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  • Uniforme distance between points

    - by Reonarudo
    Hello, How could I, having a path defined by several points that are not in a uniform distance from each other, redefine along the same path the same number of points but with a uniform distance. I'm trying to do this in Objective-C with NSArrays of CGPoints but so far I haven't had any luck with this. Thank you for any help. EDIT I was wondering if it would help to reduce the number of points, like when detecting if 3 points are collinear we could remove the middle one, but I'm not sure that would help.

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  • How to know the line joining two points?

    - by dafero
    I have two points and I want to know the line which is joining them. I don't want to draw the line. I want to create a matrix with all the points which formed the line. In the future, I want to solve if two points belong or not to a shape. And this is the first part.

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  • Uniform distance between points

    - by Reonarudo
    Hello, How could I, having a path defined by several points that are not in a uniform distance from each other, redefine along the same path the same number of points but with a uniform distance. I'm trying to do this in Objective-C with NSArrays of CGPoints but so far I haven't had any luck with this. Thank you for any help. EDIT I was wondering if it would help to reduce the number of points, like when detecting if 3 points are collinear we could remove the middle one, but I'm not sure that would help.

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  • How to represent a list of points in R

    - by Guido
    I am working with a large list of points (each point has three dimensions x,y,z). I am pretty new with R, so I would like to know what is the best way to represent that kind of information. As far as I know, an array allows me to represent any multidimensional data, so currently I am using: > points<-array( c(1,2,0,1,3,0,2,4,0,2,5,0,2,7,0,3,8,0), dim=c(3,6) ) > points [,1] [,2] [,3] [,4] [,5] [,6] [1,] 1 1 2 2 2 3 -- x dim [2,] 2 3 4 5 7 8 -- y dim [3,] 0 0 0 0 0 0 -- z dim The aim is to perform some computations to calculate the euclidean distance between two sets of points (any hint in this sense would also be highly appreciated)

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  • Setting up multiple wireless access points on same network

    - by SqlRyan
    I'd like to add wireless to my network, and I need multiple access points to cover the whole area. I'd like to set them up so that there's only one "wireless network" that the clients see, and it switches them as seamlessly as possible between access points as they wander around (if that's not possible, then at least have it so that they don't need to set up the security by hand on each one the first time, if possible). I've searched online, and there are quite a few sets of mixed instructions (same vs different SSID, frequency, does the security need to match exactly, etc.). Can somebody who has some experience doing this please let me know what they did? I imagine it's pretty simple, but there seems to be no clear cut "yes, you can do this" online, even though I know you can. I have a mid-size LAN with about 20 workstations and two Domain Controllers on it. Also, I'll be doing this with consumer wireless components, if it makes a difference, not enterprise-level components (ie. Linksys rather than Cisco).

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  • Aligning Numbered Bullet Points in Word 2007

    - by FrustratedwithWord
    I am putting together a very large business manual which incorporates numbered headings, steps to follow, diagrams, etc. When using the bullet points, they align perfectly as I work through the processes. However when I include a diagram, or something different from the "norm" of text, the alignment changes. I would like all the bullet points to be aligned in the whole document regardless of where they appear in the document. Is there a way to save the settings so that the bullets always appear in the same position? Currently I am having to reset the indents by dragging the tabs on the ruler. This will be a large document, so I don't want to manually adjust the numbered bullets every time. Help would be greatly appreciated. Thanks very much.

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  • System Restore Points

    - by Ross Lordon
    Currently I am investigating how to schedule an automatic initiation of a system restore point for all of the workstations in my office. It seems that Task Scheduler already has some nice defaults (screenshots below). Even the history for this task verifies that it is running successfully. However, when I go to Recovery in the Control Panel it only lists the System Restore Points one for every previous for only 3 weeks back even if I check the show more restore points box. Why don't the additional ones appear? Would there be a better way to implement a solution, like via group policy or a script? Is there any documentation on this online? I've been having a hard time tracking anything down except how to subvert group policy disabling this feature.

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  • Aligning Numbered Bullet Points in Word 2007

    - by Frustratedwithbullets
    Hello, I am putting together a very large business manual which incorportaes numbered heading, steps to follow, diagrams, etc. When using the bullet points, they align perfectly as I work through the processes. However when I include a diagram, or something different from the "norm" of text, the alignment changes. I would like all the bullets points to be aligned in the whole document regardless of where they appear in the document. Is there a way to save the settings so that the bullets always appear in the same position? Currently I am having to reset the indents by dragging the tabs on the ruler. This will be a large document, so I don't want to manually adjust the numbered bullets every time. Help would be greatly appreciated. Thanks very much.

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  • How do I build a matrix to translate one set of points to another?

