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 u_{n+1} = \frac{u_n+2}{u_n+1} converges to ±√2, and setting U_n = \frac{u_n-\sqrt 2}{u_n+\sqrt 2} 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 x \mapsto x^2-2):

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

This method requires iterating v_{n+1} = \frac 1 2 (v_n + 2/v_n), 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…)

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23 February 2009 at 5:23 pm Leave a comment

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 h^1(\mathcal O_X) = 3, 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 MA \cdot MB = k^2 where A and B are fixed foci, while Descartes define Cartesian ovals by the equation a \cdot MA + b \cdot MB = k (these equations are equivalently to algebraic quartic equations, using appropriate squarings).

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2 January 2009 at 5:33 pm 1 comment

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.

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26 December 2008 at 11:27 pm Leave a comment


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