North-Holland, Amsterdam Google Scholar. Brady M a Representing Shape. Brady M b Criteria for rerpresentations of shape. In: Rosenfeld, Beck eds Human and machine Vision. Bresenham JE Algorithm for computer control of a digital plotter. Brodlie KW Methods for drawing curves.

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In: Earnshaw RA ed Fundamental algorithms for computer graphics. Springer, pp — Google Scholar. Cameron SA Modelling solids in motion. Carpenter L Computer rendering of fractal curves and surfaces. Cohen E Some mathematical tools for a modeller's workbench. Cole AJ Compaction techniques for raster scan graphics using space-filling curves.

Comput J 30 1 —92 Google Scholar. Courant R, Hilbert D Methods of mathematical physics. Wiley Google Scholar.

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Dubrulle AA A class of numerical methods for the computation of pythagorean sums. Duff T Numerical methods for computer graphics. Earnshaw RA A review of curve drawing algorithms. Pfaff GE ed User interface management systems. Forrest AR Computational geometry in practice. Franklin WR Problems with raster graphics algorithms. Goldfeather J, Fuchs H Quadratic surface rendering on a logic-enhanced frame-buffer memory. Goldstein H Classical Mechanics. Addison-Wesley Google Scholar. Prentice-Hall Google Scholar. Hall R Color reproduction and illumination models. Hanrahan P Ray tracing algebraic surfaces.

Hartshorne R Algebraic geometry. Kajiya JT Transform Theory. Kajiya JT Differential and algebraic geometry. King M Towards an integrated computer art system. Lansdown RJ Requirements for knowledge-based systems in design. Without one, the program would crash, caught inside of an infinite loop. The same can be said about recursion. This circles example is rather trivial; it could easily be achieved through simple iteration. However, for scenarios in which a function calls itself more than once, recursion becomes wonderfully elegant. For every circle displayed, draw a circle half its size to the left and right of that circle.

Where do we begin? Well, we know that the Cantor set begins with a line. The above cantor function draws a line that starts at pixel coordinate x,y with a length of len. The line is drawn horizontally here, but this is an arbitrary decision.

So if we called that function, saying:. Now, the Cantor rule tells us to erase the middle third of that line, which leaves us with two lines, one from the beginning of the line to the one-third mark, and one from the two-thirds mark to the end of the line.

We can now add two more lines of code to draw the second pair of lines, moving the y-location down a bunch of pixels so that we can see the result below the original line. While this is a fine start, such a manual approach of calling line for each line is not what we want. Here is where recursion comes and rescues us. Instead of calling the line function directly, we can simply call the cantor function itself. After all, what does the cantor function do?

It draws a line at an x,y location with a given length! And so:. And since the cantor function is called recursively, the same rule will be applied to the next lines and to the next and to the next as cantor calls itself again and again! Using drawCircle and the Cantor set as models, generate your own pattern with recursion. Here is a screenshot of one that uses lines. Writing a function that recursively calls itself is one technique for generating a fractal pattern on screen.

However, what if you wanted the lines in the above Cantor set to exist as individual objects that could be moved independently? The recursive function is simple and elegant, but it does not allow you to do much besides simply generating the pattern itself. However, there is another way we can apply recursion in combination with an ArrayList that will allow us to not only generate a fractal pattern, but keep track of all its individual parts as objects.

Here are the rules. Note that it starts the same way as the Cantor set, with a single line. If the length of the original starting line is one, the first iteration of the Koch curve will yield a line of length four-thirds each segment is one-third the length of the starting line. Do it again and you get a length of sixteen-ninths. As you iterate towards infinity, the length of the Koch curve approaches infinity. Yet it fits in the tiny finite space provided right here on this paper or screen! We could proceed in the same manner as we did with the Cantor set, and write a recursive function that iteratively applies the Koch rules over and over.

Nevertheless, we are going to tackle this problem in a different manner by treating each segment of the Koch curve as an individual object. This will open up some design possibilities. For example, if each segment is an object, we could allow each segment to move independently from its original location and participate in a physics simulation. In addition, we could use a random color, line thickness, etc.

In order to accomplish our goal of treating each segment as an individual object, we must first decide what this object should be in the first place. What data should it store? What functions should it have? Now that we have our KochLine class, we can get started on the main program. Then in draw , all KochLine objects just one right now can be displayed in a loop. Remember the Game of Life cellular automata? In that simulation, we always kept track of two generations: current and next. When we were finished computing the next generation, next became current and we moved on to computing the new next generation.

We are going to apply a similar technique here. We have an ArrayList that keeps track of the current set of KochLine objects at the start of the program, there is only one.

## M.Sc. Curriculum

By calling generate over and over again for example, each time the mouse is pressed , we recursively apply the Koch curve rules to the existing set of KochLine objects. How do we break one line segment into four as described by the rules? While this can be accomplished with some simple arithmetic and trigonometry, since our KochLine object uses PVector , this is a nice opportunity for us to practice our vector math.

