RE: Plotting ellipses and other functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg37046] RE: [mg37026] Plotting ellipses and other functions*From*: "David Park" <djmp at earthlink.net>*Date*: Mon, 7 Oct 2002 05:24:57 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

David, To plot equations like that, simply use ImplicitPlot. Needs["Graphics`ImplicitPlot`"] ImplicitPlot[(x - 2)^2 + 2(y - 3)^2 == 6, {x, -1, 5}, {y, 1, 5}]; ImplicitPlot[x^3*y + y^3 == 9, {x, -10, 10}, {y, -10, 10}]; You may have to fish a little to obtain the appropriate x and y ranges. Start by making them larger and then narrow down to the region that you want. I have put a new package at my web site for solving conic section problems in the plane. You can solve for complete information on any conic section and obtain a parametric representation for plotting it. The package also comes with complete Help documentation and examples. Using your first example (the second is not a conic). Needs["ConicSections`ConicSections`"] eqn = (x - 2)^2 + 2(y - 3)^2 == 6; The routine ParseConic will take any quadratic equation and return the scale a, eccentricity e, a parametrization, and rotation matrix P, translation T and reflection matrix R that transforms the conic from standard position to its actual position. (In standard position the conic has its foci and verticies on the x-axis with the center at zero.) {{a, e}, curve[t_], {P, T, R}} = ParseConic[eqn] {{Sqrt[6], 1/Sqrt[2]}, {2 + Sqrt[6]*Cos[t], 3 + Sqrt[3]*Sin[t]}, {{{1, 0}, {0, 1}}, {2, 3}, {{1, 0}, {0, 1}}}} We could then plot the curve using ParametricPlot, which is more efficient and controllable. ParametricPlot[Evaluate[curve[t]], {t, -Pi, Pi}, AspectRatio -> Automatic, Frame -> True, Axes -> None, PlotLabel -> eqn]; Knowing a and e we can use the StandardConic routine to obtain all the information about the conic in standard position as a set of rules. standarddata = StandardConic[{a, e}] {conictype -> "Ellipse", conicequation -> x^2/6 + y^2/3 == 1, coniccurve -> {Sqrt[6]*Cos[t], Sqrt[3]*Sin[t]}, coniccurvedomain -> {-Pi, Pi}, coniccenter -> {0, 0}, conicfocus -> {{Sqrt[3], 0}, {-Sqrt[3], 0}}, conicdirectrix -> {x == -2*Sqrt[3], x == 2*Sqrt[3]}, conicvertex -> {{Sqrt[6], 0}, {-Sqrt[6], 0}}} routine to obtain the same information for the conic in its actual position. standarddata // TransformEllipseRules[P, T, R] {conictype -> "Ellipse", conicequation -> (1/6)*((-2 + x)^2 + 2*(-3 + y)^2) == 1, coniccurve -> {2 + Sqrt[6]*Cos[t], 3 + Sqrt[3]*Sin[t]}, coniccurvedomain -> {-Pi, Pi}, coniccenter -> {2, 3}, conicfocus -> {{2 + Sqrt[3], 3}, {2 - Sqrt[3], 3}}, conicdirectrix -> {2*Sqrt[3] + x == 2, x == 2*(1 + Sqrt[3])}, conicvertex -> {{2 + Sqrt[6], 3}, {2 - Sqrt[6], 3}}} David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: David [mailto:davidol at hushmail.com] To: mathgroup at smc.vnet.net How can I plot functions like: (x-2)^2 + 2(y-3)^2 = 6 and x^3y + y^3 = 9 using Mathematica?