    - by dotminic
    I've got 3 points in space that define a triangle. I've also got a vertex buffer made up of three vertices, that also represent a triangle that I will refer to as a "model". How can I can I find the matrix M that will transform vertex in my buffer to those 3 points in space ? For example, let's say my three points A, B, C are at locations: A.x = 10, A.y = 16, A.z = 8 B.x = 12, B.y = 11, B.z = 1 C.x = 19, C.y = 12, C.z = 3 given these coordinates how can I build a matrix that will translate and rotate my model such that both triangles have the exact same world space ? That is, I want the first vertex in my triangle model to have the same coordinates as A, the second to have the same coordinates as B, and same goes for C. nb: I'm using instanced rendering so I can't just give each vertex the same position as my 3 points. I have a set of three points defining a triangle, and only three vertices in my vertex buffer.

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  • Linq to find pair of points with longest length?

    - by Chris
    I have the following code: foreach (Tuple<Point, Point> pair in pointsCollection) { var points = new List<Point>() { pair.Value1, pair.Value2 }; } Within this foreach, I would like to be able to determine which pair of points has the most significant length between the coordinates for each point within the pair. So, let's say that points are made up of the following pairs: (1) var points = new List<Point>() { new Point(0,100), new Point(100,100) }; (2) var points = new List<Point>() { new Point(150,100), new Point(200,100) }; So I have two sets of pairs, mentioned above. They both will plot a horizontal line. I am interested in knowing what the best approach would be to find the pair of points that have the greatest distance between, them, whether it is vertically or horizontally. In the two examples above, the first pair of points has a difference of 100 between the X coordinate, so that would be the point with the most significant difference. But if I have a collection of pairs of points, where some points will plot a vertical line, some points will plot a horizontal line, what would be the best approach for retrieving the pair from the set of points whose difference, again vertically or horizontally, is the greatest among all of the points in the collection? Thanks! Chris

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  • Any reliable polygon normal calculation code?

    - by Jenko
    I'm currently calculating the normal vector of a polygon using this code, but for some faces here and there it calculates a wrong normal. I don't really know what's going on or where it fails but its not reliable. Do you have any polygon normal calculation that's tested and found to be reliable? // calculate normal of a polygon using all points var n:int = points.length; var x:Number = 0; var y:Number = 0; var z:Number = 0 // ensure all points above 0 var minx:Number = 0, miny:Number = 0, minz:Number = 0; for (var p:int = 0, pl:int = points.length; p < pl; p++) { var po:_Point3D = points[p] = points[p].clone(); if (po.x < minx) { minx = po.x; } if (po.y < miny) { miny = po.y; } if (po.z < minz) { minz = po.z; } } for (p = 0; p < pl; p++) { po = points[p]; po.x -= minx; po.y -= miny; po.z -= minz; } var cur:int = 1, prev:int = 0, next:int = 2; for (var i:int = 1; i <= n; i++) { // using Newell method x += points[cur].y * (points[next].z - points[prev].z); y += points[cur].z * (points[next].x - points[prev].x); z += points[cur].x * (points[next].y - points[prev].y); cur = (cur+1) % n; next = (next+1) % n; prev = (prev+1) % n; } // length of the normal var length:Number = Math.sqrt(x * x + y * y + z * z); // turn large values into a unit vector if (length != 0){ x = x / length; y = y / length; z = z / length; }else { throw new Error("Cannot calculate normal since triangle has an area of 0"); }

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  • How to manage mounted partitions (fstab + mount points) from puppet

    - by Cristian Ciupitu
    I want to manage the mounted partitions from puppet which includes both modifying /etc/fstab and creating the directories used as mount points. The mount resource type updates fstab just fine, but using file for creating the mount points is bit tricky. For example, by default the owner of the directory is root and if the root (/) of the mounted partition has another owner, puppet will try to change it and I don't want this. I know that I can set the owner of that directory, but why should I care what's on the mounted partition? All I want to do is mount it. Is there a way to make puppet not to care about the permissions of the directory used as the mount point? This is what I'm using right now: define extra_mount_point( $device, $location = "/mnt", $fstype = "xfs", $owner = "root", $group = "root", $mode = 0755, $seltype = "public_content_t" $options = "ro,relatime,nosuid,nodev,noexec", ) { file { "${location}/${name}": ensure => directory, owner => "${owner}", group => "${group}", mode => $mode, seltype => "${seltype}", } mount { "${location}/${name}": atboot => true, ensure => mounted, device => "${device}", fstype => "${fstype}", options => "${options}", dump => 0, pass => 2, require => File["${location}/${name}"], } } extra_mount_point { "sda3": device => "/dev/sda3", fstype => "xfs", owner => "ciupicri", group => "ciupicri", $options = "relatime,nosuid,nodev,noexec", } In case it matters, I'm using puppet-0.25.4-1.fc13.noarch.rpm and puppet-server-0.25.4-1.fc13.noarch.rpm.

