plotting a line from a parameter form

Hi all

I'm trying to calculate the intersection between a line and an ellipsoid, and i would like to visualize it as well. What i have so far ist this:

syms x y z s
laenge = 3;
breite = 1;
hoehe = 1;
ellipsoid(0,0,0, laenge,breite,hoehe);
axis equal
xlabel('X-Achse')
ylabel('Y-Achse')
zlabel('Z-Achse')
hold on

% Parameterequation Ellipsoid
e1 = (x^2/laenge^2) + (y^2/breite^2) + (z^2/hoehe^2) == 1;

% Parameterequation Line
g2 = [x;y;z] == [4;0;0] + s.*[-1;0;0];

E = [e1;g2]
t = solve(E, x, y, z, s);
x = t.x
y = t.y
z = t.z
s = t.s

g3 = isolate(g2(1),s)
%snew = subs(x)

%s = [2;2];
%snew = subs(g2(1),s,[2;2])

So i'm able to calculate the intersection correctly, but now i'm not sure how to plot the vector, because i can't replace s in the equations for some reason.

Thanks in advance!

Answer 1

[Star Strider]

In your isolate call, s no longer exists as a variable. You can create ‘g3’ eartlier in the code (when it does exist), and then use ‘g3’ later. At the end of the code, all the variables have been solved for, so I am not sure what you want to do with ‘g3’ at that point.

syms x y z s
laenge = 3;
breite = 1;
hoehe = 1;
ellipsoid(0,0,0, laenge,breite,hoehe);
axis equal
xlabel('X-Achse')
ylabel('Y-Achse')
zlabel('Z-Achse')
hold on

% Parameterequation Ellipsoid
e1 = (x^2/laenge^2) + (y^2/breite^2) + (z^2/hoehe^2) == 1;

% Parameterequation Line
g2 = [x;y;z] == [4;0;0] + s.*[-1;0;0];
g3 = isolate(g2(1),s)
E = [e1;g2];
t = solve(E, x, y, z, s);

x = t.x
y = t.y
z = t.z
s = t.s

%snew = subs(x)

%s = [2;2];
%snew = subs(g2(1),s,[2;2])

With respect to plotting the vector, use the hold function and the fplot3 function.

.

[Carsten]

What i'm wondering is, if i don't include g3 in the solve function like this:

% Parameterequation Line
g2 = [x;y;z] == [4;0;0] + s.*[-1;0;0];
g3 = x1 == 4 - s;
E = [e1;g2];
t = solve(E, x, y, z, s);

s = 2;
g3

and define s with a value, why am i not getting a result (4-2 => g3=2) when i'm printing g3?

[Torsten]

To substitute a numerical value for a symbolic variable, use "subs":

syms x y
y = x + 5;
x = 2;
y           % does not work
subs(y,x)   % works

[Star Strider]

You do.

It is simply not as straightforward as that. It requires creating ‘g3’ as a function of ‘s’ and then substituting and solving.

Example —

syms x y z s
laenge = 3;
breite = 1;
hoehe = 1;
ellipsoid(0,0,0, laenge,breite,hoehe);
axis equal
xlabel('X-Achse')
ylabel('Y-Achse')
zlabel('Z-Achse')
hold on

% Parameterequation Ellipsoid
e1 = (x^2/laenge^2) + (y^2/breite^2) + (z^2/hoehe^2) == 1;

% Parameterequation Line
g2 = [x;y;z] == [4;0;0] + s.*[-1;0;0];
g3(s) = isolate(g2(1),s)
E = [e1;g2];
t = solve(E, x, y, z, s);

x = t.x, char(x)

ans = '[3; -3]'

y = t.y, char(y)

ans = '[0; 0]'

z = t.z, char(z)

ans = '[0; 0]'

s = t.s, char(s)

ans = '[1; 7]'

%snew = subs(x)

%s = [2;2];
%snew = subs(g2(1),s,[2;2])

s = 2;
g3s = solve(g3(s))                          % Substitute & Solve
char(g3s)

ans = '2'

EDIT —

Added the char calls because the symbolic output of submitted posts is still not showing.

.

[Carsten]

Thanks alot for the explanation.

[Star Strider]

As always, my pleasure!