Posts Tagged 'Maps'

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 “thatsmaths” at irishtimes.com].

JFK-PEK-Gnomonic

Great circle route from New York to Beijing (gnomonic projection).

On a gnomonic projection (as above) each point on the Earth’s surface is projected from the centre of the Earth onto a plane tangent to the globe. On this map, great circles appear as straight lines.

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. WM-vs-Merc-Detail Continue reading ‘Maps on the Web’

You Can Do Maths

Bragging about mathematical ineptitude is not cool. There is nothing admirable about ignorance and incompetence. Moreover, everyone thinks mathematically all the time, even if they are not aware of it. Can we all do maths? Yes, we can!  [See this week’s That’s Maths column (TM064) or search for “thatsmaths” at irishtimes.com].

Topological map of the London Underground network

When you use a map of the underground network, you are doing topology.

Continue reading ‘You Can Do Maths’

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 to the network, yielding Steiner’s minimal tree. This generally results in a reduction of the overall length of the network.

Solution of Steiner 5-point problem with soap film [from Courant and Robbins].

A solution of Steiner 5-point problem with soap film [from Courant and Robbins].

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 {\mathbb{R}^2} so that it generates a plane curve as it moves. This, a curve results from a mapping {\mathbf{\gamma} : [a,b]\longrightarrow \mathbb{R}^2}. 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 triangle may add up to more than two right angles, or to less.

10 Deutschmark currency note

10 Deutschmark currency note (front)

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.

Santa-and-Moon

Continue reading ‘Santa’s Fractal Journey’


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