This is a method for finding all the roots of a polynomial simultaneously, by applying a sort of Newton-Raphson method. It gets its name from everybody associated wth it: Weierstrass published a version of the algorithm in 1891, and then rediscovered (independently) by Durand in 1960 and Kerner in 1966, as you can see …
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Recap, and background

Two posts ago we showed how, given four points in the plane in general position, but with a few restrictions, it was possible to find two parabolas through those points. We used computer algebra.

The steps were:

1. Create four equations

   \[ (Ax_i+By_i)^2+Cx_i+Dy_i+E=0 \]

   for each of the four …

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It is (well?) known that if \(x = at^2+bt+c\) and \(y=pt^2+qt+r\), then

\[ (Ax+By)^2+Cx+Dy+E=0 \]

where

\[\begin{align*} A&=p\\ B&=-a\\ C&=qv_2-2pv_1\\ D&=-bv_2+2av_1\\ E&=v_1^2-v_2v_3 \end{align*}\]

with \(\langle v1, v2, v3\rangle =\langle a,b,c\rangle \times \langle p,q,r\rangle\); that is, the …

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Introduction

It is (or should be) well known that a parabola has the cartesian form

\[ (Ax+By)^2+Cx+Dy+E = 0. \]

This looks as though there are five values needed, but we can divide through in such a way as to make any of the coefficients we like equal to 1:

\[ (Px+Qy)^2+Rx+Sy+1 = 0. \]

and so we see that only four …

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Although the method is simple to describe, the algebra becomes messy when written in full generality. For example, suppose we use the second method, with three points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) none of which are at the origin.

The three equations are

\begin{gather} (Ax_1+By_1)^2+Cx_1+Dy_1=0\\ …

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A bicentric heptagon is one for which all vertices lie on a circle, and for which all edges are tangential to another circle. If \(R\) and \(r\) are the radii of the outer and inner circles respectively, and \(d\) is the distance between their centres, there is an expression which relates the three values when a …
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Introduction

Poncelet's porism or Poncelet's closure theorem is one of the most remarkable results in plane geometry. It is most easily described in terms of circles: suppose we have two circles \(C\) and \(D\), with \(D\) lying entirely inside \(C\). Pick a point \(p_0\) on \(C\), and find the tangent from \(p_0\) …

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We have seen a number of different error diffusion matrices (and there are others we haven't discussed); the simplest of our matrices has been "Sierra Lite"

\[\frac{1}{4}\begin{bmatrix} 0& *& 2\\ 1& 1& 0 \end{bmatrix}\]

However, there is an even simpler one which seems to give very good …

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We have seen a number of different error diffusion matrices (and there are others we haven't discussed); the simplest of our matrices has been "Sierra Lite"

\[\frac{1}{4}\begin{bmatrix} 0& *& 2\\ 1& 1& 0 \end{bmatrix}\]

However, there is an even simpler one which seems to give very good …

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A totally different approach to dithering is error diffusion. Here, the image is scanned pixel by pixel. Each pixel is thresholded t0 1 or 0 depending on whether the pixel value is greater than 0.5 or not, and the error - the difference between the pixel value and its threshold - is diffuse across neighbouring …
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