The following video shows a result of my last year study on fractal surfaces:
It is based on surface modeling ideas that I already developed 5 years ago - as you can see in the very short video in the previous post. However, this time I have pushed the algorithm to its limits and optimized for deeper zoom levels and scaling factors. And yeah, video is longer too!
On a side note, as you can judge by the frequency of my posts here, I don't have much time lately for my spare time projects, but hey, I'm still doing "something".
Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts
Saturday, 14 March 2015
Thursday, 28 October 2010
Fractal surfaces
I was working hard recently on a new fractal surface generator and renderer. It is based on my theoretical research in generating fractal surfaces. Here is a very short movie, the first test:
It is rendered in 3D (stereoscopy). Unfortunately YouTube doesn't support NVidia 3D Vision glasses. Anaglyphic looks ok though.
It is rendered in 3D (stereoscopy). Unfortunately YouTube doesn't support NVidia 3D Vision glasses. Anaglyphic looks ok though.
Tuesday, 1 September 2009
New York, New York...
This summer, I work with my friend Ken at Brown University. I'm in his home country first time, so we decided to visit as Frank Sinatra sings "New York, New York...", the so called Big Apple, and it was even close enough to get there (2-3 hours by train). And what do you think was the most exciting part for us in such anomalously big city ?
Geometry, of course!
We decided to go to Empire State Building observatory during the night, and it was the right thing to do, the city lights are just breath-taking! You can enjoy the full power of human civilisation in just one spot!
Now, I consider sunset to be even more exciting time to go, especially if you wait after it will get completely dark and lights start to appear slowly - as far as my imagination is right about it (just be prepared for a very long waiting line or simply pay extra fee for VIP pass through).
After listening to interesting audio tour with a lot of nice, but sometimes a bit fake/artificial impressions, like "I just looove this city, as a young boy I loooved to walk on Brooklyn Bridge, etc..." (similar style of making audio tour you can experience at Boston's Prudential Tower aka Skywalk Observatory), I noticed that, there are two interesting aspects you should take into account when developing... NY-like city generation algorithm.
Can you spot so called Flatiron Building on the photo ? This was the most inspiring example to me. Generally, I divided buildings into two categories: first contains bulidngs that shape fits into street design, the second category contains the rest (buildings that for some reasons, use inefficient amount of space - as Ken noticed). And what is apparent in New York, there are many buildings that efficiently and as many that are inefficiently occuping space between streets (and in general, there are too many of them ;)). Flatiron has sharp angle and is efficient, while there are some with angle more than 90 deg. (where Broadway cross Avenues at more than 90 deg. angle, just note the small building on the bottom-right), some are just square-shaped even if there is so much room around them. I was so excited about this discovery that I almost forgot to enjoy the view in ehm.. humanistic kind of way :)
Geometry, of course!
We decided to go to Empire State Building observatory during the night, and it was the right thing to do, the city lights are just breath-taking! You can enjoy the full power of human civilisation in just one spot!
Now, I consider sunset to be even more exciting time to go, especially if you wait after it will get completely dark and lights start to appear slowly - as far as my imagination is right about it (just be prepared for a very long waiting line or simply pay extra fee for VIP pass through).
After listening to interesting audio tour with a lot of nice, but sometimes a bit fake/artificial impressions, like "I just looove this city, as a young boy I loooved to walk on Brooklyn Bridge, etc..." (similar style of making audio tour you can experience at Boston's Prudential Tower aka Skywalk Observatory), I noticed that, there are two interesting aspects you should take into account when developing... NY-like city generation algorithm.
Can you spot so called Flatiron Building on the photo ? This was the most inspiring example to me. Generally, I divided buildings into two categories: first contains bulidngs that shape fits into street design, the second category contains the rest (buildings that for some reasons, use inefficient amount of space - as Ken noticed). And what is apparent in New York, there are many buildings that efficiently and as many that are inefficiently occuping space between streets (and in general, there are too many of them ;)). Flatiron has sharp angle and is efficient, while there are some with angle more than 90 deg. (where Broadway cross Avenues at more than 90 deg. angle, just note the small building on the bottom-right), some are just square-shaped even if there is so much room around them. I was so excited about this discovery that I almost forgot to enjoy the view in ehm.. humanistic kind of way :)
Monday, 14 July 2008
Closures and boundaries
Today, I was working on some technical report (an internal publication for the university) for my PhD. I was dealing with topology of sets in metric spaces, which is a special case of general topology, well suited just for my needs, a computer graphics field.
I had minor troubles with a definition of a set boundary, but I think I finally got it right and actually some things surprised me a bit - that's why I'm sharing it here!
At first I was saying about boundary of the set A without mentioning about the metric space X that contains it. Usually I'm operating in R^n euclidean space, and all of my sets was subsets of R^n. But this time, I was dealing with a space constructed from a subset of R^n, specifically, a unitary box: [0..1]^n, further called I^n space, I={x|0<=x<=1}, where x is a real number. Let Y be a continuously deformed ball that is a subset of I^n . I was "intuitively" (by analogy to R^n space) thinking that the boundary of Y should be always connected (or even simply-connected). You can not imagine a ball that has disconnected boundary, right ?
As usual, it all depends on the definition on the boundary.
In topology, the boundary can be defined several ways that are equivalent for metric spaces. We can for example say, that a boundary of a set Y included by a metric space X is a set of points in X for which any open ball (with radius > 0) contains points both inside Y and outside Y.
