# circle

## The Nephoid – a caustic in a coffee cup

The Nephoid is the curve formed by the envelope of rays reflected on a circular surface from a set of initial parallel rays.  In other words, the curve which owns all the rays as tangents. The geomtery of each ray path is relatively straight forward, involving properties of parallel lines and isocelels triangles. It is …

## The Cardioid as the Envelope of Pencil Curves

Draw a circle and divide the circumference into and even number of equally spaced points.  In this example I have used 44: Draw following chords between numbers in such a way that when viewed as a sequemce, the second point of each pair moves twice as fast around the circle.: and and on completion, sketch …

## t-Formulae and parameterisation of the circle

t-Formulae are used in integration to tackle rational expressions of tigonometric functions.  After a spell in the cold, when they were not included in some A level specifications, they are now back in sixth form lessons. It all starts with the subsitution, from which the following functions can be derived, ,  ,  . These derivations …

## Area of a circle – proof by exhuastion

Once has been defined as the ratio circumference to diameter, the area of a circle must be . A proof relies on an infinite, limiting process which paves the way to some calculus-like ideas. A circle is cut up into 6 sectors which are then rearranged into a near rectangle, What doesn’t look too close …