## Posts Tagged 'Geometry'

### Geodesics on the Spheroidal Earth – I

Both Quito in Ecuador and Singapore are on the Equator. One can fly due eastward from Singapore and reach Quito in due course. However, this is not the shortest route. The equatorial trans-Pacific route from Singapore to Quito is not a geodesic on Earth! Why not?

A drastically flattened spheroid. Clearly, the equatorial route between the blue and red points is not the shortest path.

### The Evolute: Envelope of Normals

Every curve in the plane has several other curves associated with it. One of the most interesting and important of these is the evolute.

Sin t (blue) and its evolute (red).

### Doughnuts and Tonnetze

The circle of fifths is a remarkably useful diagram for the analysis of music. It shows the twelve notes of the chromatic scale arranged in a circle, with notes that are harmonically related (like C and G) being close together and notes that are discordant (like C and C) more distant from each other.

The Tonnetz diagram (note that the arrangement here is inverted relative to that used in the text.  It appears that there is no rigid standard, and several arrangements are in use) [Image from WikimediaCommons].

### Vanishing Hyperballs

Spherical ball contained within a cubic region

We all know that the area of a disk — the interior of a circle — is ${\pi r^2}$ where ${r}$ is the radius. Some of us may also remember that the volume of a ball — the interior of a sphere — is ${\frac{4}{3}\pi r^3}$.

### A Symbol for Global Circulation

The recycling symbol consisting of three bent arrows is found on bottles, cartons and packaging of all kinds. It originated in 1970 when the Chicago-based Container Corporation of America (CCA) held a competition for the design of a symbol suitable for printing on cartons, to encourage recycling and re-use of packaging materials.

Original (Moebius) and a variation (3-twist) of the universal recycling symbol.

### Malfatti’s Circles

Sometimes the “obvious” answer to a mathematical problem is not the correct one. The case of Malfatti’s circles is an example of this. In an equilateral triangle of unit side length, we must draw three non-overlapping circles such that the total area of the circles is maximal.

The solution seems obvious: draw three identical circles, each one tangent to two sides and to the other two circles (above figure, left). This is certainly the most symmetric arrangement possible. However, it turns out not to be the optimal solution. There is another arrangement (above figure, right) for which the three circles have greater total area.

### Drawing Multi-focal Ellipses: The Gardener’s Method

Common-or-Garden Ellipses

In an earlier post we saw how a gardener may set out oval flower-beds using a well-known property of ellipses: the sum of the distances from any point on the ellipse to the two foci is always the same value, ${2a}$, the length of the major axis. The gardener puts down two stakes and loops a piece of rope around them. Using a stick, he pulls the loop taut, marking the points around a curve. This is illustrated here.

Gardener’s method of drawing an ellipse [Image Wikimedia].

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