DOCSLIB.ORG

Camera Based Equation Solver for Android Devices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Camera Based Equation Solver for Android Devices Camera Based Equation Solver for Android Devices Aman Sikka Benny Wu Department of Electrical Engineering Department of Applied Physics Stanford University Stanford University Stanford, California 94305 Stanford, California 94305 Email: [email protected] Email: [email protected] Abstract—In this paper we demonstrate an Android based detected equation, prompts for confirmation, and displays the mobile application equation solver that takes a camera image of solution. The second is a server running the image processing, an equation and displays the solution. The program is capable of text recognition, and equation solving algorithms. The overall solving simple arithmetic equations (addition, subtraction, and multiplication) and systems of linear equations of up to two system flow is shown in Figure 1. variables written by hand or in computer font. I. INTRODUCTION Android Phone Server Solve Segment Correct OCR: Mathematical equations are a ubiquitous part of student life. Binarize Printed Line Skew Tesseract Camera Generally, one has to make use of a calculator or computer to Image Solve Hand Segment OCR: Binarize solve complex equations. Depending on the assistive device, Written Char SVM the equations needs to be entered in a specific format. Our goal Recognized Text is to develop a mobile based application that bridges the gap Confirm / Edit MATLAB between technology and the traditional pen and paper approach Solver in an intuitive manner. The user can capture a camera image Solution of a mathematical expression - both in computer generated font or in handwritten text, and a solution will be calculated and displayed on the mobile device. Fig. 1. Overall system flow showing processes running on the Android Initially, we limit the scope of the task to solving simple platform and on the server. arithmetic equations, quadratic equations, and systems of linear equations of up to two variables. Mobile applications III. IMAGE CAPTURE for solving such equations do exist. However, these apps are essentially templates for pre-defined equations where users Within the mobile application, the user is given two options enter numbers in static text boxes. While the constrained - ”Solve Printed” or ”Solve Hand Written” which capture an interface ensures robust capture of the input, it is cumbersome image and initiate two different PHP scripts on the server. The for the user, who must click multiple times to input an image is downsized to 640 x 480 pixels from its original size equation. A more intuitive approach would be for the user to speed up transmission to the server. to snap a picture of the written expression. IV. BINARIZATION AND LINE SEGMENTATION II. SYSTEM OVERVIEW Binarization and segmentation are crucial steps in identify- The general approach for solving an equation contained in ing the regions of interests. Otsu’s adaptive thresholding was an image is as follows: initially used. However, the processing time was fairly slow and the binarization was unsatisfactory for certain lighting 1) Capture image. conditions. Instead, the image is binarized using Maximally 2) Binarize image. Stable Extremal Regions (MSER) from the VL FEAT toolbox 3) Segment expressions. [2]. Regions that are either too small or too large are excluded 4) Correct perspective distortion. from the candidate list. Regions in the binarized image are 5) Recognize text. labeled, and their centroids and bounding boxes calculated. 6) Prompt for user confirmation. Individual lines (a single equation or expression) are seg- 7) Solve equation. mented out by plotting the y-centroids of the regions in a Due to limitations in the text detection accuracy, a prompt histogram with a bin size of 20 pixels, as shown in Figure for confirmation from the user was added between steps 5 and 2(b). The average y-centroid of each line is assumed to be the 7 to ensure that the correct equation is passed to the solver. maximum of the well-separated peaks. The midpoint between The overall system consists of two subsystems. The first is an each line is then calculated and used to segment the binarized Android application that captures the input image, displays the image into images of individual lines. (a) (b) Histogram of Y-Centroid used to correct for the perspective distortion. The resulting image after correction, as shown in 3(d), is then passed to the 6 Tesseract OCR engine. Y-Centroid Y-Centroid of Line 1 of Line 2 4 (a) (c) Detected Counts in Bin Counts 2 RANSAC Fit Desired RANSAC Fit for Skew Estimate (b) 240 0 0 100 200 300 400 500 260 (c) Y-Centroid of Region 280 300 Y Centroid Position Y Centroid (d) Baseline 320 Topline RANSAC Fit Baseline RANSAC Fit Topline 340 200 250 300 350 400 450 X Centroid Position Fig. 2. (a) Camera image taken by Android phone. (b) After binarization with MSER and region labeling, a histogram of the y-centroid of the regions is Fig. 3. (a) Segmented line image showing perspective distortion. (b) plotted. Regions from separate expressions form distinct peaks. (c) Using the RANSAC fits to