File:Lagrange simple.svg

Summary

DescriptionLagrange simple.svg	English: derivative of https://en.wikipedia.org/wiki/File:Lagrange_simple.jpg
Date	7 August 2012
Source	File:Lagrange_simple.jpg This diagram was created with MATLAB.
Author	vector User:Zerodamage original User:Jacobmelgaard

Source Code

The graph was made in Matlab with the following source code

% Function data
[X,Y] = meshgrid(-2:0.1:2, -2:0.1:2);
Z = (X.^2).*Y;

% Constraint data
t = -pi:0.01:pi;
xc = sqrt(3)*cos(t);
yc = sqrt(3)*sin(t);
zc = xc.^2 .* yc;

% Plot graph and constraint
figure();
surface = surfc(X,Y,Z);

hold on;
plot3(xc, yc, zc, 'k', 'LineWidth', 2);
plot3(xc, yc ,-10*ones(1,length(t)), 'k', 'LineWidth', 1);
set(gcf, 'Color', [1 1 1]);
set(gcf, 'Position', [46 107 767 682]);
% Axis labels
text(0.2, -3, -10, 'x');
text(2.5, -.5, -10, 'y');
text(-2.4, -2, 0, 'z');

% Annotations of the max and min points
% Point indicating max #1
plot3(sqrt(2), 1, 2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(sqrt(2), 1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
text('Interpreter', 'latex',...
        'String', '$$(\sqrt{2}, 1, 2)$$',...
        'Position', [1.1 .1 2.1],...
        'FontSize', 20);
% Point indicating max #2
plot3(-sqrt(2), 1, 2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(-sqrt(2), 1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
text('Interpreter', 'latex',...
        'String', '$$(-\sqrt{2}, 1, 2)$$',...
        'Position', [-1.6 .1 2.1],...
        'FontSize', 20);
% Point indicating min #1
plot3(sqrt(2), -1, -2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(sqrt(2), -1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
plot3(0,1,-3,'kx')
text('Interpreter', 'latex',...
        'String', '$$(\sqrt{2}, -1, -2)$$',...
        'Position', [.5 -.8 -1.4],...
        'FontSize', 20);
% Point indicating min #2
plot3(-sqrt(2), -1, -2, 'kx', 'MarkerSize', 14, 'LineWidth', 2);
plot3(-sqrt(2), -1, -10, 'kx', 'MarkerSize', 10, 'LineWidth', 2);
text('Interpreter', 'latex',...
        'String', '$$(-\sqrt{2}, -1, -2)$$',...
        'Position', [-1.9 -.8 -1],...
        'FontSize', 20);
hold off;
shading interp;
view([25 18]);
%plot2svg must be retrieved from http://www.zhinst.com/blogs/schwizer/
plot2svg;

Corresponding Mathematica code:

Block[{x, y, f = #^2 #2 &, g = #^2 + #2^2 &, c = 3},
 sol = With[{max = MaxValue[{f[x, y], g[x, y] == c}, {x, y}]},
   Solve[{f[x, y] == max, g[x, y] == c}, {x, y}, Reals]
   ];
 Show[
  Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, Mesh -> {{c}}, 
   MeshFunctions -> g, PlotRange -> All, Boxed -> False, 
   AspectRatio -> 1, PlotPoints -> 30, MeshStyle -> Cyan, 
   PerformanceGoal -> "Quality", ColorFunction -> "DeepSeaColors"],
  Graphics3D[{Red, PointSize[Large], Point[{x, y, f[x, y]} /. sol]}]
  ]
 ]

This is a retouched picture, which means that it has been digitally altered from its original version. Modifications: Vectorization. Modifications made by Zerodamage.

Licensing

	This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
	You are free: to share – to copy, distribute and transmit the work to remix – to adapt the work Under the following conditions: attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. share alike – If you alter, transform, or build upon this work, you may distribute the resulting work only under the same or similar license to this one. https://creativecommons.org/licenses/by-sa/3.0 CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue