Visualize conic with Matlab or Octave - matlab

Question

I like to visualize conics with Matlab or Octave. The (general) conic is given by the equation 0 = ax² + bxy + cy² +dxz +eyz+f*z² for a point p=(x,y,z). How can I plot this with Matlab or octave if I know the parameters a,b,c,d,e and f? Or respectively, how can I find the points that satisfy this equation?

Answer 1

Since you are asking for the conics, I understand that you are probably referring to the 2D contours of the general conic equation. I will also cover how to visualize this equation in other different ways.
For all the following examples, I have set the conic constants so that I obtain a hiperboloid.
My code is written in MATLAB syntax. If you are using Octave, it might differ slightly.
Visualizing 2D Conics with CONTOUR
I have isolated z in terms of x and y, from the general conic equation:
z = (1/2)*(-d*x-e*y±sqrt(-4*a*f*x.^2-4*b*f*x.*y-4*c*f*y.^2+d^2*x.^2+2*d*e*x.*y+e^2*y.^2))/f;
Since z is a piecewise function due to (± sqrt), I need to make sure that I plot both hemispheres. I designate z1 for +sqrt, and z2 for -sqrt.
Finally, I plot the contours for z1 and z2 that will yield the set of conics in 2D. This conics will be circles of different radius.
Code:
clear all;
clc;
% Conic constants.
a = 1;
b = 0;
c = 1;
d = 0;
e = 0;
f = -1;
% Value for x and y domain.
v = 10;
% Domain for x and y.
x = linspace(-v,v);
y = linspace(-v,v);
% Generate a 2D mesh with x and y.
[x,y] = meshgrid(x,y);
% Isolate z in terms of x and y.
z1 = (1/2)*(-d*x-e*y+sqrt(-4*a*f*x.^2-4*b*f*x.*y-4*c*f*y.^2+d^2*x.^2+2*d*e*x.*y+e^2*y.^2))/f;
z2 = (1/2)*(-d*x-e*y-sqrt(-4*a*f*x.^2-4*b*f*x.*y-4*c*f*y.^2+d^2*x.^2+2*d*e*x.*y+e^2*y.^2))/f;
% Find complex entries in z.
i = find(real(z1)~=z1);
j = find(real(z2)~=z2);
% Replace complex entries with NaN.
z1(i) = NaN;
z2(j) = NaN;
figure;
subplot(1,2,1);
% Draw lower hemisphere.
contour(x,y,z1,'ShowText','on');
% Adjust figure properties.
title('2D Conics: Lower hemishphere');
xlabel('x-axis');
ylabel('y-axis');
axis equal;
grid on;
box on;
axis([-10 10 -10 10]);
subplot(1,2,2);
% Draw upper hemisphere.
contour(x,y,z2,'ShowText','on');
hold off;
% Adjust figure properties.
title('2D Conics: Upper hemishphere');
xlabel('x-axis');
ylabel('y-axis');
axis equal;
grid on;
box on;
axis([-10 10 -10 10]);
Output:
Visualizing 3D Conics with CONTOUR3
Same as on the previous example, but now we plot the set of conics in 3D.
Code:
clear all;
clc;
% Conic constants.
a = 1;
b = 0;
c = 1;
d = 0;
e = 0;
f = -1;
% Value for x and y domain.
v = 10;
% Domain for x and y.
x = linspace(-v,v);
y = linspace(-v,v);
% Generate a 2D mesh with x and y.
[x,y] = meshgrid(x,y);
% Isolate z in terms of x and y.
z1 = (1/2)*(-d*x-e*y+sqrt(-4*a*f*x.^2-4*b*f*x.*y-4*c*f*y.^2+d^2*x.^2+2*d*e*x.*y+e^2*y.^2))/f;
z2 = (1/2)*(-d*x-e*y-sqrt(-4*a*f*x.^2-4*b*f*x.*y-4*c*f*y.^2+d^2*x.^2+2*d*e*x.*y+e^2*y.^2))/f;
% Find complex entries in z.
i = find(real(z1)~=z1);
j = find(real(z2)~=z2);
% Replace complex entries with NaN.
z1(i) = NaN;
z2(j) = NaN;
% Lower hemisphere. Draw 20 conics.
contour3(x,y,z1,20);
hold on;
% Upper hemisphere. Draw 20 conics.
contour3(x,y,z2,20);
hold off;
% Adjust figure properties.
title('3D Conics');
xlabel('x-axis');
ylabel('y-axis');
zlabel('z-axis');
axis equal;
grid on;
box on;
axis([-10 10 -10 10 -10 10]);
Output:
Visualizing Quadrics with ISOSURFACE
I have isolated f in terms of x, y and z, from the general conic equation, and renamed it to f_eq:
f_eq = -(a*x.^2+b*x.*y+c*y.^2+d*x.*z+e*y.*z)./z.^2;
Finally, I obtain the set of points that satisfy the equation f_eq = f, which is in fact an isosurface that yields a quadric; in this example a hiperboloid.
Code:
clear all;
clc;
% Conic constants.
a = 1;
b = 0;
c = 1;
d = 0;
e = 0;
f = -1;
% Value for x, y and z domain.
v = 10;
% Domain for x ,y and z.
x = linspace(-v,v);
y = linspace(-v,v);
z = linspace(-v,v);
% Generate a 3D mesh with x, y and z.
[x,y,z] = meshgrid(x,y,z);
% Evaluate function (3D volume of data).
f_eq = -(a*x.^2+b*x.*y+c*y.^2+d*x.*z+e*y.*z)./z.^2;
% Draw the surface that matches f_eq = f.
p = patch(isosurface(x,y,z,f_eq,f));
isonormals(x,y,z,f_eq,p)
p.FaceColor = 'red';
p.EdgeColor = 'none';
% Adjust figure properties.
title('Quadric');
xlabel('x-axis');
ylabel('y-axis');
zlabel('z-axis');
axis equal;
grid on;
box on;
axis([-10 10 -10 10 -10 10]);
camlight left;
lighting phong;
Output: