An Exploration to Locate the Optimal Place That Maximizes the Viewing Angle That Maximizes the Size of an Object

An Exploration to Locate the Optimal Place That Maximizes the Viewing Angle That Maximizes the Size of an Object

Document details
Category: Mathematics Essay
Subcategory: Statistics and Probability
Words: 2611
Pages: 18

Whilst reading The Penguin Dictionary of Curious and Interesting Geometry by David Wells (1991) I encountered an interesting inquiry. German mathematician Johannes Regiomontanus posed the following question From what position along a horizontal line can a statue best be viewed (Figure 1)? The answer lies in optimization where Calculus and Geometry merge to determine the location that maximizes the viewing angle and causes the object to appear at maximum size. In searching for an area of research I realized that this maximization problem can be used to work out the optimal distance from the try line to position the ball for a conversion kick in rugby union. Figure 1 Determining the ideal position for viewing a statueMultiple mathematicians have produced similar results when investigating this problem. Hughes (1978) was the first to explore the scenario, proposing that the locus of points from which to take conversions ishalf of each of the branches of a rectangular hyperbola, and on this the kicker should place his ellipsoid of revolution before sending it on its parabolic way. Worsnup (1989) later developed Hughes discovery, advocating the asymptote of the hyperbola instead. De Villiers (1999) then accounted for the clearance of the crossbar in his study. Since these studies have generated different methods, further investigation of the situation seems appropriate.Furthermore, successful conversions yield two points and can often decide the outcome of a rugby union much. Over a season, these points could be vital in securing promotion or causing relegation. Thus, the problem becomes significant and worth of investigation from a professional point of view.This exploration aims to locate an optimal place that maximizes the angle subtended by the bases of the goalposts. I decided to conduct a series of explorations which encompasses locating the ideal kicking position through geometric constructions. Then I employed a combination of calculus and trigonometry to verify a common ideal position from which to kick a conversion. Background InformationBefore formal investigation and modelling could begin...

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