Circle plotting on different Planes

Hey,

I am trying to plot a series of 2D circle in a 3D plot, which are all on slightly different planes (orientated at different angles).

The information I have is;

Centers of the circles;

X = [2 4 5 7];

Y = [0 2 1 0];

Z = [1 3 5 7];

The radii for the circles are the same at 0.5 units.

When I plot these circles I want to see them orientated so that the normal from each circle is pointing in the direction of the next circle. In other words, the plane the circle is drawn on will be perpendicular to the line joining the centers.

the final picture would look like a small section of a pipe.

How could I achieve this?

All help is greatly appreciated.

Answer 1

[Attila]

Hi,

so your problem is more like math-related right?

If you already have the coordinates of the unoriented circle points, you should multiply them with the result of this, like

Rotated = Rx * Ry * Rz * original

(first circle with the first 3 matrices, etc):

clear all, close all, clc

X = [2 4 5 7];
Y = [0 2 1 0];
Z = [1 3 5 7];

r = 0.5

Coords = [X;Y;Z]

Coords_inc = Coords;
Coords_inc(:,3) = Coords(:,4) - Coords(:,3);
Coords_inc(:,2) = Coords(:,3) - Coords(:,2);
Coords_inc(:,1) = Coords(:,2) - Coords(:,1)

x = 1;
y = 2;
z = 3;

hold on

color = ['r' 'g' 'b' 'k'];

theta = -atan(Coords_inc(y,:)./Coords_inc(z,:)); % angle around x
phi = atan(Coords_inc(x,:)./Coords_inc(z,:));% angle 2 (y)
psi = atan(Coords_inc(y,:)./Coords_inc(x,:));% angle 3 (z)

for i=1:4

      Rx = [1 0 0; 0 cos(theta(i)) -sin(theta(i)); 0 sin(theta(i)) cos(theta(i))]
      Ry = [cos(phi(i)) 0 sin(phi(i)); 0 1 0; -sin(phi(i)) 0 cos(phi(i))]
      Rz = [cos(psi(i)) -sin(psi(i)) 0; sin(psi(i)) cos(psi(i)) 0; 0 0 1]

      t = 0:0.1:2*pi
      x = cos(t)
      y = sin(t)
      z = zeros(1,length(t))

      tol = [X(i); Y(i); Z(i)]
      base = [x;y;z]
      rotated = Rx*Ry*Rz*base
      plot3(rotated(1,:)+X(i),rotated(2,:)+Y(i),rotated(3,:)+Z(i), color(i))

      axis equal

end

This code snippet calculates the required orientations and then builds the required rotation matrices.

[Attila]

Ok, it works now.

[Mazhar]

Ahh... this looks like the stuff!

Going to play around with it and get to understand it correctly.

[Mazhar]

Now for the radii of the circles not being a single value.

What could we do then?

for example I have this:

clear all
close all
clc

X = [2 4 5 7];

Y = [0 2 1 0];

Z = [1 3 5 7];

radii = [0.45 0.46 0.56 0.50 0.48 0.40 0.54 0.50 0.48 0.52;

         0.46 0.48 0.50 0.51 0.48 0.45 0.40 0.45 0.52 0.42;

         0.48 0.46 0.40 0.45 0.50 0.52 0.52 0.48 0.45 0.42;

         0.45 0.46 0.56 0.50 0.48 0.40 0.54 0.50 0.48 0.52];

 angel=(linspace(360/size(radii,2),360,size(radii,2)));

 for k=1:size(radii,1)

     raw_depth = radii(k,:);

     x = ((raw_depth).*(cosd(angel)));
     y = ((raw_depth).*(sind(angel)));
     z = zeros(4,10);

     x = x + X(k);
     y = y + Y(k);
     z = z + Z(k);

     hold all

    h = plot3(x,y,z,'-b');
 end

 grid on

How and where could we add in the rotate command to rotate these "circles"?

[Attila]

I'll look at the problem tomorrow afternoon, if I can find the time.

[Mazhar]

Please, and thanks again.

I'm almost there, just stuck on this last part.

[Mazhar]

Ok so I think I kind of have it...almost.

Just finding it hard to understand one part of your code.

The rotation matrices you used (Rx, Ry, Rz), in your code, you have changed the sign (-/+) on the sin, to that given in other online sources: http://www.siggraph.org/education/materials/HyperGraph/modeling/mod_tran/3drota.htm

The graph produced by your code is the correct one, and if you replace the sign (-/+) on the sin, then the graph is not what I am looking for.

You have it correct, but I am wondering how you knew to change that?

Just want to understand your thinking behind it, so I can understand this better :D

[Mazhar]

Got it to work :D

Thanks for your help!

Answer 2

[Matt J]

You could start by plotting a prototype circle in the xy plane. Then roto-translate them in 3D using a transformation

 R*points + t

where R is a 3x3 rotation matrix and t is a translation vector. This FEX file might help with that

http://www.mathworks.com/matlabcentral/fileexchange/30864-3d-rotation-about-shifted-axis

Finally, use scatter3() to plot the transformed points.

[Mazhar]

Iv'e been trying to read through the code and understand it.

How exactly will I be able to implement this in to my file?

I'm not sure I have all the inputs it requires.

[Matt J]

For example, in the following, I plot circles of radius 5 in a 45 degree tilted plane. Just adjust the transformation parameters to your liking.

r=5;
theta=linspace(0,2*pi,1000);
t=reshape(0:4,1,1,[]); %translations

xyz=[r*cos(theta);r*sin(theta); zeros(1,1000)]; %prototype circles

   xyz=bsxfun(@plus,xyz,t);
   xyz=reshape(xyz,3,[]);

XYZ=num2cell(AxelRot(xyz,45,[1,0,0],[]),2); %transformed circles

scatter3(XYZ{:});

[Mazhar]

OK yeah, I see how it works now.

Is it possible for to create a code that will calculate the angel (rotation) required from the center points data??

[Mazhar]

Also, If my circle does not have a set radius, but I have several radii data for each point of the data. How can I add in a loop to be able to plot such a circle (or shape, as it might not be completely circular)

Answer 3

[Grzegorz Knor]

Try this code:

X = [2 4 5 7];
Y = [0 2 1 0];
Z = [1 3 5 7];

r = 0.5;
[x,y,z] = cylinder(r*ones(size(X)),100);

hold on
for k=length(X):-1:1
    x(k,:) = x(k,:)+X(k);
    y(k,:) = y(k,:)+Y(k);
    z(k,:) = z(k,:)+Z(k);
    h(k) = plot3(x(k,:),y(k,:),z(k,:),'r-');
    direction = rand(1,3);
    alpha = randi(90);
    rotate(h(k),direction,alpha)
end
hold off
view(3)
axis equal

I've used random rotation directions and angles. Just calculate the proper values and replace in the code above.

[Mazhar]

Its the calculating the direction and rotation that I am struggling to achieve.

How exactly does your code work? It looks pretty close.

If you add the center line to the end of that;

plot3(X,Y,Z,'-k')

you are almost there.

Are you able to do this without random rotation, but the correct amount to have it sit correctly on the center line?

[Grzegorz Knor]

First of all there is a small mistake, should be:

z(k,:) = Z(k);

instead of:

z(k,:) = z(k,:)+Z(k);

Each circle should be perpendicular to the line connecting its center point with the center point of the next circle, right? And what about last circle and its orientation?

[Mazhar]

Almost, just the other way round.

The first circle should be horizontal (flat) and the corresponding circle should be perpendicular to the line joining its center and the one before.