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  • A Taxonomy of Numerical Methods v1

    - by JoshReuben
    Numerical Analysis – When, What, (but not how) Once you understand the Math & know C++, Numerical Methods are basically blocks of iterative & conditional math code. I found the real trick was seeing the forest for the trees – knowing which method to use for which situation. Its pretty easy to get lost in the details – so I’ve tried to organize these methods in a way that I can quickly look this up. I’ve included links to detailed explanations and to C++ code examples. I’ve tried to classify Numerical methods in the following broad categories: Solving Systems of Linear Equations Solving Non-Linear Equations Iteratively Interpolation Curve Fitting Optimization Numerical Differentiation & Integration Solving ODEs Boundary Problems Solving EigenValue problems Enjoy – I did ! Solving Systems of Linear Equations Overview Solve sets of algebraic equations with x unknowns The set is commonly in matrix form Gauss-Jordan Elimination http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination C++: http://www.codekeep.net/snippets/623f1923-e03c-4636-8c92-c9dc7aa0d3c0.aspx Produces solution of the equations & the coefficient matrix Efficient, stable 2 steps: · Forward Elimination – matrix decomposition: reduce set to triangular form (0s below the diagonal) or row echelon form. If degenerate, then there is no solution · Backward Elimination –write the original matrix as the product of ints inverse matrix & its reduced row-echelon matrix à reduce set to row canonical form & use back-substitution to find the solution to the set Elementary ops for matrix decomposition: · Row multiplication · Row switching · Add multiples of rows to other rows Use pivoting to ensure rows are ordered for achieving triangular form LU Decomposition http://en.wikipedia.org/wiki/LU_decomposition C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-lu-decomposition-for-solving.html Represent the matrix as a product of lower & upper triangular matrices A modified version of GJ Elimination Advantage – can easily apply forward & backward elimination to solve triangular matrices Techniques: · Doolittle Method – sets the L matrix diagonal to unity · Crout Method - sets the U matrix diagonal to unity Note: both the L & U matrices share the same unity diagonal & can be stored compactly in the same matrix Gauss-Seidel Iteration http://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method C++: http://www.nr.com/forum/showthread.php?t=722 Transform the linear set of equations into a single equation & then use numerical integration (as integration formulas have Sums, it is implemented iteratively). an optimization of Gauss-Jacobi: 1.5 times faster, requires 0.25 iterations to achieve the same tolerance Solving Non-Linear Equations Iteratively find roots of polynomials – there may be 0, 1 or n solutions for an n order polynomial use iterative techniques Iterative methods · used when there are no known analytical techniques · Requires set functions to be continuous & differentiable · Requires an initial seed value – choice is critical to convergence à conduct multiple runs with different starting points & then select best result · Systematic - iterate until diminishing returns, tolerance or max iteration conditions are met · bracketing techniques will always yield convergent solutions, non-bracketing methods may fail to converge Incremental method if a nonlinear function has opposite signs at 2 ends of a small interval x1 & x2, then there is likely to be a solution in their interval – solutions are detected by evaluating a function over interval steps, for a change in sign, adjusting the step size dynamically. Limitations – can miss closely spaced solutions in large intervals, cannot detect degenerate (coinciding) solutions, limited to functions that cross the x-axis, gives false positives for singularities Fixed point method http://en.wikipedia.org/wiki/Fixed-point_iteration C++: http://books.google.co.il/books?id=weYj75E_t6MC&pg=PA79&lpg=PA79&dq=fixed+point+method++c%2B%2B&source=bl&ots=LQ-5P_taoC&sig=lENUUIYBK53tZtTwNfHLy5PEWDk&hl=en&sa=X&ei=wezDUPW1J5DptQaMsIHQCw&redir_esc=y#v=onepage&q=fixed%20point%20method%20%20c%2B%2B&f=false Algebraically rearrange a solution to isolate a variable then apply incremental method Bisection method http://en.wikipedia.org/wiki/Bisection_method C++: http://numericalcomputing.wordpress.com/category/algorithms/ Bracketed - Select an initial interval, keep bisecting it ad midpoint into sub-intervals and then apply incremental method on smaller & smaller intervals – zoom in Adv: unaffected by function gradient à reliable Disadv: slow convergence False Position Method http://en.wikipedia.org/wiki/False_position_method C++: http://www.dreamincode.net/forums/topic/126100-bisection-and-false-position-methods/ Bracketed - Select an initial interval , & use the relative value of function at interval end points to select next sub-intervals (estimate how far between the end points the solution might be & subdivide based on this) Newton-Raphson method http://en.wikipedia.org/wiki/Newton's_method C++: http://www-users.cselabs.umn.edu/classes/Summer-2012/csci1113/index.php?page=./newt3 Also known as Newton's method Convenient, efficient Not bracketed – only a single initial guess is required to start iteration – requires an analytical expression for the first derivative of the function as input. Evaluates the function & its derivative at each step. Can be extended to the Newton MutiRoot method for solving multiple roots Can be easily applied to an of n-coupled set of non-linear equations – conduct a