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  • Neural Network 0 vs -1

    - by Louis
    I have seen a few times people using -1 as opposed to 0 when working with neural networks for the input data. How is this better and does it effect any of the mathematics to implement it? Edit: Using feedforward and back prop Edit 2: I gave it a go but the network stopped learning so I assume the maths would have to change somewhere?

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  • How to calculate the cycles that change one permutation into another?

    - by fortran
    Hi, I'm looking for an algorithm that given two permutations of a sequence (e.g. [2, 3, 1, 4] and [4, 1, 3, 2]) calculates the cycles that are needed to convert the first into the second (for the example, [[0, 3], [1, 2]]). The link from mathworld says that Mathematica's ToCycle function does that, but sadly I don't have any Mathematica license at hand... I'd gladly receive any pointer to an implementation of the algorithm in any FOSS language or mathematics package. Thanks!

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  • Better algorithm for estimating download time

    - by Scott Smith
    We've all seen the download time running estimate that initially says something like "7 days", but keeps dropping wildly (e.g. "23 hours", "45 minutes", "1 min. 50 sec", etc) with each successive estimation as the chunks are downloaded. To avoid these initial (alarming) estimates, there are techniques one could try like suppressing display of the first n estimates, or waiting for the delta between estimates to drop below some threshold before you start displaying them, but these don't seem like a general, robust solution. There are corner cases involving too few samples, or samples that actually are wildly varying... I think I recall a general solution for this kind of thing in mathematics (statistics?) that reduced or eliminated these wild values. Does anyone know?

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  • Simple encryption - Sum of Hashes in C

    - by Dogbert
    I am attempting to demonstrate a simple proof of concept with respect to a vulnerability in a piece of code in a game written in C. Let's say that we want to validate a character login. The login is handled by the user choosing n items, (let's just assume n=5 for now) from a graphical menu. The items are all medieval themed: eg: _______________________________ | | | | | Bow | Sword | Staff | |-----------|-----------|-------| | Shield | Potion | Gold | |___________|___________|_______| The user must click on each item, then choose a number for each item. The validation algorithm then does the following: Determines which items were selected Drops each string to lowercase (ie: Bow becomes bow, etc) Calculates a simple string hash for each string (ie: `bow = b=2, o=15, w=23, sum = (2+15+23=40) Multiplies the hash by the value the user selected for the corresponding item; This new value is called the key Sums together the keys for each of the selected items; this is the final validation hash IMPORTANT: The validator will accept this hash, along with non-zero multiples of it (ie: if the final hash equals 1111, then 2222, 3333, 8888, etc are also valid). So, for example, let's say I select: Bow (1) Sword (2) Staff (10) Shield (1) Potion (6) The algorithm drops each of these strings to lowercase, calculates their string hashes, multiplies that hash by the number selected for each string, then sums these keys together. eg: Final_Validation_Hash = 1*HASH(Bow) + 2*HASH(Sword) + 10*HASH(Staff) + 1*HASH(Shield) + 6*HASH(Potion) By application of Euler's Method, I plan to demonstrate that these hashes are not unique, and want to devise a simple application to prove it. in my case, for 5 items, I would essentially be trying to calculate: (B)(y) = (A_1)(x_1) + (A_2)(x_2) + (A_3)(x_3) + (A_4)(x_4) + (A_5)(x_5) Where: B is arbitrary A_j are the selected coefficients/values for each string/category x_j are the hash values for each string/category y is the final validation hash (eg: 1111 above) B,y,A_j,x_j are all discrete-valued, positive, and non-zero (ie: natural numbers) Can someone either assist me in solving this problem or point me to a similar example (ie: code, worked out equations, etc)? I just need to solve the final step (ie: (B)(Y) = ...). Thank you all in advance.

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  • Create a model that switches between two different states using Temporal Logic?

