The Stony Brook Algorithm Repository

An Intuitive Explanation of Fourier Theory Steven Lehar slehar@cns.bu.edu Fourier theory is pretty complicated mathematically. But there are some beautifully simple holistic concepts behind Fourier theory which are relatively easy to explain intuitively. There are other sites on the web that can give you the mathematical formulation of the Fourier transform. Basic Principles: How space is represented by frequency Higher Harmonics: "Ringing" effects An Analog Analogy: The Optical Fourier Transform Fourier Filtering: Image Processing using Fourier Transforms Basic Principles Fourier theory states that any signal, in our case visual images, can be expressed as a sum of a series of sinusoids. These three values capture all of the information in the sinusoidal image. The magnitude of the sinusoid corresponds to its contrast, or the difference between the darkest and brightest peaks of the image. There is also a "DC term" corresponding to zero frequency, that represents the average brightness across the whole image. Back to top

Introduction to Programming in Java: An Interdisciplinary Approach Spatial references, coordinate systems, projections, datums, ell People are often mixing the above as if they were one and the same, so here’s a recap of them. One of the things you often find people saying is that “my data is in the WGS84 coordinate system”. This doesn’t really make sense, but I will get back to this later. This is a very confusing subject, and I might have gotten a few things wrong myself, so please add a comment and I’ll update it ASAP. Coordinate systems A coordinate system is simply put a way of describing a spatial property relative to a center. The Geocentric coordinate system is based on a normal (X,Y,Z) coordinate system with the origin at the center of Earth. Sidenote: The geocentric coordinate system is strictly speaking a cartesian coordinate system too, but this is the general terms I've seen used the most when talking about world coordinate systems. Datums and ellipsoids This poses two immediate problems: Where is the center of the earth What is the shape of the earth? Read more on Datums and Spheroids. Projections

Magical Square Root Implementation In Quake III Any 3D engine draws it’s power and speed from the mathematical models and implementations within, and trust John Carmack of ID software for using really good hacks. As it turns out, a very interesting hack is used in Quake III to calculate an inverse square root. Preface ID software has recently released the source code of Quake III engine with a GPL license. Carmack’s Unusual Inverse Square Root A fast glance at the file game/code/q_math.c reveals many interesting performance hacks. Observe the original function from q_math.c: float Q_rsqrt( float number ) { long i; float x2, y; const float threehalfs = 1.5F; x2 = number * 0.5F; y = number; i = * ( long * ) &y; // evil floating point bit level hacking i = 0x5f3759df - ( i >> 1 ); // what the fuck? Not only does it work, on some CPU’s Carmack’s Q_rsqrt runs up to 4 times faster than (float)(1.0/sqrt(x), eventhough sqrt() is usually implemented using the FSQRT assembley instruction! Newton’s Approximation of Roots A Witchcraft Number

Archimedean Solid The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types arranged in the same way about each vertex with all sides the same length (Cromwell 1997, pp. 91-92). The Archimedean solids are distinguished by having very high symmetry, thus excluding solids belonging to a dihedral group of symmetries (e.g., the two infinite families of regular prisms and antiprisms), as well as the elongated square gyrobicupola (because that surface's symmetry-breaking twist allows vertices "near the equator" and those "in the polar regions" to be distinguished; Cromwell 1997, p. 92). The Archimedean solids are sometimes also referred to as the semiregular polyhedra. The Archimedean solids are illustrated above. Nets of the Archimedean solids are illustrated above. The following table lists the uniform, Schläfli, Wythoff, and Cundy and Rollett symbols for the Archimedean solids (Wenninger 1989, p. 9). , edges where

Evil Mad Scientist Laboratories - Iterative Algorithmic Plastic One of our favorite shapes is the Sierpinski triangle. In one sense, a mere mathematical abstraction, on the other, a pattern that naturally emerges in real life from several different simple algorithms. On paper, one can play the Chaos Game to generate the shape (or cheat and just use the java applet). You can also generate a Sierpinski triangle in what is perhaps a more obvious way: by exploiting its fractal self-similarity. We begin with a few packages of polymer clay– two colors of Fimo Soft, in this case. Form the two clay colors into long triangular shapes. Press the stack of triangles together to make sure that the edges fuse well. Cut the stretched “first iteration” piece into four pieces of equal length. Again stretch the result from the previous iteration, cut into four pieces of equal length and set one aside. By now, you should have the hang of the iterative algorithm for making the fractal. The fifth iteration has 243 dark triangles.