Taylor Series expansion of a function, dropping terms of order n, rewrite as a Jacobian matrix of PDs & convert to simultaneous linear equations !!! Secant Method http://en.wikipedia.org/wiki/Secant_method C++: http://forum.vcoderz.com/showthread.php?p=205230 Unlike N-R, can estimate first derivative from an initial interval (does not require root to be bracketed) instead of inputting it Since derivative is approximated, may converge slower. Is fast in practice as it does not have to evaluate the derivative at each step. Similar implementation to False Positive method Birge-Vieta Method http://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/polynomial%20methods/bv%20method.html C++: http://books.google.co.il/books?id=cL1boM2uyQwC&pg=SA3-PA51&lpg=SA3-PA51&dq=Birge-Vieta+Method+c%2B%2B&source=bl&ots=QZmnDTK3rC&sig=BPNcHHbpR_DKVoZXrLi4nVXD-gg&hl=en&sa=X&ei=R-_DUK2iNIjzsgbE5ID4Dg&redir_esc=y#v=onepage&q=Birge-Vieta%20Method%20c%2B%2B&f=false combines Horner's method of polynomial evaluation (transforming into lesser degree polynomials that are more computationally efficient to process) with Newton-Raphson to provide a computational speed-up Interpolation Overview Construct new data points for as close as possible fit within range of a discrete set of known points (that were obtained via sampling, experimentation) Use Taylor Series Expansion of a function f(x) around a specific value for x Linear Interpolation http://en.wikipedia.org/wiki/Linear_interpolation C++: http://www.hamaluik.com/?p=289 Straight line between 2 points à concatenate interpolants between each pair of data points Bilinear Interpolation http://en.wikipedia.org/wiki/Bilinear_interpolation C++: http://supercomputingblog.com/graphics/coding-bilinear-interpolation/2/ Extension of the linear function for interpolating functions of 2 variables – perform linear interpolation first in 1 direction, then in another. Used in image processing – e.g. texture mapping filter. Uses 4 vertices to interpolate a value within a unit cell. Lagrange Interpolation http://en.wikipedia.org/wiki/Lagrange_polynomial C++: http://www.codecogs.com/code/maths/approximation/interpolation/lagrange.php For polynomials Requires recomputation for all terms for each distinct x value – can only be applied for small number of nodes Numerically unstable Barycentric Interpolation http://epubs.siam.org/doi/pdf/10.1137/S0036144502417715 C++: http://www.gamedev.net/topic/621445-barycentric-coordinates-c-code-check/ Rearrange the terms in the equation of the Legrange interpolation by defining weight functions that are independent of the interpolated value of x Newton Divided Difference Interpolation http://en.wikipedia.org/wiki/Newton_polynomial C++: http://jee-appy.blogspot.co.il/2011/12/newton-divided-difference-interpolation.html Hermite Divided Differences: Interpolation polynomial approximation for a given set of data points in the NR form - divided differences are used to approximately calculate the various differences. For a given set of 3 data points , fit a quadratic interpolant through the data Bracketed functions allow Newton divided differences to be calculated recursively Difference table Cubic Spline Interpolation http://en.wikipedia.org/wiki/Spline_interpolation C++: https://www.marcusbannerman.co.uk/index.php/home/latestarticles/42-articles/96-cubic-spline-class.html Spline is a piecewise polynomial Provides smoothness – for interpolations with significantly varying data Use weighted coefficients to bend the function to be smooth & its 1st & 2nd derivatives are continuous through the edge points in the interval Curve Fitting A generalization of interpolating whereby given data points may contain noise à the curve does not necessarily pass through all the points Least Squares Fit http://en.wikipedia.org/wiki/Least_squares C++: http://www.ccas.ru/mmes/educat/lab04k/02/least-squares.c Residual – difference between observed value & expected value Model function is often chosen as a linear combination of the specified functions Determines: A) The model instance in which the sum of squared residuals has the least value B) param values for which model best fits data Straight Line Fit Linear correlation between independent variable and dependent variable Linear Regression http://en.wikipedia.org/wiki/Linear_regression C++: http://www.oocities.org/david_swaim/cpp/linregc.htm Special case of statistically exact extrapolation Leverage least squares Given a basis function, the sum of the residuals is determined and the corresponding gradient equation is expressed as a set of normal linear equations in matrix form that can be solved (e.g. using LU Decomposition) Can be weighted - Drop the assumption that all errors have the same significance –-> confidence of accuracy is different for each data point. Fit the function closer to points with higher weights Polynomial Fit - use a polynomial basis function Moving Average http://en.wikipedia.org/wiki/Moving_average C++: http://www.codeproject.com/Articles/17860/A-Simple-Moving-Average-Algorithm Used for smoothing (cancel fluctuations to highlight longer-term trends & cycles), time series data analysis, signal processing filters Replace each data point with average of neighbors. Can be simple (SMA), weighted (WMA), exponential (EMA). Lags behind latest data points – extra weight can be given to more recent data points. Weights can decrease arithmetically or exponentially according to distance from point. Parameters: smoothing factor, period, weight basis Optimization Overview Given function with multiple variables, find Min (or max by minimizing –f(x)) Iterative approach Efficient, but not necessarily reliable Conditions: noisy