    - by NLed
    Im trying to design a model that can manage different requests for different water sources. Platform : MAC OSX, using latest Python with TuLip module installed. For example, Definitions : Two water sources : w1 and w2 3 different requests : r1,r2,and r3 - Specifications : Water 1 (w1) is preferred, but w2 will be used if w1 unavailable. Water 2 is only used if w1 is depleted. r1 has the maximum priority. If all entities request simultaneously, r1's supply must not fall below 50%. - The water sources are not discrete but rather continuous, this will increase the difficulty of creating the model. I can do a crude discretization for the water levels but I prefer finding a model for the continuous state first. So how do I start doing that ? Some of my thoughts : Create a matrix W where w1,w2 ? W Create a matrix R where r1,r2,r3 ? R or leave all variables singular without putting them in a matrix I'm not an expert in coding so that's why I need help. Not sure what is the best way to start tackling this problem. I am only interested in the model, or a code sample of how can this be put together. edit Now imagine I do a crude discretization of the water sources to have w1=[0...4] and w2=[0...4] for 0, 25, 50, 75,100 percent respectively. == means implies Usage of water sources : if w1[0]==w2[4] -- meaning if water source 1 has 0%, then use 100% of water source 2 etc if w1[1]==w2[3] if w1[2]==w2[2] if w1[3]==w2[1] if w1[4]==w2[0] r1=r2=r3=[0,1] -- 0 means request OFF and 1 means request ON Now what model can be designed that will give each request 100% water depending on the values of w1 and w2 (w1 and w2 values are uncontrollable so cannot define specific value, but 0...4 is used for simplicity )

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  • Should I Teach My Son Programming? What approaches should I take? [closed]

    - by DaveDev
    I was wondering if it's a good idea to teach object oriented programming to my son? I was never really good at math as a kid, but I think since I've started programming it's given me a greater ability to understand math by being better able to visualise relationships between abstract models. I thought it might give him a better advantage in learning & applying logical & mathematical concepts throughout his life if he was able to take advantage of the tools available to programmers. what would be the best programming fields, techniques and/or concepts? What approach should I take? what concepts should I avoid? what fields of mathematics would he find this benfits him most? He's only 2 now so it wouldn't be for another few years before I do this, (and even at that, only from a very high level point of view). I thought I'd put it to the programming community and see what you guys thought? Possible Duplicate: What are some recommended programming resources for pre-teens?

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  • Runge-Kutta (RK4) integration for game physics

    - by Kai
    Gaffer on Games has a great article about using RK4 integration for better game physics. The implementation is straightforward but the math behind it confuses me. I understand derivatives and integrals on a conceptual level but I haven't manipulated equations in a long time. Here's the brunt of Gaffer's implementation: void integrate(State &state, float t, float dt) { Derivative a = evaluate(state, t, 0.0f, Derivative()); Derivative b = evaluate(state, t+dt*0.5f, dt*0.5f, a); Derivative c = evaluate(state, t+dt*0.5f, dt*0.5f, b); Derivative d = evaluate(state, t+dt, dt, c); const float dxdt = 1.0f/6.0f * (a.dx + 2.0f*(b.dx + c.dx) + d.dx); const float dvdt = 1.0f/6.0f * (a.dv + 2.0f*(b.dv + c.dv) + d.dv) state.x = state.x + dxdt * dt; state.v = state.v + dvdt * dt; } Can anybody explain in simple terms how RK4 works? Specifically, why are we averaging the derivatives at 0.0f, 0.5f, 0.5f, and 1.0f? How is averaging derivatives up to the 4th order different from doing a simple euler integration with a smaller timestep? After reading the accepted answer below, and several other articles, I have a grasp on how RK4 works. To answer my own questions: Can anybody explain in simple terms how RK4 works? RK4 takes advantage of the fact that we can get a much better approximation of a function if we use its higher-order derivatives rather than just the first or second derivative. That's why the Taylor series converges much faster than Euler approximations. (take a look at the animation on the right side of that page) Specifically, why are we averaging the derivatives at 0.0f, 0.5f, 0.5f, and 1.0f? The Runge-Kutta method is an approximation of a function that samples derivatives of several points within a timestep, unlike the Taylor series which only samples derivatives of a single point. After sampling these derivatives we need to know how to weigh each sample to get the closest approximation possible. An easy way to do this is to pick constants that coincide with the Taylor series, which is how the constants of a Runge-Kutta equation are determined. This article made it clearer for me: http://web.mit.edu/10.001/Web/Course%5FNotes/Differential%5FEquations%5FNotes/node5.html. Notice how (15) is the Taylor series expansion while (17) is the Runge-Kutta derivation. How is averaging derivatives up to the 4th order different from doing a simple euler integration with a smaller timestep? Mathematically it converges much faster than doing many Euler approximations. Of course, with enough Euler approximations we can gain equal accuracy to RK4, but the computational power needed doesn't justify using Euler.

