Plane in 3D space

I'm trying to define a plane in 3D for interpolation. Since it needs to be able to be any plan, my though was to first define x and y and then rotate them. Then define z, independently of x and y. That way I wanted to z-plane tilt around the x axis and rotate with x and y. In short, it would all be defined by a center point and two angles (rotation and plane tilt). However, this doesn't work when i start rotating x and y. Tilt is working fine, but the rotation seems to cause z to be place wrong or smth. Can anyone help a bit? I posted the code below...

% Set variables (later to be function input
cp    = [0.0 -0.0 0.04]; % Center point coordinats              [m]
xsize = 0.08;            % Size along x-dim of the sample plane [m]
ysize = 0.08;            % Size aling y-dim of the sample plane [m]
N     = 500;             % Spatiel resolution                   [px]
angR  = 90;              % Rotational angle                     [deg]
angT  = 90;              % Tilt angle                           [deg]

% Define standard x and y
x = linspace(-xsize/2, xsize/2, N);
y = linspace(-ysize/2*cosd(angT), ysize/2*cosd(angT), N);

% Set rotation matrix
R = [cosd(angR) -sind(angR); ...
     sind(angR)  cosd(angR)];

% Rotate x and y coords
XY = R*[x; y];
X  = cp(1)+repmat(XY(1,:), [N 1]);
Y  = cp(2)+repmat(XY(2,:)',  [1 N]);

% Define Z (does not depend on X or Y)
Z  = linspace((-ysize/2)*sind(angT), (ysize/2)*sind(angT), N);
Z  = cp(3)+repmat(Z', [1 N]);

Answer 1

[Geoff]

Without really trying to look at your maths, let me suggest that you will have less issues if you treat your 3D objects in a standard way and work backwards from there.

The normal way to define a plane in 3D is to specify 3 vectors: centre, up, and right. The last two define the plane's axis system and are usually unit vectors (to obtain the forward-vector of the axis you take the cross-product of up and right).

These can go into a 3x3 transformation matrix as follows:

T = [right;forward;up];

Normally, you use a 4x4 (or sometimes 4x3) transformation matrix, but let's stick with the 3x3.

You can define your rotation (rotation about Z-axis) as you already have, but in matrix form:

rotZ = @(x) [cosd(x) -sind(x) 0; sind(x) cosd(x) 0; 0 0 1];

And a the tilt is rotation about the X-axis:

rotX = @(x) [1 0 0; 0 cosd(x) -sind(x); 0 sind(x) cosd(x)];

[by the way, your rotations all seem to be clockwise, whereas anticlockwise is more standard -- I've stuck with clockwise here]

Now your initial plane transformation is the unit. Z is up, and the X- and Y-axes are on the plane.

T = diag([1 1 1]);  % Unit transform

You're ready to rotate and tilt your plane:

% First tilt by 90 degrees, then rotate by 45 degrees.
S = rotZ(45) * rotX(90) * T;

You could pull your axes out like so:

X = S(1,:);
Y = S(2,:);
Z = S(3,:);

... and then linspace a mesh by projecting onto these new axes (remembering they should be unit-length)...

But remember that an arbitrary transformation will not define a mesh with a 3 simple vectors of x,y, and z co-ordinates. Every single x,y,z vector in a mesh could be unique. So that would be the wrong thing to do.

Instead you need to rotate the whole mesh. First, generate the original mesh:

x = linspace(-xsize,xsize,N) / 2;
y = linspace(-ysize,ysize,N) / 2;
[mx,my,mz] = meshgrid(x,y,0);
M = [mx(:) my(:) mz(:)];

M contains co-ords of untransformed mesh points organised in rows.

Mtrans = M * S;

Finally you add your centre point back to the transformed mesh.

Mfinal = Mtrans + repmat(cp, size(M,1), 1);

Mfinal now contains your whole mesh as a collection of points, rotated and translated however you like. It's up to you to work out what to do with it. =)

[Jakob Sørensen]

Thanks for the help. However, I can't seem to figure out how to use this for making my plan of sample points. Also, no matter what i change the angle to when calculating S, it doesn't seem to change the values of Mfinal. Am I doing something wrong here?

[Jakob Sørensen]

Never mind, i figured it out myself :p

[Geoff]

Oh yeah, sorry about that. I was meant to use Mtrans in that last calculation.