data, constraints, non-linear models Detection via sign of first derivative - Derivative of saddle points will be 0 Local minima Bisection method Similar method for finding a root for a non-linear equation Start with an interval that contains a minimum Golden Search method http://en.wikipedia.org/wiki/Golden_section_search C++: http://www.codecogs.com/code/maths/optimization/golden.php Bisect intervals according to golden ratio 0.618.. Achieves reduction by evaluating a single function instead of 2 Newton-Raphson Method Brent method http://en.wikipedia.org/wiki/Brent's_method C++: http://people.sc.fsu.edu/~jburkardt/cpp_src/brent/brent.cpp Based on quadratic or parabolic interpolation – if the function is smooth & parabolic near to the minimum, then a parabola fitted through any 3 points should approximate the minima – fails when the 3 points are collinear , in which case the denominator is 0 Simplex Method http://en.wikipedia.org/wiki/Simplex_algorithm C++: http://www.codeguru.com/cpp/article.php/c17505/Simplex-Optimization-Algorithm-and-Implemetation-in-C-Programming.htm Find the global minima of any multi-variable function Direct search – no derivatives required At each step it maintains a non-degenerative simplex – a convex hull of n+1 vertices. Obtains the minimum for a function with n variables by evaluating the function at n-1 points, iteratively replacing the point of worst result with the point of best result, shrinking the multidimensional simplex around the best point. Point replacement involves expanding & contracting the simplex near the worst value point to determine a better replacement point Oscillation can be avoided by choosing the 2nd worst result Restart if it gets stuck Parameters: contraction & expansion factors Simulated Annealing http://en.wikipedia.org/wiki/Simulated_annealing C++: http://code.google.com/p/cppsimulatedannealing/ Analogy to heating & cooling metal to strengthen its structure Stochastic method – apply random permutation search for global minima - Avoid entrapment in local minima via hill climbing Heating schedule - Annealing schedule params: temperature, iterations at each temp, temperature delta Cooling schedule – can be linear, step-wise or exponential Differential Evolution http://en.wikipedia.org/wiki/Differential_evolution C++: http://www.amichel.com/de/doc/html/ More advanced stochastic methods analogous to biological processes: Genetic algorithms, evolution strategies Parallel direct search method against multiple discrete or continuous variables Initial population of variable vectors chosen randomly – if weighted difference vector of 2 vectors yields a lower objective function value then it replaces the comparison vector Many params: #parents, #variables, step size, crossover constant etc Convergence is slow – many more function evaluations than simulated annealing Numerical Differentiation Overview 2 approaches to finite difference methods: · A) approximate function via polynomial interpolation then differentiate · B) Taylor series approximation – additionally provides error estimate Finite Difference methods http://en.wikipedia.org/wiki/Finite_difference_method C++: http://www.wpi.edu/Pubs/ETD/Available/etd-051807-164436/unrestricted/EAMPADU.pdf Find differences between high order derivative values - Approximate differential equations by finite differences at evenly spaced data points Based on forward & backward Taylor series expansion of f(x) about x plus or minus multiples of delta h. Forward / backward difference - the sums of the series contains even derivatives and the difference of the series contains odd derivatives – coupled equations that can be solved. Provide an approximation of the derivative within a O(h^2) accuracy There is also central difference & extended central difference which has a O(h^4) accuracy Richardson Extrapolation http://en.wikipedia.org/wiki/Richardson_extrapolation C++: http://mathscoding.blogspot.co.il/2012/02/introduction-richardson-extrapolation.html A sequence acceleration method applied to finite differences Fast convergence, high accuracy O(h^4) Derivatives via Interpolation Cannot apply Finite Difference method to discrete data points at uneven intervals – so need to approximate the derivative of f(x) using the derivative of the interpolant via 3 point Lagrange Interpolation Note: the higher the order of the derivative, the lower the approximation precision Numerical Integration Estimate finite & infinite integrals of functions More accurate procedure than numerical differentiation Use when it is not possible to obtain an integral of a function analytically or when the function is not given, only the data points are Newton Cotes Methods http://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas C++: http://www.siafoo.net/snippet/324 For equally spaced data points Computationally easy – based on local interpolation of n rectangular strip areas that is piecewise fitted to a polynomial to get the sum total area Evaluate the integrand at n+1 evenly spaced points – approximate definite integral by Sum Weights are derived from Lagrange Basis polynomials Leverage Trapezoidal Rule for default 2nd formulas, Simpson 1/3 Rule for substituting 3 point formulas, Simpson 3/8 Rule for 4 point formulas. For 4 point formulas use Bodes Rule. Higher orders obtain more accurate results Trapezoidal Rule uses simple area, Simpsons Rule replaces the integrand f(x) with a quadratic polynomial p(x) that uses the same values as f(x) for its end points, but adds a midpoint Romberg Integration http://en.wikipedia.org/wiki/Romberg's_method C++: http://code.google.com/p/romberg-integration/downloads/detail?name=romberg.cpp&can=2&q= Combines