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  • Find the "largest" dense sub matrix in a large sparse matrix

    - by BCS
    Given a large sparse matrix (say 10k+ by 1M+) I need to find a subset, not necessarily continuous, of the rows and columns that form a dense matrix (all non-zero elements). I want this sub matrix to be as large as possible (not the largest sum, but the largest number of elements) within some aspect ratio constraints. Are there any known exact or aproxamate solutions to this problem? A quick scan on Google seems to give a lot of close-but-not-exactly results. What terms should I be looking for? edit: Just to clarify; the sub matrix need not be continuous. In fact the row and column order is completely arbitrary so adjacency is completely irrelevant. A thought based on Chad Okere's idea Order the rows from largest count to smallest count (not necessary but might help perf) Select two rows that have a "large" overlap Add all other rows that won't reduce the overlap Record that set Add whatever row reduces the overlap by the least Repeat at #3 until the result gets to small Start over at #2 with a different starting pair Continue until you decide the result is good enough

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  • How would you solve this graph theory handshake problem in python?

    - by Zachary Burt
    I graduated college last year with a degree in Psychology, but I also took a lot of math for fun. I recently got the book "Introductory Graph Theory" by Gary Chartrand to brush up on my math and have some fun. Here is an exercise from the book that I'm finding particularly befuddling: Suppose you and your husband attended a party with three other married couples. Several handshakes took place. No one shook hands with himself (or herself) or with his (or her) spouse, and no one shook hands with the same person more than once. After all the handshaking was completed, suppose you asked each person, including your husband, how many hands he or she had shaken. Each person gave a different answer. a) How many hands did you shake? b) How many hands did your husband shake? Now, I've been reasoning about this for a while, and trying to draw sample graphs that could illustrate a solution, but I'm coming up empty-handed. My logic is this: there are 8 different vertices in the graph, and 7 of them have different degrees. The values for the degrees must therefore be 0, 1, 2, 3, 4, 5, 6, and x. The # of degrees for one married couple is (0, 6). Since all graphs have an even number of odd vertices, x must be either 5, 3, or 1. What's your solution to this problem? And, if you could solve it in python, how would you do it? (python is fun.) Cheers.

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  • How would you solve this graph theory handshake problem in python?

    - by Zachary Burt
    I graduated college last year with a degree in Psychology, but I also took a lot of math for fun. I recently got the book "Introductory Graph Theory" by Gary Chartrand to brush up on my math and have some fun. Here is an exercise from the book that I'm finding particularly befuddling: Suppose you and your husband attended a party with three other married couples. Several handshakes took place. No one shook hands with himself (or herself) or with his (or her) spouse, and no one shook hands with the same person more than once. After all the handshaking was completed, suppose you asked each person, including your husband, how many hands he or she had shaken. Each person gave a different answer. a) How many hands did you shake? b) How many hands did your husband shake? Now, I've been reasoning about this for a while, and trying to draw sample graphs that could illustrate a solution, but I'm coming up empty-handed. My logic is this: there are 8 different vertices in the graph, and 7 of them have different degrees. The values for the degrees must therefore be 0, 1, 2, 3, 4, 5, 6, and x. The # of degrees for one married couple is (0, 6). Since all graphs have an even number of odd vertices, x must be either 5, 3, or 1. What's your solution to this problem? And, if you could solve it in python, how would you do it? (python is fun.) Cheers.

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  • naive bayesian spam filter question

    - by Microkernel
    Hi guys, I am planning to implement spam filter using Naive Bayesian classification model. Online I see a lot of info on Naive Bayesian classification, but the problem is its a lot of mathematical stuff, than clearly stating how its done. And the problem is I am more of a programmer than a mathematician (yes I had learnt Probability and Bayesian theorem back in school, but out of touch for a long long time, and I don't have luxury of learning it now (Have nearly 3 weeks to come-up with a working prototype)). So if someone can explain or point me to location where its explained for programmers than a mathematician, it would be a great help. PS: By the way I have to implement it in C, if you want to know. :( Regards, Microkernel

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  • Maths Question: number of different permutations

