing squares and drawing connecting lines between the subdivision centers in a particular order. Hilbert curves have a topological dimension of 1 but a fractal dimension of 2, so they appear to fill a 2D space despite being a 1D line. The document provides C++ code to generate a Hilbert curve using recursion.

Uploaded by

Bhushan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Year & Branch: S.E.

Computer Engineering
Subject Name: Computer Graphics (210251)

Unit-6: Curves and Fractals


Fractals
Mr. Bhushan S. Gholap 1
Hilbert’s Curve..
• Hilbert Curves are named after the German mathematician
David Hilbert. They were first described in 1891.

• It is also called as Peano curve. (as a variant of the space-


filling Peano curves discovered by Giuseppe Peano in 1890)

• A Hilbert curve is a continuous space-filling curve. They


are also fractal and are self-similar; If you zoom in and look
closely at a section of a higher-order curve, the pattern you
see looks just the same as itself.
Mr. Bhushan S. Gholap 2
Hilbert’s Curve..
• The curve begins with initial square. The generation of
curve requires successive approximations.

• Cups and Joins: The basic elements of the Hilbert curves


are "cups" (a square with one open side) and "joins" (a
vector that joins two cups). The "open" side of a cup can be
top, bottom, left or right. In a similar vein, a join has a
direction: up, down, left or right.

Mr. Bhushan S. Gholap 3


Hilbert’s Curve..
• In the first approximation, we are dividing the square into
four quadrants and then drawing the curve which connects
the center points of each quadrant.

Mr. Bhushan S. Gholap 4


Hilbert’s Curve..
• The second approximation will be to further subdivide each
of the four quadrants and draw the curve which connects the
center points of each these finer subdivisions before moving
the next major quadrant.

Mr. Bhushan S. Gholap 5


Hilbert’s Curve..
• The third approximation again subdivides each quadrant
and process continues.

• Hilbert curves, first to third order as shown in figure.

Mr. Bhushan S. Gholap 6


Hilbert’s Curve..
• We will come to know that, the curve is not getting
overlap at any point.

• There is no limit to this subdivision. The curve fills


the square.

• Ideally, the length of curve is infinite. With each


subdivision the length increases by factor 4.

• Topological Dimension (Dt) must be equal to 1.


Mr. Bhushan S. Gholap 7
Hilbert’s Curve..
• At each subdivision the scale (S) changes by 2 i.e. at every
approximation we are dividing plane into four quadrants.

• But the length (N) of the curve changes by 4, at each


subdivision. So we need 4 scaled curves to form original
curve.
• i.e. N = SDf
4 = 2Df
Df = 2
(Df - Fractal Dimension)
Mr. Bhushan S. Gholap 8
Hilbert’s Curve..
• Hilbert curve has,
– Topological Dimension (Dt) = 1

– Fractal Dimension(Df) = 2

• So, Hilbert curve is a line only, but it is so folded


that it looks likes a 2D object.

Mr. Bhushan S. Gholap 9


University Questions
1. Explain Hilbert curve with example.

2. What is fractal ? Explain Hilbert curve in detail.

3. Explain Hilbert curve and its application in details.

Mr. Bhushan S. Gholap 10


Assignment No. 7: Write C++/Java program to generate
Hilbert curve using concept of fractals.
#include<iostream>
#include<stdio.h>
#include<graphics.h>
#include<math.h>
#include<stdlib.h>
using namespace std;

void move(int j, int h, int &x, int &y) // Move function


{
if(j==1)
y = y-h;
elseif(j==2)
x = x+h;
else if(j==3)
y = y+h;
else if(j==4)
x = x-h;

lineto(x,y); //Draws a line from current position to the point(x,y).


}
Mr. Bhushan S. Gholap 11
void hilbert(int r,int d,int l ,int u,int i,int h,int &x,int &y) int main()
{//Hillbert Curve Function {
if(i>0) int n,x1,y1;
{ int x0=50,y0=150,x,y,h=10,r=2,d=3,l=4,u=1;
i--;
cout<<"Give the value of n"; //Accept value for n
hilbert(d,r,u,l,i,h,x,y); cin>>n;
move(r,h,x,y); x=x0;
hilbert(r,d,l,u,i,h,x,y); y=y0;
move(d,h,x,y);
int driver=DETECT,mode=0;
hilbert(r,d,l,u,i,h,x,y);
initgraph(&driver,&mode,NULL);
move(l,h,x,y);
hilbert(u,l,d,r,i,h,x,y); moveto(x,y);

} hilbert(r,d,l,u,n,h,x,y);

}
delay(100000);
closegraph();
return 0;
}

Mr. Bhushan S. Gholap 12


• To be continue…

Mr. Bhushan S. Gholap 13

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