the topline and the baseline. (c) Top: The detected, distorted histogram results, the binarized image is segmented into separate expressions. bounding box. Bottom: The desired, corrected bounding box. (d) Corrected image after projective affine transformation is applied to the distorted text. V. TEXT RECOGNITION Text recognition is divided into two separate classes - B. Handwritten Text Recognition computer printed expressions and hand written expressions. Though Tesseract performed well for printed text, with a Computer printed expressions are processed with Tesseract [3], detection rate better than 85% for fonts within its database. a free Optical Character Recognition (OCR) engine originally However, its detection rates for handwritten text is below developed by HP Labs and now maintained by Google. 50%, likely due to size variations in the writing and a lack of Handwritten expressions are processes with a Support-Vector matching fonts in its database. A machine learning algorithm Machine (SVM) prediction model. based on support vector machines (SVM) was applied instead. In order to improve detection acuracy the number of charac- SVM is a supervised learning method that analyzes data ters in the database are limited to digits (0 - 9), two variables and recognizes patterns. It is often used for classification (a, b), and a subset of mathematical operators (+,-, x, and =). and regression analysis. The prediction model was created These characters are sufficient for composing both quadratic using libSVM, a set of tools for training and modeling SVM equations and the majority of a system of linear equations developed at National Taiwan University [6]. in two variables. The dictionary can be easily expanded to In order to create a training dataset, character images must include more characters in the future. first be converted into vector form. After line segmentation, region labels are used to determine the bounding box for A. Computer Text Recognition individual characters. A small amount of padding is added to Recognizing computer generated text is considerably easier the border, as shown in Figure 4(a). The segmented character than recognizing handwritten text. Due to the predictable is downsampled to 32x32 pixels and further divided into 64 structure (font style) of the characters, a template matching 4x4 regions, with region 1 at the top left and region 64 at algorithm can be used. The free OCR engine Tesseract was the bottom right. The count in each region is the determined chosen for printed text recognition. vector value, as shown in Figure 4(b). This conversion thus As demonstrated in [4], the accuracy of optical text- results in a 64 dimensional vector for each character image. recognition algorithms improves significantly if perspective distortion is first corrected. Following a similar method to (a) (b) 0 1 6 8 [4], the topline and baseline of the regions in the segmented 0 8 16 14 expression are found. The segmented line image is shown in 0 8 12 3 3(a). The RANSAC MATLAB Toolbox from the Graphics 0 8 16 16 and Media Lab of Lomonosov Moscow State University [5] is used to compute RANdom SAmple Consensus (RANSAC) fits to the topline and baseline as shown in 3(b). A bounding box for the distorted text is found, and the four corner points 0 identified. Using the leading edge of this bounding box, the Fig. 4. (a) Segmented character from input image. (b) Character downsam- desired, corrected bounding box is then calculated, as shown pled to 32 x 32 pixels. The image is then further divided into 64 4x4 regions, in 3(c). A projective affine transformation that maps the four with region 1 at the top left and region 64 at the bottom right. The count in corner points of the distorted box to the desired box is then each region is the vector value. Initially, the Optical Recognition of Handwritten Digits Data more accurate than segmenting and detecting individual char- Set from the Machine Learning Repository at the University acters. of California at Irvine [7] was used as the training set. The Tesseract performs well for fonts that exist in its database, 64 dimensional vector conversion of this data set was kindly with an accuracy in the range of 85-90%. The notable excep- provided by Sam Tsai. After training libSVM, a test of our tions were the characters ’0’,’ -’, ’=’, and ’a’. The character ’0’ handwritten digits resulted in a low prediction rate of between was at times mistakenly identified as ’()’ (prior to the removal 20-30%. We thus decided to create a new training data set of ’(’and ’)’ from the dictionary). The character ’a’ was at based on our own handwriting. A simple application for times mistakenly identified as ’3’or ’8’. The characters ’-’ and taking a camera image of a single character, segmenting and ’=’ were at times completely absent from the detection, likely downsampling the character, calculating the corresponding 64 due to the down-sampling of the camera image to 640x480 dimensional vector, and writing it to an output text file, was pixels.
Load more

Recommended publications
  • Chapter 11. Three Dimensional Analytic Geometry and Vectors
    Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1.
