## Posts tagged ‘conic’

### Rational approximations of √2

Introducing undergraduates to rational approximations of √2 can be an opportunity to insidiously tell them about many parts of mathematics they certainly don’t want to hear about. In a less pessimistic way, I would say this is a nice way to illustrate the use of several theories in abstract mathematics.

First, you may want to tell them it is not a rational number, which could be easy, unless they never heard about factoring integers.

Then, you could use classical sequences from high school classes: it is easy to check that iterating converges to ±√2, and setting they will even be able to give an explicit formula with a geometric sequence. There is of course a well-known algorithm which speeds up considerably the computation : *Newton’s method*. This can be illustrated geometrically by drawing the graph of a function having √2 as a root (for example ):

- take a rough approximation, such as x=1
- imagine the function is affine (replacing the graph by its tangent)
- use the approximation of the function to calculate an approximation of the solution
- use this newly found rough solution to iterate from step 1

This method requires iterating , and converges considerably faster. Both methods yield sequences of rational numbers converging to √2, so should be considered as methods to obtain fractional numbers giving good approximations of √2. (more…)

### Bicircular quartic curves

Working in the Euclidean (projective) plane, a bicircular quartic curve is defined to be a quartic which is singular at the circular points I and J. We are usually interested in real curves, so the type of singulaity is the same at I or J. Salmon in his *Treatise on Higher Plane Curves* and Basset in his *Elementary Treatise on Cubic and Quartic curves* deal in detail with these curves.

A plane quartic curve has arithmetic genus , but since bicircular quartics have at least two double points, they have *geometric genus* 0 or 1 (genus of the desingularized curve).

Curves having geometric genus 0 are called rational curves (formerly known as unicursal curves), and admit a parameterization by rational functions of one variable.

Families of bicircular curves were defined by Cassini and Descartes by metric properties : Cassini ovals are defined by the equation where A and B are fixed *foci*, while Descartes define Cartesian ovals by the equation (these equations are equivalently to algebraic quartic equations, using appropriate squarings).

### Casey’s *Treatise on analytical geometry*

John Casey’s *Treatise on the analytical geometry of the point, line, circle and conic sections* is a book from the end of the 19th century gathering a wide range of results and constructions in plane geometry known at those times. The *Bibliothèque Nationale de France* gives free access to a digitized version of the book (see there).

The book begins with the definitions of classical coordinate systems: cartesian coordinates, polar coordinates, projective homogeneous coordinates (which were then dubbed trilinear or barycentric). Casey has no taboo concerning points of the field of complex numbers: in particular, many statements make use of cyclic points or zero-radius circles. Much further into the book, the author introduces projections and perspectives, as well as quadratic cones, and harmonic divisions.

Recent Comments