As you can see from the above figure, we need five points a, b, c, d, and e to generate the new KochLine objects and make the new line segments ab, cb, cd, and de. Where do we get these points? Since we have a KochLine object, why not ask the KochLine object to compute all these points for us?

Now we just need to write five new functions in the KochLine class, each one returning a PVector according to Figure 8. B is one-third of the way along the line segment and D is two-thirds. Here we can make a PVector that points from start to end and shrink it to one-third the length for B and two-thirds the length for D to find these points.

The last point, C, is the most difficult one to find. However, if you recall that the angles of an equilateral triangle are all sixty degrees, this makes it a little bit easier. If we know how to find point B with a PVector one-third the length of the line, what if we were to rotate that same PVector sixty degrees and move along that vector from point B? Try animating the Koch curve. For example, can you draw it from left to right? Can you vary the visual design of the line segments?

Can you move the line segments using techniques from earlier chapters? What if each line segment were made into a spring toxiclibs or joint Box2D? The fractals we have examined in this chapter so far are deterministic, meaning they have no randomness and will always produce the identical outcome each time they are run.

They are excellent demonstrations of classic fractals and the programming techniques behind drawing them, but are too precise to feel natural. In this next part of the chapter, I want to examine some techniques behind generating a stochastic or non-deterministic fractal. Here are our production rules:. Again, we have a nice fractal with a recursive definition: A branch is a line with two branches connected to it.

Each new branch must rotate relative to the previous branch, which is rotated relative to all its previous branches. Luckily for us, Processing has a mechanism to keep track of rotations for us—the transformation matrix. And since the root starts at the bottom of the window see above , the first step requires translating to that spot:. Remember, when we rotate in Processing, we are always rotating around the point of origin, so here the point of origin must always be translated to the end of our current branch. Now that we have a branch going to the right, we need one going to the left.

We can use pushMatrix to save the transformation state before we rotate, letting us call popMatrix to restore that state and draw the branch to the left. We could keep adding more and more calls to line for more and more branches, but just as with the Cantor set and Koch curve, our code would become incredibly complicated and unwieldy.

Instead, we can use the above logic as our foundation for writing a recursive function, replacing the direct calls to line with our own function called branch.

Notice how in the above code we use pushMatrix and popMatrix around each subsequent call to branch. This is one of those elegant code solutions that feels almost like magic.

### Kundrecensioner

Each call to branch takes a moment to remember the location of that particular branch. Such components are called hyperbolic components. This problem, known as density of hyperbolicity , may be the most important open problem in the field of complex dynamics. Hypothetical non-hyperbolic components of the Mandelbrot set are often referred to as "queer" or ghost components. Note that hyperbolic components intersecting the real axis correspond exactly to periodic windows in the Feigenbaum diagram.

So this result states that such windows exist near every parameter in the diagram. Not every hyperbolic component can be reached by a sequence of direct bifurcations from the main cardioid of the Mandelbrot set. However, such a component can be reached by a sequence of direct bifurcations from the main cardioid of a little Mandelbrot copy see below. This means that the cycle contains the critical point 0, so that 0 is iterated back to itself after some iterations. It is conjectured that the Mandelbrot set is locally connected. This famous conjecture is known as MLC for Mandelbrot locally connected.

By the work of Adrien Douady and John H. Hubbard , this conjecture would result in a simple abstract "pinched disk" model of the Mandelbrot set. In particular, it would imply the important hyperbolicity conjecture mentioned above. The work of Jean-Christophe Yoccoz established local connectivity of the Mandelbrot set at all finitely renormalizable parameters; that is, roughly speaking those contained only in finitely many small Mandelbrot copies. The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points.

It is also conjectured to be self-similar around generalized Feigenbaum points e.

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## Visual Modeling and Fractal Methods in Science - IEEE Conference Publication

The Mandelbrot set in general is not strictly self-similar but it is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. The little copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set. The Hausdorff dimension of the boundary of the Mandelbrot set equals 2 as determined by a result of Mitsuhiro Shishikura. In the Blum-Shub-Smale model of real computation , the Mandelbrot set is not computable, but its complement is computably enumerable.

### Daniel Shiffman

However, many simple objects e. At present, it is unknown whether the Mandelbrot set is computable in models of real computation based on computable analysis , which correspond more closely to the intuitive notion of "plotting the set by a computer". Hertling has shown that the Mandelbrot set is computable in this model if the hyperbolicity conjecture is true. As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set.

For instance, a point is in the Mandelbrot set exactly when the corresponding Julia set is connected. This principle is exploited in virtually all deep results on the Mandelbrot set. For example, Shishikura proved that, for a dense set of parameters in the boundary of the Mandelbrot set, the Julia set has Hausdorff dimension two, and then transfers this information to the parameter plane. We can thus determine the period of a given bulb by counting these antennas.

It can be shown that the Fibonacci sequence is located within the Mandelbrot Set and that a relation exists between the main cardioid and the Farey Diagram. Upon mapping the main cardioid to a disk, one can notice that the amount of antennae that extends from the next largest Hyperbolic component, and that is located between the two previously selected components, follows suit with the Fibonacci sequence.