    [Show full text]
  • A Quick Algebra Review
    A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation.
    [Show full text]
  • Writing the Equation of a Line
    Name______________________________ Writing the Equation of a Line When you find the equation of a line it will be because you are trying to draw scientific information from it. In math, you write equations like y = 5x + 2 This equation is useless to us. You will never graph y vs. x. You will be graphing actual data like velocity vs. time. Your equation should therefore be written as v = (5 m/s2) t + 2 m/s. See the difference? You need to use proper variables and units in order to compare it to theory or make actual conclusions about physical principles. The second equation tells me that when the data collection began (t = 0), the velocity of the object was 2 m/s. It also tells me that the velocity was changing at 5 m/s every second (m/s/s = m/s2). Let’s practice this a little, shall we? Force vs. mass F (N) y = 6.4x + 0.3 m (kg) You’ve just done a lab to see how much force was necessary to get a mass moving along a rough surface. Excel spat out the graph above. You labeled each axis with the variable and units (well done!). You titled the graph starting with the variable on the y-axis (nice job!). Now we turn to the equation. First we replace x and y with the variables we actually graphed: F = 6.4m + 0.3 Then we add units to our slope and intercept. The slope is found by dividing the rise by the run so the units will be the units from the y-axis divided by the units from the x-axis.
    [Show full text]
  • The Geometry of the Phase Diffusion Equation
    J. Nonlinear Sci. Vol. 10: pp. 223–274 (2000) DOI: 10.1007/s003329910010 © 2000 Springer-Verlag New York Inc. The Geometry of the Phase Diffusion Equation N. M. Ercolani,1 R. Indik,1 A. C. Newell,1,2 and T. Passot3 1 Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA 2 Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK 3 CNRS UMR 6529, Observatoire de la Cˆote d’Azur, 06304 Nice Cedex 4, France Received on October 30, 1998; final revision received July 6, 1999 Communicated by Robert Kohn E E Summary. The Cross-Newell phase diffusion equation, (|k|)2T =∇(B(|k|) kE), kE =∇2, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order “self-dual” equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. 1. Introduction The mathematical models discussed in this paper are motivated by physical systems, far from equilibrium, which spontaneously form patterns.
    [Show full text]
  • Numerical Algebraic Geometry and Algebraic Kinematics
    Numerical Algebraic Geometry and Algebraic Kinematics Charles W. Wampler∗ Andrew J. Sommese† January 14, 2011 Abstract In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work. Contents 1 Introduction 4 2 Notation 5 I Fundamentals of Algebraic Kinematics 6 3 Some Motivating Examples 6 3.1 Serial-LinkRobots ............................... ..... 6 3.1.1 Planar3Rrobot ................................. 6 3.1.2 Spatial6Rrobot ................................ 11 3.2 Four-BarLinkages ................................ 13 3.3 PlatformRobots .................................. 18 ∗General Motors Research and Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI 48090- 9055, U.S.A. Email: [email protected] URL: www.nd.edu/˜cwample1. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and by General Motors Research and Development. †Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-4618, U.S.A. Email: [email protected] URL: www.nd.edu/˜sommese. This material is based upon work supported by the National Science Foundation under Grant DMS-0712910 and the Duncan Chair of the University of Notre Dame.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Chapter 1: Analytic Geometry
    1 Analytic Geometry Much of the mathematics in this chapter will be review for you. However, the examples will be oriented toward applications and so will take some thought. In the (x,y) coordinate system we normally write the x-axis horizontally, with positive numbers to the right of the origin, and the y-axis vertically, with positive numbers above the origin. That is, unless stated otherwise, we take “rightward” to be the positive x- direction and “upward” to be the positive y-direction. In a purely mathematical situation, we normally choose the same scale for the x- and y-axes. For example, the line joining the origin to the point (a,a) makes an angle of 45◦ with the x-axis (and also with the y-axis). In applications, often letters other than x and y are used, and often different scales are chosen in the horizontal and vertical directions. For example, suppose you drop something from a window, and you want to study how its height above the ground changes from second to second. It is natural to let the letter t denote the time (the number of seconds since the object was released) and to let the letter h denote the height. For each t (say, at one-second intervals) you have a corresponding height h. This information can be tabulated, and then plotted on the (t, h) coordinate plane, as shown in figure 1.0.1. We use the word “quadrant” for each of the four regions into which the plane is divided by the axes: the first quadrant is where points have both coordinates positive, or the “northeast” portion of the plot, and the second, third, and fourth quadrants are counted off counterclockwise, so the second quadrant is the northwest, the third is the southwest, and the fourth is the southeast.
