UBI 516Advanced Computer Graphics Fractal Geometry Methods Aydın Öztürk firstname.lastname@example.org http://www.ube.ege.edu.tr/~ozturk
Fractals • Many objects in nature aren't formed of squares or triangles, but of more complicated geometric figures. • Many natural objects - ferns, coastlines, etc. - are shaped like fractals.
Fractals(cont) • All the object representations we have considered so far used Euclidean geometry methods that is object shapes were described with equations. • Natural objects can be realistically described with fractal geometry methods, where procedures rather than equations are used.
Fractals(cont.) • A fractal has two basic charecteristics - Infinite detail at every point - Self similarity between the object parts.
Self Similarity • Geometric figures are similar if they have the same shape. • The two squares are similar. • The two rectangles are not similar. • But the two rectangles are similar.
Self Similarity • Many figures that are not fractals are self-similar. • Notice the figure to the right. The outline of the figure is a trapezoid. All the trapezoids inside make up the larger trapezoid.
Construction of Self Similar Fractals • To construct a deterministic self-similar fractal, we start with a geometric shape called initiator. • Subparts of of initiator are then replaced with a pattern called generator. Initiator Generator
Construction of Self Similar Fractals • To construct a deterministic self-similar fractal, we start with a geometric shape called initiator. • Subparts of of initiator are then replaced with a pattern called generator.
The Sierpinski Triangle(cont.) Step One Draw an equilateral triangle with sides of 2 triangle lengths each. Connect the midpoints of each side. Shade out the triangle in the center
The Sierpinski Triangle(cont.) Step Two Draw another equilateral triangle with sides of 2 triangle lengths each. Connect the midpoints of the sides and shade the triangle in the center as before. Notice the three small triangles that also need to be shaded out in each of the three triangles on each corner - three more holes.
The Sierpinski Triangle(cont.) Step Three Draw an equilateral triangle with sides of 4 triangle lengths each.
The Sierpinski Triangle(cont.) Step Four We follow the above patternand complete the Sierpinski Triangle.
Classification of Fractals • Self-similar fractals have parts that are scaled-down versions of the entire object. • Self-affinefractals have parts that are formed with different scaling parameters sx , sy ,szin different coordinate directions. • Invariant fractal setsare formed with nonlinear transformations.
Fractal Dimension • The detail variation in a fractal object can be described with a number D, calledfractal dimension, which is a measure of roughness, or fragmentation of the object. • More jagged-looking objects have larger fractal dimensions.
Fractal Dimension(cont.) A point has no dimensions –no length, no width, no height. A line has one dimension - length. It has no width and no height, but infinite A plane has two dimensions - length and width, no depth.
Fractal Dimension(cont.) • A cubehas three dimensions, length, width, and depth, extending to infinity in all three directions • Fractals can have fractional (or fractal)dimension.A fractal might have dimension of 1.6 or 2.4.
Fractal Dimension(cont.) • Aunit straight-line segment is divided into two equal length sub-parts.This gives two scaled copies of the original segment. • A unit square is divided into four sub-squares. Scaling the length and width by 2 gives four copies of the original square. • A unit cube is divided into eight sub-cubes. Scaling the length, width, and height by 2 gives eightcopies of the original cube.
Fractal Dimension(cont.) Relationship between the number of copies, scaling factor and the dimension.
Fractal Dimension(cont.) Sierpinski triangle.Scale the length of the sides of an equilateral triangle by 2. Scaling the sides gives us three copies, so 3 = 2d, where d is the dimension.Solving this equation we obtain the corresponding fractal dimension d=1.584.
Fractal Dimension(cont.) Let s be the scaling factor. Relationship between the number of copies, the scaling factor and the dimension can be written as Solwing this expression for d, we have
Fractal Dimension: Examples Generator Segment length Fractal dimension 1/√7 d=ln 3/ln(√7)=1.129 1/4 d=ln 8/ln 4=1.500 1/6 d=ln 18/ln 6=1.613
Creating an Image by Means of Iterated Function Systems • Suppose that we take an initial image I0 , and put it through a special photocopier that produces a new image I1. • I1 is nota simple copy of I0; rather, it is a superposition of several reduced versions of I0. • We then take I1and feed it back into the copier again, to produce image I2 ...etc. • We investigate whether these images converge to some image.
Multiple Reduction Copying Machine(cont.) • There are multiple lens arrangements to create multiple copies of the original. • Each lens arrangement reduces the size of the original. • The copier operates in a feedback loop, with the output of one stage the input to the next. • The initial input may be any image.
Multiple Reduction Copying Machine(cont.) Sierpinski’s Triangle is a typical example http://www.arcytech.org/java/fractals/sierpinski.shtml
Multiple Reduction Copying Machine: Sierpinski’s Triangle • The Sierpinski’s triangle is a most well known and simplest example of Iterated Function Systems(IFS). • It is comprised of three component functions(lenses), each of which shrinks the input image by one half and translates it to a new position. • This contractive property guarantees convergence of the iterative process.