    - by KingCong
    This is more of a maths question than programming but I figure a lot of people here are pretty good at maths! :) My question is: Given a 9 x 9 grid (81 cells) that must contain the numbers 1 to 9 each exactly 9 times, how many different grids can be produced. The order of the numbers doesn't matter, for example the first row could contain nine 1's etc. This is related to Sudoku and we know the number of valid Sudoku grids is 6.67×10^21, so since my problem isn't constrained like Sudoku by having to have each of the 9 numbers in each row, column and box then the answer should be greater than 6.67×10^21. My first thought was that the answer is 81! however on further reflection this assume that the 81 number possible for each cell are different, distinct number. They are not, there are 81 possible numbers for each cell but only 9 possible different numbers. My next thought was then that each of the cells in the first row can be any number between 1 and 9. If by chance the first row happened to be all the same number, say all 1s, then each cell in the second row could only have 8 possibilites, 2-9. If this continued down until the last row then number of different permutations could be calculated by 9^2 * 8^2 * 7^2 ..... * 1^2. However this doesn't work if each row doesn't contain 9 of the same number. It's been quite a while since I studied this stuff and I can't think of a way to work it out, I'd appreciate any help anyone can offer.

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  • Ruby Programming Techniques: simple yet not so simple object manipulation

    - by Shyam
    Hi, I want to create an object, let's say a Pie. class Pie def initialize(name, flavor) @name = name @flavor = flavor end end But a Pie can be divided in 8 pieces, a half or just a whole Pie. For the sake of argument, I would like to know how I could give each Pie object a price per 1/8, 1/4 or per whole. I could do this by doing: class Pie def initialize(name, flavor, price_all, price_half, price_piece) @name = name @flavor = flavor @price_all = price_all @price_half = price_half @price_piece = price_piece end end But now, if I would create fifteen Pie objects, and I would take out randomly some pieces somewhere by using a method such as getPieceOfPie(pie_name) How would I be able to generate the value of all the available pies that are whole and the remaining pieces? Eventually using a method such as: myCurrentInventoryHas(pie_name) # output: 2 whole strawberry pies and 7 pieces. I know, I am a Ruby nuby. Thank you for your answers, comments and help!

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  • Static analysis of multiple if statements (conditions)

    - by koppernickus
    I have code similar to: if conditionA(x, y, z) then doA() else if conditionB(x, y, z) then doB() ... else if conditionZ(x, y, z) then doZ() else throw ShouldNeverHappenException I would like to validate two things (using static analysis): If all conditions conditionA, conditionB, ..., conditionZ are mutually exclusive (i.e. it is not possible that two or more conditions are true in the same time). All possible cases are covered, i.e. "else throw" statement will never be called. Could you recommend me a tool and/or a way I could (easily) do this? I would appreciate more detailed informations than "use Prolog" or "use Mathematica"... ;-)

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  • Fractional to Decimal Form.

    - by ThePower
    Hi, there probably isn't an answer to this apart from "Create it yourself", but you never know, there might be some string representation for this. Basically, I would like to display number values as fractional instead of decimal when displaying the values as a string. Instead of a value displaying as: 1.1428571428571428571428571428571 I would prefer it to display as 8/7 Is there any way of doing this without writing the functionality myself? Regards Lloyd

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  • Encode complex number as RGB pixel

    - by Vi
    How is it better to encode a complex number into RGB pixel and vice versa? Probably (logarithm of) an absolute value goes to brightness and an argument goes to hue. Desaturated pixes should receive randomized argument in reverse transformation. Something like: 0 - (0,0,0) 1 - (255,0,0) -1 - (0,255,255) 0.5 - (128,0,0) i - (255,255,0) -i - (255,0,255) (0,0,0) - 0 (255,255,255) - e^(i * random) (128,128,128) - 0.5 * e^(i *random) (0,128,128) - -0.5 Are there ready-made formulas for that?

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  • Scared of Calculus - Required to pass Differential Calculus as part of my Computer science major

    - by ke3pup
    Hi guys I'm finishing my Computer science degree in university but my fear of maths (lack of background knowledge) made me to leave all my maths units til' the very end which is now. i either take them on and pass or have to give up. I've passed all my programming units easily but knowing my poor maths skills won't do i've been staying clear of the maths units. I have to pass Differential Calculus and Linear Algebra first. With a help of book named "Linear Algebra: A Modern Introduction" i'm finding myself on track and i think i can pass the Linear Algebra unit. But with differential calculus i can't find a book to help me. They're either too advanced or just too simple for what i have to learn. The things i'm required to know for this units are: Set notation, the real number line, Complex numbers in cartesian form. Complex plane, modulus. Complex numbers in polar form. De Moivre’s Theorem. Complex powers and nth roots. Definition of ei? and ez for z complex. Applications to trigonometry. Revision of domain and range of a function Working in R3. Curves and surfaces. Functions of 2 variables. Level curves.Partial derivatives and tangent planes. The derivative as a difference quotient. Geometric significance of the derivative. Discussion of limit. Higher order partial derivatives. Limits of f(x,y). Continuity. Maxima and minima of f(x,y). The chain rule. Implicit differentiation. Directional derivatives and the gradient. Limit laws, l’Hoˆpital’s rule, composition law. Definition of sinh and cosh and their inverses. Taylor polynomials. The remainder term. Taylor series. Is there a book to help me get on track with the above? Being a student i can't buy too many books hence why i'm looking for a book that covers topics I need to know. The University library has a fairly limited collection which i took as loan but didn't find useful as it was too complex.

