Algebraic Surfaces

Nice geometric shapes come from Algebraic Equations. Defining right equations, any shaped can be created. Such as; (x2+9/4y2+z2-1)3 – x2z3-9/80y2z3=0 plot Heart !!! And so on … such as mathematik/bildergalerie/gallery !!! Just look and see ^_^….

If you have MATLAB, then you can plot and have fun with these shapes, using the codes I give in below…

function AlgSur
d=-2:0.03:2; [X,Y,Z] = meshgrid(d,d,d); % Generate X and Y arrays for 3-D plots

V=X.^6 + Y.^6 +Z.^6-1; % Shape equation; must equel zero

h = patch(isosurface(X,Y,Z,V,0),'FaceColor','red','EdgeColor','none');
% you may change color
camlight; lighting phong
alpha(0.8) % Set transparency; 1 is 'opaque', 0 is 'clear' 
view(3) 
axis off
axis equal

Such as; if V= x6+y6+z6-1=0, shape become Cube. And if exponential value is increased evenly (as 2n), edge of cube become sharper ! ;) …. You can get more equations for Algebraic Surfaces in wolfram. You may change and play the values of equations and discovery new shapes :D

If equaitions are parametric, so the shapes must plot parameticly….. Here is an example of Matlab code;

function Pareq
u=linspace(0,6*pi,60); v=linspace(0,2*pi,60); [u,v]=meshgrid(u,v);
% Generate arrays for 3-D plots

% Parametric equations
x=2*(1-exp(u/(6*pi))).*cos(u).*cos(v/2).^2;
y=2*(-1+exp(u/(6*pi))).*sin(u).*cos(v/2).^2;
z=1-exp(u/(3*pi))-sin(v)+exp(u/(6*pi)).*sin(v);

surf(x,y,z,'FaceColor','interp','EdgeColor','none','FaceLighting','phong')
camlight left
axis equal
axis off

The equations are ;

And the plot of the code comes as this (of course comes one shape, but I show it in different sides for easy look, that’s why it looks 3 of them).

Note; parametric equations seem more complicated than algebraic ones. If you are not a mathematician, it may be hard to make transformations between algebraic and parametric equations. Fortunately, a lot of equations, both algebraic and parametric, are already found and displayed by senior mathematicians. Such as in here wolfram-AlgebraicSurfaces, or maxwelldemon-surfaces or  math/bildergalerie … both algebraic and parametric equations of most of shapes are given. Have fun ! ^_^

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