THE PAPERS
An attempt is made to mitigate computational expense by suppressing frequent trigonometric and root applications in Area and Perimeter calculation.
The Areas and Perimeters of convex hulls are estimated using respectively sums of radii products and sums of approximate perimeter segments.
Both estimates are improved by Corrector Constants.
In a series of spreadsheet experiments involving 50% radial variation and 20% angular variation it was found that Area and Perimeter percentage errors were respectively +3 and -1.5 for irregular quadrilaterals to +0.35 and -0.0025 for 360-sided irregular polygons.
To suppress square-rooting in perimeter summations a Method of Mean Transitions was developed.
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Whole-population arithmetic mean is a sole function of the square matrix side length and the mean of upper-right ( or lower-left ) elements. The sum of the diagonal squared-deviations is a function of that mean and side length only.
Thirdly, the sum of all the squared-deviations is a simple function only of the sum of the upper-right squared-deviations and the diagonal squared deviations.
That leads directly to an economised estimate of the sample and population variances.
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A transect is a straight line connecting different sides of a polygon. The Mean Transect is the arithmetic mean of the lengths of all such sides.
Conventionally, mean transect lengths are referent to a polygon side length of unity.
An analytic mean transect exists only for the square.
For the square an analytic value was derived and compared with four thousand random transects generated by spreadsheet. In regard to the crossing transection, a small specific defect of 0.358162% was ascribed to systematic programming error. The discrepancy between analytic and numerical cubature figures for the overall mean transect of a square was -0.0238%.
For squares and higher regular polygons, numerical cubatures based upon statistical expectation of transect length were developed. In this case the integrand transect is defined in terms of Carnot's Theorem ( the cosine rule ).
Several Romberg algorithms and Newton-Cotes formulae were comparatively employed for the actual cubatures.
Inferior estimates of the mean transect can be made using transects resulting from ( pseudo- ) random number generators. My experiments used 4096 trials for each polygon.
Appendix program TRANANAL.BAS applies the double integration whilst TRANSECT.BAS uses the random transect trials. Appendix Six compares results for twelve regular polygons with from 3 to 100 sides.
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For many-sided regular polygons it is possible to approximate the area as a function of the radius and number of sides only, because the cosine of the side-subtended angle is about unity minus half the square of the angle.
Some algebraic manipulation then shows that the mean transect can be expressed as the product of root area times some suitable function that converges to a constant as the number of sides tends to infinity.
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The intervisibility of some eminent British summits is computed as are
some lesser tops known to the Author.
The effects of planetary asphericity and of atmospheric refraction are disconsidered.
( Refraction acts to render occluded land visible ).
Some startling and entertaining facts emerge.
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Estimation of The Complete Elliptic Integral of the Fist Kind
By Approxiamting the Arithmetic-Geometric Mean
and then using the well-known formula that relates
the CIEF to the AGM.
The estimator's fidelity is assessed by reference to
Classical Solution and the Hasting's Formula,
as well as simple Simpson's Rule integration.
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| NAME OF PAPER: | Some Findings in the Statistical Estimation |
| Of the Area and Perimeter | |
| Of the General Polygon | |
| DATE OF WRITING: | 23 December 1998 |
| LENGTH: | 11 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 0 |
| EQUATIONS: | 34 |
| TABLES: | 3 |
| REFERENCES: | 0 |
| APPENDICES: | No Appendices |
| FILENAME: | statape.pdf |
| ABSTRACT: |
An attempt is made to mitigate computational expense by suppressing frequent trigonometric and root applications in Area and Perimeter calculation.
The Areas and Perimeters of convex hulls are estimated using respectively sums of radii products and sums of approximate perimeter segments.
Both estimates are improved by Corrector Constants.
In a series of spreadsheet experiments involving 50% radial variation and 20% angular variation it was found that Area and Perimeter percentage errors were respectively +3 and -1.5 for irregular quadrilaterals to +0.35 and -0.0025 for 360-sided irregular polygons.
To suppress square-rooting in perimeter summations a Method of Mean Transitions was developed.
Read this Paper
| NAME OF PAPER: | Economisation of the Computation of |
| The Momental Statistics | |
| Of Symmetric Matrices | |
| DATE OF WRITING: | 1 March 1999 |
| LENGTH: | 4 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 0 |
| EQUATIONS: | 11 |
| TABLES: | 1 |
| REFERENCES: | 0 |
| APPENDICES: | No Appendices |
| FILENAME: | symstatc.pdf |
| ABSTRACT: |
Whole-population arithmetic mean is a sole function of the square matrix side length and the mean of upper-right ( or lower-left ) elements. The sum of the diagonal squared-deviations is a function of that mean and side length only.