To define an open ball in X you surely need to define a metrics of X. If it is euclidean R^n space, a ball "looks" like a ball = it is a sphere, if not, it can be almost "anything". But stick to the euclidean first. The I^n set has a boundary in R^n. It is an "empty" box. But any open ball in euclidean I^n space will not always be an open sphere - near the boundary of I^n in R^n, it will be just a part of a sphere cut by the walls of I^n box. Now, if we take any connected subset Y of I^n that touches the walls of I^n box, the set of points at the walls of I^n box will be not a part of boundary of Y! This is why we can construct Y that will have a disconnected boundary in I^n, an example provided below:
In the figure above, a boundary is denoted as a thicker line. Note that any open ball co-centered with "a" and smaller than "a" will contain only points inside Y, thus the center of "a" is not a part of the boundary of Y. We can also clearly see that the center of "b" lies on the boundary of Y (any ball around it contains point both inside and outside Y). Obviously the boundary of Y' in R^n is connected.
I had minor troubles with a definition of a set boundary, but I think I finally got it right and actually some things surprised me a bit - that's why I'm sharing it here!
At first I was saying about boundary of the set A without mentioning about the metric space X that contains it. Usually I'm operating in R^n euclidean space, and all of my sets was subsets of R^n. But this time, I was dealing with a space constructed from a subset of R^n, specifically, a unitary box: [0..1]^n, further called I^n space, I={x|0<=x<=1}, where x is a real number. Let Y be a continuously deformed ball that is a subset of I^n . I was "intuitively" (by analogy to R^n space) thinking that the boundary of Y should be always connected (or even simply-connected). You can not imagine a ball that has disconnected boundary, right ?
As usual, it all depends on the definition on the boundary.
In topology, the boundary can be defined several ways that are equivalent for metric spaces. We can for example say, that a boundary of a set Y included by a metric space X is a set of points in X for which any open ball (with radius > 0) contains points both inside Y and outside Y.
To define an open ball in X you surely need to define a metrics of X. If it is euclidean R^n space, a ball "looks" like a ball = it is a sphere, if not, it can be almost "anything". But stick to the euclidean first. The I^n set has a boundary in R^n. It is an "empty" box. But any open ball in euclidean I^n space will not always be an open sphere - near the boundary of I^n in R^n, it will be just a part of a sphere cut by the walls of I^n box. Now, if we take any connected subset Y of I^n that touches the walls of I^n box, the set of points at the walls of I^n box will be not a part of boundary of Y! This is why we can construct Y that will have a disconnected boundary in I^n, an example provided below:
In the figure above, a boundary is denoted as a thicker line. Note that any open ball co-centered with "a" and smaller than "a" will contain only points inside Y, thus the center of "a" is not a part of the boundary of Y. We can also clearly see that the center of "b" lies on the boundary of Y (any ball around it contains point both inside and outside Y). Obviously the boundary of Y' in R^n is connected.
Tuesday, 17 June 2008
Math-fiction
I came up with a mental exercise today. I will try to predict how mathematics will look in the far future or in no time, it is a math fiction, everything is possible.
Mathematics in the XXII century
A classical mathematical notation will be replaced with a strict machine-friendly format. The concepts like sets and relations will not change, but we will construct sets and relations differently. There is a set-builder notation and it already makes a lot of troubles. A computer-friendly notation would allow automatic syntax checking, inclusion/exclusion test, or even advanced hypothesis verification. The next-generation computer-aided math package will be implemented in TeX 2.0, a new scientific publication language format that will push the cooperation between so called human inteligence to higher levels! Using new math tools we will solve most of the millenium problems with ease. We will clarify complexity theory classes, also write no more than two page long proofs for Riemann and Hodge hypothesis. In the mean time, pentagonal tiles of type 15 will be covering all of our residence floors. The new mathematical tools will be very powerful because we will proof that P is equal to NP or even we will figure out that every algorithm require not more than logarithmic to input data size steps on the so called Big-Bang machine. We will be Big-Banging parallel universes whenever sorting a column in a spreadsheet. Last but not least mathematics will hit the trenches. All those uneducated people (mostly engineers and nature "scientist") will use standarized terminology.
There was a time when "real" mathematics was done with a pencil, paper and fresh mind only, and many mathamaticians actually still believes it is the only way to do it and other tools are only unnecessary complications.
And what do you think about it ?
Mathematics in the XXII century
A classical mathematical notation will be replaced with a strict machine-friendly format. The concepts like sets and relations will not change, but we will construct sets and relations differently. There is a set-builder notation and it already makes a lot of troubles. A computer-friendly notation would allow automatic syntax checking, inclusion/exclusion test, or even advanced hypothesis verification. The next-generation computer-aided math package will be implemented in TeX 2.0, a new scientific publication language format that will push the cooperation between so called human inteligence to higher levels! Using new math tools we will solve most of the millenium problems with ease. We will clarify complexity theory classes, also write no more than two page long proofs for Riemann and Hodge hypothesis. In the mean time, pentagonal tiles of type 15 will be covering all of our residence floors. The new mathematical tools will be very powerful because we will proof that P is equal to NP or even we will figure out that every algorithm require not more than logarithmic to input data size steps on the so called Big-Bang machine. We will be Big-Banging parallel universes whenever sorting a column in a spreadsheet. Last but not least mathematics will hit the trenches. All those uneducated people (mostly engineers and nature "scientist") will use standarized terminology.
There was a time when "real" mathematics was done with a pencil, paper and fresh mind only, and many mathamaticians actually still believes it is the only way to do it and other tools are only unnecessary complications.
And what do you think about it ?
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