What's all this, then?

• Parabolas, numerically 2025-02-03
  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: Create four equations \[ (Ax_i+By_i)^2+Cx_i+Dy_i+E=0 \] for each of the four \((x_i,y_i)\) coordinates. Solve the […]
• Parameterization of the parabola 2025-01-12
  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 \(v_i\) values are the elements of the cross product of the vectors of the coefficients. In other words, two […]
• Four point parabolas 2024-12-29
  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: \[ (Px+Qy)^2+Rx+Sy+1 = 0. \] […]
• General expressions 2024-12-25
  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\\ (Ax_2+By_2)^2+Cx_2+Dy_2=0\\ (Ax_3+By_3)^2+Cx_3+Dy_3=0 \end{gather} Solving the first two for \(C\) and \(D\) produces: […]
• Bicentric heptagons 2024-10-21
  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 […]
• Poncelet's porism on non-circular conic sections 2024-10-14
  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 \(p_0\) to \(D\). Let […]
• Image dithering: a very simple error diffusion matrix 2023-07-10
  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 results. The simplest dither matrix usually given is […]
• Image dithering: a very simple error diffusion matrix 2023-07-10
  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 results. The simplest dither matrix usually given is […]
• Image dithering (2): error diffusion 2023-07-08
  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 […]
• Image dithering (1): half toning 2023-07-07
  Image dithering, also known as half-toning, is a method for reducing the number of colours in an image, while at the same time trying to retain as much of its "look and feel" as possible. Originally this was required for newspaper printing, where no shades of grey were possible, and only black and white could […]
• The Pegasus and related methods for solving equations 2023-07-06
  In the previous post, we saw that a small change to the method of false position provided much faster convergence, while retaining its bracketing. This was the Illinois method which is only one of a whole host of similar methods, some of which converge even faster. And as a reminder, here's its definition, with a […]
• The Illinois method for solving equations 2023-07-05
  Such a long time since a last post! Well, that's academic life for you ... If you look at pretty much any modern textbook on numerical methods, of which there are many, you'll find that the following methods will be given for the solution of a single non-linear equation \(f(x)=0\): direct iteration, also known as […]
• Carroll's "improved" Doublets: allowing permutations 2022-11-07
  Carroll originally invented his Doublets in 1877, they were published in "Vanity Fair" (the magazine, not the Thackeray novel) in 1879. Some years later, in an 1892 letter, Carroll added another rule: that permutations were allowed. This allows very neat chains such as: roses, noses, notes, steno, stent, scent Because the words stay the same […]
• Super Doublets: more word ladders with Julia 2022-11-05
  Apparently there's a version of Doublets (see previous post) which allows you to add or delete a letter each turn. Thus we can go from WHEAT to BREAD as WHEAT, HEAT, HEAD, READ, BREAD which is shorter than the ladder given in that previous post. However, we can easily adjust the material from that post […]
• Word ladders with Julia 2022-11-03
  Lewis Carroll's game of Doublets Such a long time since my last post! Well, that's the working life for you. Anyway, recently I was reading about Lewis Carroll - always one of my favourite people - and was reminded of his word game "Doublets" in which one word is turned into another by changing one […]
• Every academic their own text-matcher 2022-06-19
  Plagiarism, text matching, and academic integrity Every modern academic teacher is in thrall to giant text-matching systems such as Ouriginal or Turnitin. These systems are sold as "plagiarism detectors", which they are not - they are text matching systems, and they generally work by providing a report showing how much of a student's submitted work […]
• More mapping "not quite how-to" - Voronoi regions 2022-06-18
  What this post is about In the previous post we showed how to set up a simple interactive map using Python and its folium package. As the example, we used a Federal electorate situated within the city of Melbourne, Australia, and the various voting places, or polling places (also known as polling "booths") associated with […]
• A mapping "not quite how-to" 2022-06-11
  Message about the underlying software NOTE: much of the material and discussion here uses the Python package "folium", which is a front end to the Javascript package "leaflet.js". The lead developer of leaflet.js is Volodymyr Agafonkin, a Ukrainian up until recently living and working in Kyiv. Leaflet version 1.80 was released on April 18, "in […]
• Further mapping: a win and a near miss 2022-06-09
  In this post we look at two Divisions from the recent Federal election: the inner city seat of Melbourne, and the bayside seat of Macnamara. Up until the recent election, Melbourne was the only Division to have a Greens representative. Macnamara, previously known as "Melbourne Ports" has been a Labor stronghold for all of its […]
• Post-election mapping 2022-06-05
  This continues on from the previous post, trying to make some sense of the voting in my electorate of Wills and the neighbouring electorate of Cooper. Both these electorates (or more formally "Divisions"), as I mentioned in the previous post, are very similar in their geography, demography, and history. Last post I simply showed a […]