trapezoidal rule with Richardson Extrapolation Evaluates the integrand at equally spaced points The integrand must have continuous derivatives Each R(n,m) extrapolation uses a higher order integrand polynomial replacement rule (zeroth starts with trapezoidal) à a lower triangular matrix set of equation coefficients where the bottom right term has the most accurate approximation. The process continues until the difference between 2 successive diagonal terms becomes sufficiently small. Gaussian Quadrature http://en.wikipedia.org/wiki/Gaussian_quadrature C++: http://www.alglib.net/integration/gaussianquadratures.php Data points are chosen to yield best possible accuracy – requires fewer evaluations Ability to handle singularities, functions that are difficult to evaluate The integrand can include a weighting function determined by a set of orthogonal polynomials. Points & weights are selected so that the integrand yields the exact integral if f(x) is a polynomial of degree <= 2n+1 Techniques (basically different weighting functions): · Gauss-Legendre Integration w(x)=1 · Gauss-Laguerre Integration w(x)=e^-x · Gauss-Hermite Integration w(x)=e^-x^2 · Gauss-Chebyshev Integration w(x)= 1 / Sqrt(1-x^2) Solving ODEs Use when high order differential equations cannot be solved analytically Evaluated under boundary conditions RK for systems – a high order differential equation can always be transformed into a coupled first order system of equations Euler method http://en.wikipedia.org/wiki/Euler_method C++: http://rosettacode.org/wiki/Euler_method First order Runge–Kutta method. Simple recursive method – given an initial value, calculate derivative deltas. Unstable & not very accurate (O(h) error) – not used in practice A first-order method - the local error (truncation error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size In evolving solution between data points xn & xn+1, only evaluates derivatives at beginning of interval xn à asymmetric at boundaries Higher order Runge Kutta http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods C++: http://www.dreamincode.net/code/snippet1441.htm 2nd & 4th order RK - Introduces parameterized midpoints for more symmetric solutions à accuracy at higher computational cost Adaptive RK – RK-Fehlberg – estimate the truncation at each integration step & automatically adjust the step size to keep error within prescribed limits. At each step 2 approximations are compared – if in disagreement to a specific accuracy, the step size is reduced Boundary Value Problems Where solution of differential equations are located at 2 different values of the independent variable x à more difficult, because cannot just start at point of initial value – there may not be enough starting conditions available at the end points to produce a unique solution An n-order equation will require n boundary conditions – need to determine the missing n-1 conditions which cause the given conditions at the other boundary to be satisfied Shooting Method http://en.wikipedia.org/wiki/Shooting_method C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-shooting-method-for-solving.html Iteratively guess the missing values for one end & integrate, then inspect the discrepancy with the boundary values of the other end to adjust the estimate Given the starting boundary values u1 & u2 which contain the root u, solve u given the false position method (solving the differential equation as an initial value problem via 4th order RK), then use u to solve the differential equations. Finite Difference Method For linear & non-linear systems Higher order derivatives require more computational steps – some combinations for boundary conditions may not work though Improve the accuracy by increasing the number of mesh points Solving EigenValue Problems An eigenvalue can substitute a matrix when doing matrix multiplication à convert matrix multiplication into a polynomial EigenValue For a given set of equations in matrix form, determine what are the solution eigenvalue & eigenvectors Similar Matrices - have same eigenvalues. Use orthogonal similarity transforms to reduce a matrix to diagonal form from which eigenvalue(s) & eigenvectors can be computed iteratively Jacobi method http://en.wikipedia.org/wiki/Jacobi_method C++: http://people.sc.fsu.edu/~jburkardt/classes/acs2_2008/openmp/jacobi/jacobi.html Robust but Computationally intense – use for small matrices < 10x10 Power Iteration http://en.wikipedia.org/wiki/Power_iteration For any given real symmetric matrix, generate the largest single eigenvalue & its eigenvectors Simplest method – does not compute matrix decomposition à suitable for large, sparse matrices Inverse Iteration Variation of power iteration method – generates the smallest eigenvalue from the inverse matrix Rayleigh Method http://en.wikipedia.org/wiki/Rayleigh's_method_of_dimensional_analysis Variation of power iteration method Rayleigh Quotient Method Variation of inverse iteration method Matrix Tri-diagonalization Method Use householder algorithm to reduce an NxN symmetric matrix to a tridiagonal real symmetric matrix vua N-2 orthogonal transforms     Whats Next Outside of Numerical Methods there are lots of different types of algorithms that I’ve learned over the decades: Data Mining – (I covered this briefly in a previous post: http://geekswithblogs.net/JoshReuben/archive/2007/12/31/ssas-dm-algorithms.aspx ) Search & Sort Routing Problem Solving Logical Theorem Proving Planning Probabilistic Reasoning Machine Learning Solvers (eg MIP) Bioinformatics (Sequence Alignment, Protein Folding) Quant Finance (I read Wilmott’s books – interesting) Sooner or later, I’ll cover the above topics as well.

