Ireland’s Mapping Grid in Harmony with GPS

The earthly globe is spherical; more precisely, it is an oblate spheroid, like a ball slightly flattened at the poles. More precisely still, it is a triaxial ellipsoid that closely approximates a “geoid”, a surface of constant gravitational potential [TM199 or search for “thatsmaths” at irishtimes.com]. Mapping the Globe Positions on the globe are given by … Continue reading Ireland’s Mapping Grid in Harmony with GPS →

Trigonometric Comfort Blankets on Hilltops

On a glorious sunny June day we reached the summit of Céidín, south of the Glen of Imall, to find a triangulation station or trig pillar. These concrete pillars are found on many prominent peaks throughout Ireland, and were erected to aid in surveying the country [see TM142, or search for “thatsmaths” at irishtimes.com]. The pillars are about … Continue reading Trigonometric Comfort Blankets on Hilltops →

From Sailing on a Rhumb to Flying on a Geodesic

If you fly 14,500 km due westward from New York you will come to Beijing. The two cities are on the fortieth parallel of latitude. However, by flying a great circle route over the Arctic, you can reach Beijing in 11,000 km, saving 3,500 km and much time and aviation fuel. [TM124 or search for … Continue reading From Sailing on a Rhumb to Flying on a Geodesic →

Maps on the Web

In a nutshell: In web maps, geographical coordinates are projected as if the Earth were a perfect sphere. The results are great for general use but not for high-precision applications. Mercator's Projection Cartographers have devised numerous projections, each having advantages and drawbacks. No one projection is ideal. Mercator's projection is based on a cylinder tangent … Continue reading Maps on the Web →

The Steiner Minimal Tree

Steiner's minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner's problem allows for additional points – now called Steiner points – to be added … Continue reading The Steiner Minimal Tree →

New Curves for Old: Inversion

Special Curves A large number of curves, called special curves, have been studied by mathematicians. A curve is the path traced out by a point moving in space. To keep things simple, we assume that the point is confined to two-dimensional Euclidean space $latex {\mathbb{R}^2}&fg=000000$ so that it generates a plane curve as it moves. … Continue reading New Curves for Old: Inversion →

Gauss’s Great Triangle and the Shape of Space

In the 1820s Carl Friedrich Gauss carried out a surveying experiment to measure the sum of the three angles of a large triangle. Euclidean geometry tells us that this sum is always 180º or two right angles. But Gauss himself had discovered other geometries, which he called non-Euclidean. In these, the three angles of a … Continue reading Gauss’s Great Triangle and the Shape of Space →

Santa’s Fractal Journey

The article in this week’s That’s Maths column in the Irish Times ( TM035 ) is about the remarkable Christmas Eve journey of Santa Claus. Dimensions & Fractals How far must Santa travel on Christmas Eve? At a broad scale, he visits all the continents. In more detail he travels to every country. Zooming in, … Continue reading Santa’s Fractal Journey →

Ireland’s Fractal Coastline

Reports of the length of Ireland's coastline vary widely. The World Factbook of the Central Intelligence Agency gives a length of 1448 km. The Ordnance Survey of Ireland has a value of 3,171 km (http://www.osi.ie). The World Resources Institute, using data from the United States Defense Mapping Agency, gives 6,347km (see Wikipedia article [3]). Fractals … Continue reading Ireland’s Fractal Coastline →