The amount of antennae also correlates with the Farey Diagram and the denominator amounts within the corresponding fractional values, of which relate to the distance around the disk. Thus, the Fibonacci sequence of 1, 2, 3, 5, 8, 13, and 21 can be found within the Mandelbrot set. The Mandelbrot set shows more intricate detail the closer one looks or magnifies the image, usually called "zooming in". The following example of an image sequence zooming to a selected c value gives an impression of the infinite richness of different geometrical structures and explains some of their typical rules.

The magnification of the last image relative to the first one is about 10 10 to 1. Relating to an ordinary monitor, it represents a section of a Mandelbrot set with a diameter of 4 million kilometers. Its border would show an astronomical number of different fractal structures. The seahorse "body" is composed by 25 "spokes" consisting of two groups of 12 "spokes" each and one "spoke" connecting to the main cardioid.

These two groups can be attributed by some kind of metamorphosis to the two "fingers" of the "upper hand" of the Mandelbrot set; therefore, the number of "spokes" increases from one "seahorse" to the next by 2; the "hub" is a so-called Misiurewicz point. Between the "upper part of the body" and the "tail" a distorted small copy of the Mandelbrot set called satellite may be recognized.

The central endpoint of the "seahorse tail" is also a Misiurewicz point. Part of the "tail" — there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a simply connected set, which means there are no islands and no loop roads around a hole.

## Fractal geometry : mathematical methods, algorithms, applications

The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center. Open this location in an interactive viewer. Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head". Double-spirals with satellites of second order — analogously to the "seahorses", the double-spirals may be interpreted as a metamorphosis of the "antenna".

In the outer part of the appendices, islands of structures may be recognized; they have a shape like Julia sets J c ; the largest of them may be found in the center of the "double-hook" on the right side. Detail of the spiral. The islands above seem to consist of infinitely many parts like Cantor sets , as is [ clarification needed ] actually the case for the corresponding Julia set J c.

However, they are connected by tiny structures, so that the whole represents a simply connected set. The tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value of c for the corresponding J c is not that of the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in the 6th zoom step. Multibrot sets are bounded sets found in the complex plane for members of the general monic univariate polynomial family of recursions.

For an integer d, these sets are connectedness loci for the Julia sets built from the same formula. A parameter is in the cubic connectedness locus if both critical points are stable. The Multibrot set is obtained by varying the value of the exponent d. There is no perfect extension of the Mandelbrot set into 3D. This is because there is no 3D analogue of the complex numbers for it to iterate on.

However, there is an extension of the complex numbers into 4 dimensions, called the quaternions , that creates a perfect extension of the Mandelbrot set and the Julia sets into 4 dimensions. Of particular interest is the tricorn fractal, the connectedness locus of the anti-holomorphic family. The tricorn also sometimes called the Mandelbar was encountered by Milnor in his study of parameter slices of real cubic polynomials.

It is not locally connected. This property is inherited by the connectedness locus of real cubic polynomials. There are many programs used to generate the Mandelbrot set and other fractals, some of which are described in fractal-generating software. These programs use a variety of algorithms to determine the color of individual pixels and achieve efficient computation. The simplest algorithm for generating a representation of the Mandelbrot set is known as the "escape time" algorithm. A repeating calculation is performed for each x , y point in the plot area and based on the behavior of that calculation, a color is chosen for that pixel.

The x and y locations of each point are used as starting values in a repeating, or iterating calculation described in detail below. The result of each iteration is used as the starting values for the next. The values are checked during each iteration to see whether they have reached a critical "escape" condition, or "bailout".

If that condition is reached, the calculation is stopped, the pixel is drawn, and the next x , y point is examined. For some starting values, escape occurs quickly, after only a small number of iterations. For starting values very close to but not in the set, it may take hundreds or thousands of iterations to escape.

For values within the Mandelbrot set, escape will never occur. The programmer or user must choose how much iteration, or "depth", they wish to examine. The higher the maximal number of iterations, the more detail and subtlety emerge in the final image, but the longer time it will take to calculate the fractal image. Escape conditions can be simple or complex. Because no complex number with a real or imaginary part greater than 2 can be part of the set, a common bailout is to escape when either coefficient exceeds 2.

A more computationally complex method that detects escapes sooner, is to compute distance from the origin using the Pythagorean theorem , i. If this value exceeds 2, or equivalently, when the sum of the squares of the real and imaginary parts exceed 4, the point has reached escape. More computationally intensive rendering variations include the Buddhabrot method, which finds escaping points and plots their iterated coordinates. The color of each point represents how quickly the values reached the escape point. Often black is used to show values that fail to escape before the iteration limit, and gradually brighter colors are used for points that escape.

This gives a visual representation of how many cycles were required before reaching the escape condition. To render such an image, the region of the complex plane we are considering is subdivided into a certain number of pixels.