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  • Algorithm to determine which points should be visible on a map based on zoom

    - by lgratian
    Hi! I'm making a Google Maps-like application for a course at my Uni (not something complex, it should load the map of a city for example, not the whole world). The map can have many layers, including markers (restaurants, hospitals, etc.) The problem is that when you have many points and you zoom out the map it doesn't look right. At this zoom level only some points need to be visible (and at the maximum map size, all points). The question is: how can you determine which points should be visible for a specified zoom level? Because I have implemented a PR Quadtree to speed up rendering I thought that I could define some "high-priority" markers (that are always visible, defined in the map editor) and put them in a queue. At each step a marker is removed from the queue and all it's neighbors that are at least D units away (D depends on the zoom levels) are chosen and inserted in the queue, and so on. Is there any better way than the algorithm I thought of? Thanks in advance!

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  • Matlab - Propagate points orthogonally on to the edge of shape boundaries

    - by Graham
    Hi I have a set of points which I want to propagate on to the edge of shape boundary defined by a binary image. The shape boundary is defined by a 1px wide white edge. I also have the coordinates of these points stored in a 2 row by n column matrix. The shape forms a concave boundary with no holes within itself made of around 2500 points. I want to cast a ray from each point from the set of points in an orthogonal direction and detect at which point it intersects the shape boundary at. What would be the best method to do this? Are there some sort of ray tracing algorithms that could be used? Or would it be a case of taking orthogonal unit vector and multiplying it by a scalar and testing after multiplication if the end point of the vector is outside the shape boundary. When the end point of the unit vector is outside the shape, just find the point of intersection? Thank you very much in advance for any help!

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  • How toget a list of "fastest miles" from a set of GPS Points

    - by santiagobasulto
    I'm trying to solve a weird problem. Maybe you guys know of some algorithm that takes care of this. I have data for a cargo freight truck and want to extract some data. Suppose I've got a list of sorted points that I get from the GPS. That's the route for that truck: [ { "lng": "-111.5373066", "lat": "40.7231711", "time": "1970-01-01T00:00:04Z", "elev": "1942.1789265256325" }, { "lng": "-111.5372056", "lat": "40.7228762", "time": "1970-01-01T00:00:07Z", "elev": "1942.109892409177" } ] Now, what I want to get is a list of the "fastest miles". I'll do an example: Given the points: A, B, C, D, E, F the distance from point A to point B is 1 mile, and the cargo took 10:32 minutes. From point B to point D i've got other mile, and the cargo took 10 minutes, etc. So, i need a list sorted by time. Similar to: B -> D: 10 A -> B: 10:32 D -> F: 11:02 Do you know any efficient algorithm that let me calculate that? Thank you all. PS: I'm using Python. EDIT: I've got the distance. I know how to calculate it and there are plenty of posts to do that. What I need is an algorithm to tokenize by mile and get speed from that. Having a distance function is not helpful enough: results = {} for point in points: aux_points = points.takeWhile(point>n) #This doesn't exist, just trying to be simple for aux_point in aux_points: d = distance(point, aux_point) if d == 1_MILE: time_elapsed = time(point, aux_point) results[time_elapsed] = (point, aux_point) I'm still doing some pretty inefficient calculations.

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  • How to calculate the normal of points on a 3D cubic Bézier curve given normals for its start and end points?

