A Note on Standard Deviation and Root Mean Square (RMS)
Technical paper by R.E. Deakin and D.G. Kildea detailing the difference between standard deviation and RMS (6 pages)
St_dev.pdf
Catenary Curve
The catenary is the curve in which a uniform chain or cable hangs freely under the force of gravity from two supports. It is a U-shaped curve symmetric about a vertical axis through its low-point and was first described mathematically by Leibniz, Huygens and Johann Bernoulli in 1691 responding to a challenge put out by Jacob Bernoulli (Johann’s elder brother) to find the equation of the ‘chain curve’. Every person viewing power lines hanging between supporting poles is seeing a catenary, a curve whose name is derived from the Latin word catena, meaning chain, and the catenary’s mathematical ‘discovery’ is due to Galileo’s claim – proved incorrect by Bernoulli and others – that a hanging chain was parabolic. This paper gives a mathematical derivation of the catenary with examples.
Catenary Curve.pdf
Centroids
Technical paper by R.E. Deakin, R.I. Grenfell and S.C. Bird on the definition and calculation of centroids including the determination of various centroids of Victoria.
MSIA_Centroid.pdf
Chi-squared distribution and Confidence Intervals of the Variance
These notes (8 pages) give an explanation of the chi-squared distribution and its use in estimating confidence intervals of the variance. Examples are provided and the equation of the distribution is developed.
Chi-squared distribution.pdf
Horizontal Positional Accuracy from Directional Antenna
Technical paper presented at Victorian Regional Survey Conference, Traralgon, 23-25 May, 2003 (18 pages).
Positional Accuracy 2.pdf
HP35s Surveying Programs
A suite of surveying computation programs for the Hewlett Packard (HP) 35s calculator that will be useful in the field and office. Some (Closure, Radiations, Joins, Offsets) have a heritage extending back to HP desktop-computer programs from the 1970’s written by Bodo Taube of Francis O’Halloran, Surveyors. And Bodo Taube’s programs were (and are) models of efficiency. Others are more recent. Each program has a set of User Instructions, with examples and relevant formula and HP35s Program Sheets listing the program steps (that you may key into your calculator), storage registers used and program notes.
HP35s Surveying Programs.pdf
Newton-Raphson Iteration
Notes and historical discussion of this important numerical method of solving equations (27 pages).
Newton-Raphson Iteration.pdf
Routh’s Theorem and solution to Curly’s Conundrum No. 20
The Institution of Surveyors Victoria (ISV) has a news bulletin Traverse that is published quarterly and circulated to members. In Traverse 120 (Nov., 1991) a puzzle was published as Curly’s Conundrum No. 20. In the next issue, rather than the solution, the following note appeared:
Well, Curly has finally tripped himself up! I am embarrassed! A solution to Conundrum No. 20 is beyond me. The puzzle was published in The Australian Mind of the Year Contest for 1989 and was one of the questions posed to the five finalists. The answer, published the following week was five (5) square units.
PS: Please send me a solution as this problem has been driving me crazy for years.
Well, here’s some information on Routh’s Theorem and the solution that Curly’s been searching for.
Routh’s Theorem.pdf
Shrine of Remembrance, Melbourne
One of the best known features of the Shrine of Remembrance in Melbourne is the beam of sunlight that falls on the Stone of Remembrance on Armistice day at the 11th hour of the 11th day of the 11th month each year (clouds permitting); commemorating the moment of the end of the First World War in 1918.
Each year the Shrine trustees arrange a Remembrance Day service, but since the introduction of daylight saving in the summer of 1971-72, the natural event of the beam of sunlight passing across the Stone of Remembrance occurs at 12 pm rather than at the commemoration time of 11 am. The problem caused by daylight saving was overcome by ‘bending’ the Sun’s rays by reflections from an inclined and a horizontal mirror. The document Bending the Beam explains Frank Johnston’s solution to the problem (Frank is a retired staff member of RMIT Surveying Department).
In 2014, The Age newspaper had a feature article on the Shrine (see below) and interviewed Frank whilst the inclined mirror was being set a few days before Remembrance Day. The article and a video movie of 3 min duration is available below.
The Shrine Mirror Problem was a favourite assignment topic for surveying students studying Astronomy in the 1970’s and 1980’s and was reproduced for surveying students in 2010 as an exercise in spherical trigonometry. The assignment and worked solution are shown below. This might trigger buried memories of ‘Astro’ for those old enough to remember.
BENDING THE BEAM.pdf
The Age Sat-08-Nov-2014.pdf
Shrine movie
Shrine Assignment.pdf
ShrineAssignmentSolution2010.pdf
Student’s t-distribution and Confidence Intervals of the Mean
These notes (10 pages) give an explanation of the t-distribution and its use in estimating confidence intervals. Examples are provided and the equation of the distribution is developed.
t distribution.pdf
Tau–distribution and testing residuals
Technical paper (17 pages) by R.E Deakin and M.N. Hunter on the tau-distribution and testing residuals for outliers. This paper describes the original tau statistic (Thompson 1935) and a modern interpretation if this statistic by Pope (1976). The paper defines the relationship between the tau-distribution and Student’s t-distribution and has a derivation of the probability density function of the tau distribution. The properties of the tau-distribution are explained and tables of areas under the tau-curve are provided for statistical testing. Computer simulations are used to verify these tabulated values and several examples are provided to help explain the use of the tau-distribution for detecting outliers in lists of observations/residuals.
Tau distribution.pdf
The Logistic Function
Technical paper (44 pages) on the Logistic function including a brief history of Pierre Verhulst’s la courbe logistique and a derivation of the function. The properties of the logistic curve are derived and a general equation developed with a special case, the sigmoid curve. This is followed by the derivation of the Logistic distribution. Logistic regression is discussed and the method of least squares is employed to give a solution for the parameters of a logistic curve that is the best fit of the outcomes of binary dependent variables. Two examples of this technique are shown. Finally, the connection between a sport rating system (Elo rating system) and the logistic curve is shown. References for further reading and two appendices are included.
The Logistic Function.pdf
Total Increment Theorem
Notes on the total increment theorem (total differentials) and their practical use in some surveying applications (9 pages).
Total Increment Theorem.pdf
Traverse Analysis
Technical paper on a method of assessing traverse quality using propagation of variances (11 pages)
TRAVERSE ANALYSIS(GEOM2091).pdf
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