    [Show full text]
  • The Evolution of Equation-Solving: Linear, Quadratic, and Cubic
    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2006 The evolution of equation-solving: Linear, quadratic, and cubic Annabelle Louise Porter Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Mathematics Commons Recommended Citation Porter, Annabelle Louise, "The evolution of equation-solving: Linear, quadratic, and cubic" (2006). Theses Digitization Project. 3069. https://scholarworks.lib.csusb.edu/etd-project/3069 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. THE EVOLUTION OF EQUATION-SOLVING LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degre Master of Arts in Teaching: Mathematics by Annabelle Louise Porter June 2006 THE EVOLUTION OF EQUATION-SOLVING: LINEAR, QUADRATIC, AND CUBIC A Project Presented to the Faculty of California State University, San Bernardino by Annabelle Louise Porter June 2006 Approved by: Shawnee McMurran, Committee Chair Date Laura Wallace, Committee Member , (Committee Member Peter Williams, Chair Davida Fischman Department of Mathematics MAT Coordinator Department of Mathematics ABSTRACT Algebra and algebraic thinking have been cornerstones of problem solving in many different cultures over time. Since ancient times, algebra has been used and developed in cultures around the world, and has undergone quite a bit of transformation. This paper is intended as a professional developmental tool to help secondary algebra teachers understand the concepts underlying the algorithms we use, how these algorithms developed, and why they work.
    [Show full text]
  • Geometry Formula Sheet ─ Page 1
    KEYSTONE RefEFERENCE GEOMETRY FORMULA SHEET ─ PAGE 1 Formulas that you may need to solve questions on this exam are found below. You may use calculator π or the number 3.14. Properties of Circles Right Triangle Formulas Angle measure is represented by x. Arc measure is represented Pythagorean Theorem: by m and n. Lengths are given by a, b, c, and d. c If a right triangle has legs with a measures a and b and hypotenuse with measure c, then... n° Inscribed Angle b 2 + 2 = 2 x° 1 a b c x = n 2 Trigonometric Ratios: opposite sin θ = x° Tangent-Chord hypotenuse n° = 1 x n hypotenuse adjacent 2 opposite cos θ = hypotenuse θ adjacent opposite tan θ = 2 Chords adjacent d a · = · m°x° n° a b c d c 1 b x = (m + n) 2 Coordinate Geometry Properties a x° Tangent-Secant = – 2 + – 2 b Distance Formula: d (x2 x 1) (y2 y 1) n° 2 = + a b (b c) m° x + x y + y 1 1 2 , 1 2 x = (m − n) Midpoint: c 2 2 2 y − y = 2 1 Slope: m − x 2 x 1 a 2 Secants Point-Slope Formula: (y − y ) = m(x − x ) b + = + 1 1 b (a b) d (c d ) m° n° x° 1 c d x = (m − n) Slope Intercept Formula: y = mx + b 2 Standard Equation of a Line: Ax + By = C a 2 Tangents a = b m° n° x° 1 x = (m − n) b 2 Copyright © 2011 by the Pennsylvania Department of Education. The materials contained in this publication may be duplicated by Pennsylvania educators for local classroom use.
    [Show full text]
  • Analytic Geometry
    INTRODUCTION TO ANALYTIC GEOMETRY BY PEECEY R SMITH, PH.D. N PROFESSOR OF MATHEMATICS IN THE SHEFFIELD SCIENTIFIC SCHOOL YALE UNIVERSITY AND AKTHUB SULLIVAN GALE, PH.D. ASSISTANT PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ROCHESTER GINN & COMPANY BOSTON NEW YORK CHICAGO LONDON COPYRIGHT, 1904, 1905, BY ARTHUR SULLIVAN GALE ALL BIGHTS RESERVED 65.8 GINN & COMPANY PRO- PRIETORS BOSTON U.S.A. PEE FACE In preparing this volume the authors have endeavored to write a drill book for beginners which presents, in a manner conform- ing with modern ideas, the fundamental concepts of the subject. The subject-matter is slightly more than the minimum required for the calculus, but only as much more as is necessary to permit of some choice on the part of the teacher. It is believed that the text is complete for students finishing their study of mathematics with a course in Analytic Geometry. The authors have intentionally avoided giving the book the form of a treatise on conic sections. Conic sections naturally appear, but chiefly as illustrative of general analytic methods. Attention is called to the method of treatment. The subject is developed after the Euclidean method of definition and theorem, without, however, adhering to formal presentation. The advan- tage is obvious, for the student is made sure of the exact nature of each acquisition. Again, each method is summarized in a rule stated in consecutive steps. This is a gain in clearness. Many illustrative examples are worked out in the text. Emphasis has everywhere been put upon the analytic side, that is, the student is taught to start from the equation.