    - by Robert
    I'm trying to render a "3D ribbon" using a single 3D cubic Bézier curve to describe it (the width of the ribbon is some constant). The first and last control points have a normal vector associated with them (which are always perpendicular to the tangents at those points, and describe the surface normal of the ribbon at those points), and I'm trying to smoothly interpolate the normal vector over the course of the curve. For example, given a curve which forms the letter 'C', with the first and last control points both having surface normals pointing upwards, the ribbon should start flat, parallel to the ground, slowly turn, and then end flat again, facing the same way as the first control point. To do this "smoothly", it would have to face outwards half-way through the curve. At the moment (for this case), I've only been able to get all the surfaces facing upwards (and not outwards in the middle), which creates an ugly transition in the middle. It's quite hard to explain, I've attached some images below of this example with what it currently looks like (all surfaces facing upwards, sharp flip in the middle) and what it should look like (smooth transition, surfaces slowly rotate round). Silver faces represent the front, black faces the back. Incorrect, what it currently looks like: Correct, what it should look like: All I really need is to be able to calculate this "hybrid normal vector" for any point on the 3D cubic bézier curve, and I can generate the polygons no problem, but I can't work out how to get them to smoothly rotate round as depicted. Completely stuck as to how to proceed!

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  • Forming triangles from points and relations

    - by SiN
    Hello, I want to generate triangles from points and optional relations between them. Not all points form triangles, but many of them do. In the initial structure, I've got a database with the following tables: Nodes(id, value) Relations(id, nodeA, nodeB, value) Triangles(id, relation1_id, relation2_id, relation3_id) In order to generate triangles from both nodes and relations table, I've used the following query: INSERT INTO Triangles SELECT t1.id, t2.id , t3.id, FROM Relations t1, Relations t2, Relations t3 WHERE t1.id < t2.id AND t3.id > t1.id AND ( t1.nodeA = t2.nodeA AND (t3.nodeA = t1.nodeB AND t3.nodeB = t2.nodeB OR t3.nodeA = t2.nodeB AND t3.nodeB = t1.nodeB) OR t1.nodeA = t2.nodeB AND (t3.nodeA = t1.nodeB AND t3.nodeB = t2.nodeA OR t3.nodeA = t2.nodeA AND t3.nodeB = t1.nodeB) ) It's working perfectly on small sized data. (~< 50 points) In some cases however, I've got around 100 points all related to each other which leads to thousands of relations. So when the expected number of triangles is in the hundreds of thousands, or even in the millions, the query might take several hours. My main problem is not in the select query, while I see it execute in Management Studio, the returned results slow. I received around 2000 rows per minute, which is not acceptable for my case. As a matter of fact, the size of operations is being added up exponentionally and that is terribly affecting the performance. I've tried doing it as a LINQ to object from my code, but the performance was even worse. I've also tried using SqlBulkCopy on a reader from C# on the result, also with no luck. So the question is... Any ideas or workarounds?

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  • Problem with Quotas and File Screening on Mount Points in Windows 2008

    - by James P
    Hello, I have a Windows 2008 Server running the File Server Role and I would like to use mount points for my volumes instead of drive letters. However, I need to use the quota and file screening features of File Server Resource Manager, and it seems that they do not apply correctly to mount point folders. I am able to upload oversized files and excluded file types without any warnings. Could someone help me with a fix or workaround for this issue? Thanks, Jamie

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  • adding many points to a personalized google map

    - by Tal Galili
    Hi all, I wish to create a personalized google map (as is shown here), I see it is possible to import a geo file (KML, KMZ or GeoRSS) with many points. I would love to use that but don't know how to create such a geo file in the first place. Which one should I use? What is the best way to create them? Thanks.

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  • Zenoss need to get freespace threshold and alerts on windows "mount points"

    - by agilenoob
    I have a WMI query that will give me all the data I need to do this but I can't figure out how to get this working in Zenoss. I know I need to set data points and a threshold, and optionaly a graph. The problem is examples of how to do this with WMI are few and very confusing. Could anyone atleast point me to documention on how to do this? WMI Query(WQL): "SELECT Caption, Capacity, Freespace FROM Win32_Volume WHERE DriveLetter IS NULL"

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  • Algorithm shortest path between all points

    - by Jeroen
    Hi, suppose I have 10 points. I know the distance between each point. I need to find the shortest possible route passing trough all points. I have tried a couple of algorithms (Dijkstra, Floyd Warshall,...) and the all give me the shortest path between start and end, but they don't make a route with all points on it. Permutations work fine, but they are to resource expensive. What algorithms can you advise me to look into for this problem? Or is there a documented way to do this with the above mentioned algorithms? Tnx Jeroen

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  • Distributing points over a surface within boundries

    - by vise
    I'm interested in a way (algorithm) of distributing a predefined number of points over a 4 sided surface like a square. The main issue is that each point has got to have a minimum and maximum proximity to each other (random between two predefined values). Basically the distance of any two points should not be closer than let's say 2, and a further than 3. My code will be implemented in ruby (the points are locations, the surface is a map), but any ideas or snippets are definitely welcomed as all my ideas include a fair amount of brute force.

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