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  • Python, dictionaries, and chi-square contingency table

    - by rohanbk
    I have a file which contains several lines in the following format (word, time that the word occurred in, and frequency of documents containing the given word within the given instance in time): #inputfile <word, time, frequency> apple, 1, 3 banana, 1, 2 apple, 2, 1 banana, 2, 4 orange, 3, 1 I have Python class below that I used to create 2-D dictionaries to store the above file using as the key, and frequency as the value: class Ddict(dict): ''' 2D dictionary class ''' def __init__(self, default=None): self.default = default def __getitem__(self, key): if not self.has_key(key): self[key] = self.default() return dict.__getitem__(self, key) wordtime=Ddict(dict) # Store each inputfile entry with a <word,time> key timeword=Ddict(dict) # Store each inputfile entry with a <time,word> key # Loop over every line of the inputfile for line in open('inputfile'): word,time,count=line.split(',') # If <word,time> already a key, increment count try: wordtime[word][time]+=count # Otherwise, create the key except KeyError: wordtime[word][time]=count # If <time,word> already a key, increment count try: timeword[time][word]+=count # Otherwise, create the key except KeyError: timeword[time][word]=count The question that I have pertains to calculating certain things while iterating over the entries in this 2D dictionary. For each word 'w' at each time 't', calculate: The number of documents with word 'w' within time 't'. (a) The number of documents without word 'w' within time 't'. (b) The number of documents with word 'w' outside time 't'. (c) The number of documents without word 'w' outside time 't'. (d) Each of the items above represents one of the cells of a chi-square contingency table for each word and time. Can all of these be calculated within a single loop or do they need to be done one at a time? Ideally, I would like the output to be what's below, where a,b,c,d are all the items calculated above: print "%s, %s, %s, %s" %(a,b,c,d)

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  • Rotation Interpolation

    - by Rob
    Hello, NB: I'll present this question in degrees purely for simplicity, radians, degrees, different zero-bearing, the problem is essentially the same. Does anyone have any ideas on the code behind rotational interpolation? Given a linear interpolation function: Lerp(from, to, amount), where amount is 0...1 which returns a value between from and to, by amount. How could I apply this same function to a rotational interpolation between 0 and 360 degrees? Given that degrees should not be returned outside 0 and 360. Given this unit circle for degrees: where from = 45 and to = 315, the algorithm should take the shortest path to the angle, i.e. it should go through zero, to 360 and then to 315 - and not all the way round 90, 180, 270 to 315. Is there a nice way to achieve this? Or is it going to just be a horrid mess of if() blocks? Am I missing some well understood standard way of doing this? Any help would be appreciated.

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  • Calculating negative fractions in Objective C

    - by Mark Reid
    I've been coding my way through Steve Kochan's Programming in Objective-C 2.0 book. I'm up to an exercise in chapter 7, ex 4, in case anyone has the book. The question posed by the exercise it will the Fraction class written work with negative fractions such as -1/2 + -2/3? Here's the implementation code in question - @implementation Fraction @synthesize numerator, denominator; -(void) print { NSLog(@"%i/%i", numerator, denominator); } -(void) setTo: (int) n over: (int) d { numerator = n; denominator = d; } -(double) convertToNum { if (denominator != 0) return (double) numerator / denominator; else return 1.0; } -(Fraction *) add: (Fraction *) f { // To add two fractions: // a/b + c/d = ((a * d) + (b * c)) / (b * d) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = (numerator * f.denominator) + (denominator * f.numerator); resultDenom = denominator * f.denominator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(Fraction *) subtract: (Fraction *) f { // To subtract two fractions: // a/b - c/d = ((a * d) - (b * c)) / (b * d) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = numerator * f.denominator - denominator * f.numerator; resultDenom = denominator * f.denominator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(Fraction *) multiply: (Fraction *) f { // To multiply two fractions // a/b * c/d = (a*c) / (b*d) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = numerator * f.numerator; resultDenom = denominator * f.denominator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(Fraction *) divide: (Fraction *) f { // To divide two fractions // a/b / c/d = (a*d) / (b*c) // result will store the result of the addition Fraction *result = [[Fraction alloc] init]; int resultNum, resultDenom; resultNum = numerator * f.denominator; resultDenom = denominator * f.numerator; [result setTo: resultNum over: resultDenom]; [result reduce]; return result; } -(void) reduce { int u = numerator; int v = denominator; int temp; while (v != 0) { temp = u % v; u = v; v = temp; } numerator /= u; denominator /= u; } @end My question to you is will it work with negative fractions and can you explain how you know? Part of the issue is I don't know how to calculate negative fractions myself so I'm not too sure how to know. Many thanks.