Quantum Random Bit Generator Service The trouble with five December 2007 We are all familiar with the simple ways of tiling the plane by equilateral triangles, squares, or hexagons. These are the three regular tilings: each is made up of identical copies of a regular polygon — a shape whose sides all have the same length and angles between them — and adjacent tiles share whole edges, that is, we never have part of a tile's edge overlapping part of another tile's edge. Figure 1: The three regular tilings. In this collection of tilings by regular polygons the number five is conspicuously absent. Figure 2: Three pentagons arranged around a point leave a gap, and four overlap. But there is no reason to give up yet: we can try to find other interesting tilings of the plane involving the number five by relaxing some of the constraints on regular tilings. Is it now possible to find a set of shapes with five-fold symmetry that together will tile the plane? Going for simple shapes Figure 3: Constructing a tiling piece by piece. Dividing monsters forever

Handy Mathematics Facts for Graphics email scd@cs.brown.edu with suggested additions or corrections Eric Weisstein's world of Mathematics (which used to be called Eric's Treasure Trove of Mathematics) is an extremely comprehensive collection of math facts and definitions. Eric has other encyclopedias at www.treasure-troves.com S.O.S. Mathematics has a variety of algebra, trigonometry, calculus, and differential equations tutorial pages. Dave Eberly has a web site called Magic Software with several pages of descriptions and code that answers questions from comp.graphics.algorithms. Steve Hollasch at Microsoft has a very comprehensive page of graphics notes which he would like to turn into a graphics encyclopedia. Vector math identities and algorithms from Japan. Paul Bourke has a variety of pages with useful tidbits, many of which are linked to from Steve Hollasch's page. The graphics group at UC Davis also has notes about computer graphics. Peter H. Josh Levenberg has a page of links to yet more graphics algorithm resources. e pi

A 10 minute tutorial for solving Math problems with Maxima Posted by Antonio Cangiano in Essential Math, Software on June 4th, 2007 | 132 responses About 50,000 people read my article 3 awesome free Math programs. Chances are that at least some of them downloaded and installed Maxima. Maxima as a calculator You can use Maxima as a fast and reliable calculator whose precision is arbitrary within the limits of your PC’s hardware. (%i1) 9+7; (%o1) (%i2) -17*19; (%o2) (%i3) 10/2; (%o3) Maxima allows you to refer to the latest result through the % character, and to any previous input or output by its respective prompted %i (input) or %o (output). (%i4) % - 10; (%o4) (%i5) %o1 * 3; (%o5) For the sake of simplicity, from now on we will omit the numbered input and output prompts produced by Maxima’s console, and indicate the output with a => sign. float(1/3); => float(26/4); => As mentioned above, big numbers are not an issue: float((7/3)^35); => Constants and common functions Here is a list of common constants in Maxima, which you should be aware of: log(%e); =>

Project Euler 3 awesome free Math programs Posted by Antonio Cangiano in Software on June 2nd, 2007 | 109 responses Mathematical software can be very expensive. Programs like Mathematica, Maple and Matlab are incredibly powerful, flexible and usually well documented and supported. Their price tags however are a big let down for many people, even if there are cheap (in some cases crippled) versions available for educational purposes (if you are a student or a teacher). 1. A general purpose CAS (Computer Algebra System) is a program that’s able to perform symbolic manipulation for the resolution of common problems. Valuable mentions are: 2. Matlab is the standard for numerical computing, but there are a few clones and valid alternatives that are entirely free. Valid alternatives are: For statistical computing and analysis in the Open Source world, it doesn’t get any better than R. As usual, please feel free to share your experiences and add your suggestions to enrich the discussion.

Ulam spiral Ulam spiral of size 200×200. Black dots represent prime numbers. Diagonal, vertical, and horizontal lines with a high density of prime numbers are clearly visible. The Ulam spiral, or prime spiral (in other languages also called the Ulam Cloth) is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. It was discovered by the mathematician Stanislaw Ulam in 1963, while he was doodling during the presentation of a "long and very boring paper" at a scientific meeting. In an addendum to the Scientific American column, Gardner mentions work of the herpetologist Laurence M. Construction[edit] Ulam constructed the spiral by writing down a regular rectangular grid of numbers, starting with 1 at the center, and spiraling out: He then circled all of the prime numbers and he got the following picture: To his surprise, the circled numbers tended to line up along diagonal lines. Variants[edit]