    [Show full text]
  • Asce Standardized Reference Evapotranspiration Equation
    THE ASCE STANDARDIZED REFERENCE EVAPOTRANSPIRATION EQUATION Task Committee on Standardization of Reference Evapotranspiration Environmental and Water Resources Institute of the American Society of Civil Engineers January, 2005 Final Report ASCE Standardized Reference Evapotranspiration Equation Page i THE ASCE STANDARDIZED REFERENCE EVAPOTRANSPIRATION EQUATION PREPARED BY Task Committee on Standardization of Reference Evapotranspiration of the Environmental and Water Resources Institute TASK COMMITTEE MEMBERS Ivan A. Walter (chair), Richard G. Allen (vice-chair), Ronald Elliott, Daniel Itenfisu, Paul Brown, Marvin E. Jensen, Brent Mecham, Terry A. Howell, Richard Snyder, Simon Eching, Thomas Spofford, Mary Hattendorf, Derrell Martin, Richard H. Cuenca, and James L. Wright PRINCIPAL EDITORS Richard G. Allen, Ivan A. Walter, Ronald Elliott, Terry Howell, Daniel Itenfisu, Marvin Jensen ENDORSEMENTS Irrigation Association, 2004 ASCE-EWRI Task Committee Report, January, 2005 ASCE Standardized Reference Evapotranspiration Equation Page ii ABSTRACT This report describes the standardization of calculation of reference evapotranspiration (ET) as recommended by the Task Committee on Standardization of Reference Evapotranspiration of the Environmental and Water Resources Institute of the American Society of Civil Engineers. The purpose of the standardized reference ET equation and calculation procedures is to bring commonality to the calculation of reference ET and to provide a standardized basis for determining or transferring crop coefficients for agricultural and landscape use. The basis of the standardized reference ET equation is the ASCE Penman-Monteith (ASCE-PM) method of ASCE Manual 70. For the standardization, the ASCE-PM method is applied for two types of reference surfaces representing clipped grass (a short, smooth crop) and alfalfa (a taller, rougher agricultural crop), and the equation is simplified to a reduced form of the ASCE–PM.
    [Show full text]
  • The Roots of Any Polynomial Equation
    The roots of any polynomial equation G.A.Uytdewilligen, Bergen op Zoomstraat 76, 5652 KE Eindhoven. [email protected] Abstract We provide a method for solving the roots of the general polynomial equation n n−1 a ⋅x + a ⋅x + . + a ⋅x + s 0 n n−1 1 (1) To do so, we express x as a powerseries of s, and calculate the first n-1 coefficients. We turn the polynomial equation into a differential equation that has the roots as solutions. Then we express the powerseries’ coefficients in the first n-1 coefficients. Then the variable s is set to a0. A free parameter is added to make the series convergent. © 2004 G.A.Uytdewilligen. All rights reserved. Keywords: Algebraic equation The method The method is based on [1]. Let’s take the first n-1 derivatives of (1) to s. Equate these derivatives to zero. di Then find x ( s ) in terms of x(s) for i from 1 to n-1. Now make a new differential equation dsi n 1 n 2 d − d − m1⋅ x(s) + m2⋅ x(s) + . + m ⋅x(s) + m 0 n−1 n−2 n n+1 ds ds (2) di and fill in our x ( s ) in (2). Multiply by the denominator of the expression. Now we have a dsi polynomial in x(s) of degree higher then n. Using (1) as property, we simplify this polynomial to the degree of n. Set it equal to (1) and solve m1 .. mn+1 in terms of s and a1 .. an Substituting these in (2) gives a differential equation that has the zeros of (1) among its solutions.
    [Show full text]