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  • Is it possible to implement bitwise operators using integer arithmetic?

    - by Statement
    Hello World! I am facing a rather peculiar problem. I am working on a compiler for an architecture that doesn't support bitwise operations. However, it handles signed 16 bit integer arithmetics and I was wondering if it would be possible to implement bitwise operations using only: Addition (c = a + b) Subtraction (c = a - b) Division (c = a / b) Multiplication (c = a * b) Modulus (c = a % b) Minimum (c = min(a, b)) Maximum (c = max(a, b)) Comparisons (c = (a < b), c = (a == b), c = (a <= b), et.c.) Jumps (goto, for, et.c.) The bitwise operations I want to be able to support are: Or (c = a | b) And (c = a & b) Xor (c = a ^ b) Left Shift (c = a << b) Right Shift (c = a b) (All integers are signed so this is a problem) Signed Shift (c = a b) One's Complement (a = ~b) (Already found a solution, see below) Normally the problem is the other way around; how to achieve arithmetic optimizations using bitwise hacks. However not in this case. Writable memory is very scarce on this architecture, hence the need for bitwise operations. The bitwise functions themselves should not use a lot of temporary variables. However, constant read-only data & instruction memory is abundant. A side note here also is that jumps and branches are not expensive and all data is readily cached. Jumps cost half the cycles as arithmetic (including load/store) instructions do. On other words, all of the above supported functions cost twice the cycles of a single jump. Some thoughts that might help: I figured out that you can do one's complement (negate bits) with the following code: // Bitwise one's complement b = ~a; // Arithmetic one's complement b = -1 - a; I also remember the old shift hack when dividing with a power of two so the bitwise shift can be expressed as: // Bitwise left shift b = a << 4; // Arithmetic left shift b = a * 16; // 2^4 = 16 // Signed right shift b = a >>> 4; // Arithmetic right shift b = a / 16; For the rest of the bitwise operations I am slightly clueless. I wish the architects of this architecture would have supplied bit-operations. I would also like to know if there is a fast/easy way of computing the power of two (for shift operations) without using a memory data table. A naive solution would be to jump into a field of multiplications: b = 1; switch (a) { case 15: b = b * 2; case 14: b = b * 2; // ... exploting fallthrough (instruction memory is magnitudes larger) case 2: b = b * 2; case 1: b = b * 2; } Or a Set & Jump approach: switch (a) { case 15: b = 32768; break; case 14: b = 16384; break; // ... exploiting the fact that a jump is faster than one additional mul // at the cost of doubling the instruction memory footprint. case 2: b = 4; break; case 1: b = 2; break; }

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  • How to convert closed bezier curves to Bitmaps?

    - by Sorush Rabiee
    I need an algorithm to convert a closed bezier curve (perhaps self-crossing) to a binary bitmap: 0 for inside pixels and 1 for outside. I'm writing a code that needs to implement some operations on bezier curves, could anybody give me some resources or tutorials about beziere? Wikipedia and others didn't say anything about optimization, subtracting, union, knot insertion and deletion and other operations :-)

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  • Mapping a BigInteger to a circle

    - by Martin
    I have a C# system using 160 bit numbers, stored in a BigInteger. I want to display these things on a circle, which means mapping the 0-2^160 range into the 0-2Pi range. How would I do this? The approach that jumps instantly to mind is angle = (number / pow(2, 160)) * TwoPi; However, that has complexities because the division will truncate the result into an integer.

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