Thirdly, the sum of all the squared-deviations is a simple function only of the sum of the upper-right squared-deviations and the diagonal squared deviations.
That leads directly to an economised estimate of the sample and population variances.
Read this Paper
| NAME OF PAPER: | Some Approaches to the |
| Determination of the Mean Transect of | |
| A Regular Polygon | |
| DATE OF WRITING: | 3 January 1998 |
| LENGTH: | 21 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 1 |
| EQUATIONS: | 38 |
| TABLES: | 0 |
| REFERENCES: | 3 |
| APPENDICES: | All Appendices included in the Internet Version |
| FILENAME: | transectf.pdf |
| ABSTRACT: |
A transect is a straight line connecting different sides of a polygon. The Mean Transect is the arithmetic mean of the lengths of all such sides.
Conventionally, mean transect lengths are referent to a polygon side length of unity.
An analytic mean transect exists only for the square.
For the square an analytic value was derived and compared with four thousand random transects generated by spreadsheet. In regard to the crossing transection, a small specific defect of 0.358162% was ascribed to systematic programming error. The discrepancy between analytic and numerical cubature figures for the overall mean transect of a square was -0.0238%.
For squares and higher regular polygons, numerical cubatures based upon statistical expectation of transect length were developed. In this case the integrand transect is defined in terms of Carnot's Theorem ( the cosine rule ).
Several Romberg algorithms and Newton-Cotes formulae were comparatively employed for the actual cubatures.
Inferior estimates of the mean transect can be made using transects resulting from ( pseudo- ) random number generators. My experiments used 4096 trials for each polygon.
Appendix program TRANANAL.BAS applies the double integration whilst TRANSECT.BAS uses the random transect trials. Appendix Six compares results for twelve regular polygons with from 3 to 100 sides.
Read this Paper
| NAME OF PAPER: | Corroborative Approximation of |
| The Mean Transect for Many-Sided Regular Polygons | |
| DATE OF WRITING: | 14 December 1997 |
| LENGTH: | 3 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 0 |
| EQUATIONS: | 20 |
| TABLES: | 0 |
| REFERENCES: | 0 |
| APPENDICES: | No Appendices |
| FILENAME: | tranmanyc.pdf |
| ABSTRACT: |
For many-sided regular polygons it is possible to approximate the area as a function of the radius and number of sides only, because the cosine of the side-subtended angle is about unity minus half the square of the angle.
Some algebraic manipulation then shows that the mean transect can be expressed as the product of root area times some suitable function that converges to a constant as the number of sides tends to infinity.
Read this Paper
| NAME OF PAPER: | The Intervisibility of Eminences Upon a Sphere |
| DATE OF WRITING: | 31 August 2006 |
| LENGTH: | 24 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 7 ( including 4 photographs ) |
| EQUATIONS: | 28 |
| TABLES: | 4 ( in appendices ) |
| REFERENCES: | 7 |
| APPENDICES: | Appendices included in the Internet Version |
| FILENAME: | ivspheree.pdf |
| ABSTRACT: |
The intervisibility of some eminent British summits is computed as are
some lesser tops known to the Author.
The effects of planetary asphericity and of atmospheric refraction are disconsidered.
( Refraction acts to render occluded land visible ).
Some startling and entertaining facts emerge.
Read this Paper
| NAME OF PAPER: | A Simple but Accurate Approximator of |
| The Complete Elliptic Integral of the Fist Kind | |
| DATE OF WRITING: | 31 October 2013 |
| LENGTH: | 16 A4 Pages at Times New Roman 12 |
| ILLUSTRATIONS: | 2 |
| EQUATIONS: | 32 |
| TABLES: | 5 |
| REFERENCES: | 5 |
| APPENDICES: | 1 Appendix included in the Internet Version |
| FILENAME: | ceif-compressed.pdf |
| ABSTRACT: |
Estimation of The Complete Elliptic Integral of the Fist Kind
By Approxiamting the Arithmetic-Geometric Mean
and then using the well-known formula that relates
the CIEF to the AGM.
The estimator's fidelity is assessed by reference to
Classical Solution and the Hasting's Formula,
as well as simple Simpson's Rule integration.
Read this Paper
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