## Posts Tagged 'Trigonometry'

### Image Processing Emerges from the Shadows

Satellite images are of enormous importance in military contexts. A battery of mathematical and image-processing techniques allows us to extract information that can play a critical role in tactical planning and operations. The information in an image may not be immediately evident. For example, an overhead image gives no direct information about the height of buildings or industrial installations, but shadows, together with the time, date and basic trigonometry, enable heights to be determined  [TM233 or search for “thatsmaths” at irishtimes.com].

### Al Biruni and the Size of the Earth

Al Biruni at Persian Scholars Pavilion in Vienna.

The 11th century Persian mathematician Abu Rayhan al-Biruni used simple trigonometric results to estimate the radius and circumference of the Earth. His estimate has been quoted as 6,340 km, which is within 1% of the mean radius of 6,371 km. While al-Biruni’s method was brilliant and, for its era, spectacular, the accuracy claimed must be regarded with suspicion.

Al-Biruni assumed that the Earth is a perfect sphere of (unknown) radius ${a}$. He realised that because of the Earth’s curvature the horizon, as viewed from a mountain-top, would appear to be below the horizontal direction. This direction is easily obtained as being orthogonal to the vertical, which is indicated by a plumb line.

### Elliptic Trigonometry: Fun with “sun”, “cun” and “dun”

Introduction

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ using the integral

$\displaystyle u = \int_0^{\phi} \frac{\mathrm{d}\phi}{\sqrt{1-k^2\sin^2\phi}} \,.$

He called ${\phi}$ the amplitude and wrote ${\phi = \mathop\mathrm{am} u}$. It can be difficult to understand what motivated his definitions. We will define the elliptic functions ${\mathop\mathrm{sn} u}$, ${\mathop\mathrm{cn} u}$, ${\mathop\mathrm{dn} u}$ in a more intuitive way, as simple ratios associated with an ellipse.