\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)

Omar Hayyam Method

This plot explains Omar Khayyam's graphical method to solve the cubic equation x3+a2x=b, for a,b>0:
(http://riotorto.users.sourceforge.net/Maxima/gnuplot/omar/index.html)

(Realised with wxmaxima rel. 17.10.1)

(%i1) load(draw)$
\[\mbox{}\\\mbox{0 errors, 0 warnings}\]
(%i10) a: 3$
b: 26$
r: b/(2·a^2)$
(%i11) draw2d( General settings ·/
      xrange        = [3,3],
      yrange        = [2,4],
      axis_top      = false,
      axis_right    = false,
      axis_left     = false,
      axis_bottom   = false,
      transparent   = true,
      user_preamble = "unset tics",
      title         = "Omar Khayyam solution to the cubic x^3 + a^2 x = b",
      dimensions    = [450,450],
      terminal      = svg,
      file_name     = "omar",

      draw parabola y=x^2/a ·/
      explicit(x^2/a,x,3,3),
      head_length   = 0.1,
      vector([1.626,2],[0.5,0.5]),
      label(["y=x^2/a",1.5,2.1]),

      draw circle: radius r & center (r,0) ·/
      ellipse(r,0,r,r,0,360),
      vector([r,0],[0,r]),
      label(["r=b/(2 a^2)",r0.5,r·2/3]),

      draw axis ·/
      points_joined = true,
      color         = black,
      point_size    = 0,
      points([[0,3.5],[0,1.5]]),
      points([[2.5,0],[2.5,0]]),

      Mark points P e Q·/
      color         = black,
      point_type    = 7,
      point_size    = 1,
      points([[2,4/3]]),
      label(["P",2.25,1.42]),
      points([[2,0]]),
      label(["Q",2.25,0.23]),

      Draw parallel to parabola axis ·/
      point_size    = 0,
      color         = blue,
      line_type     = dots,
      points([[2,2],[2,0.5]]),

      Get the solution ·/
      line_width    = 4,
      color         = red,
      line_type     = solid,
      points([[0,0],[2,0]]),
      label(["The length of this red segment is the solution",1,0.5])) $
-->

Figure 1:E:\Dynamic_Content\Maxima Programs\omar.png
Diagram


Created with wxMaxima.