Correct view-space transform

Question

I am attempting to model a simple graphics pipeline (i.e. Local->Word->View->Screen->2D spaces).

I've been looking at the algorithm required to transform from world to view-space and using the following transform

Although I have been using U,V,N,C notation rather than Xc,Yc,Zc,e.

And my matrix transformation at first glance, appears correct: When I plot the vectors onto world space, I get a vector from the camera position to the point we're looking at, and two correctly placed new X and Y vectors (see the left hand image below).

By looking along the line of sight vector (such that it disappears,and the focus & camera points overlap), as in the middle figure, should (I believe?) give an accurate example of what the resultant viewspace transformation should look like, with the green vertical and right-pointing vectors showing the new X & Y vectors.

However - when I actually do the view space transform I don't get the middle figure, instead, the resultant transform seems to be looking at the object from a different position possibly aligned with one of the axes? This is shown in the right hand figure - which is actually looking at the object from a slightly lower perspective

My question is essentially what have I done wrong, or am I misunderstanding both the transform and thus my results?

Thanks very much,

David

UPDATES

After stepping through the code - I discovered that my source book was having me convert the line-of-sight vector to spherical coords and then they were being converted straight back again to create R (viewspace coordination matrix). I've eliminated this redundancy (and potential source of errors) but it hasn't solved the problem...

clear; clc; close all;
%======Create World Space (hard-coded values for demo) ===========
ws_vtx = [0,2,0,2,1,0.5,1.5,0.5,1.5,0.5,1.5,0.5,1.5;
    0,0,0,0,-2,0,0,2,2,0,0,2,2;
    0,0,2,2,1,0.5,0.5,0.5,0.5,1.5,1.5,1.5,1.5];
ws_fcs = [1,2,4,3,3,1,6,6,6,6,7,7,9,8,8,8,10,10;
    2,4,3,1,4,4,9,8,7,11,9,13,8,12,10,6,11,13;
    5,5,5,5,1,2,7,9,11,10,13,11,13,13,12,10,13,12];
%==================Create view matrix===================
focus = [1.5,0,1.5]; %The point we're looking at
Cx = 3; Cy = -3; Cz = 3; %Position of camera
Vspec = [0;0;1]; %Specified up direction for view space (i.e. camera orientation)
vector = focus - [Cx,Cy,Cz]; %Vector camera to focus point 
p = sqrt(vector(1)^2 + vector(2)^2 + vector(3)^2); %Total magnitude of vector
N = vector'/p;
V = Vspec - cross(cross(Vspec,N),N); %Create new "up" direction
U = cross(N,V); %Create new x-axis
%Create rotational matrix to view space
R= [U(1),U(2),U(3),0; % U is direction of camera space X axis
    V(1),V(2),V(3),0; % V is direction of camera space Y axis
    N(1),N(2),N(3),0;
    0   ,   0,   0,1];
Tr = [1,0,0,-Cx;
      0,1,0,-Cy;
      0,0,1,-Cz;
      0,0,0,  1];
T = R*Tr; %Total view transform = rotation * translation
%============Plot the camera vectors & World Space=================
grid on; hold on; xlabel('x'); ylabel('y'); zlabel('z');
scatter3(Cx,Cy,Cz,'s'); %Plot the camera position
plot3([Cx, Cx+p*N(1)],[Cy, Cy+p*N(2)],[Cz, Cz+p*N(3)],'g'); %Plot camera vectors
plot3([Cx, Cx+V(1)],[Cy, Cy+V(2)],[Cz, Cz+V(3)],'g');
plot3([Cx, Cx+U(1)],[Cy, Cy+U(2)],[Cz, Cz+U(3)],'g');
scatter3(ws_vtx(1,:),ws_vtx(2,:),ws_vtx(3,:)) %Plot all the points
patch('Faces',ws_fcs','Vertices',ws_vtx','Facecolor', 'none');
for i = 1:length(ws_vtx)
    str = sprintf('%d',i);
    text(ws_vtx(1,i),ws_vtx(2,i),ws_vtx(3,i), str,'FontSize',14, 'Color','r');
end
%====================Viewspace Transform===========================
ws_vtx = T*vertcat(ws_vtx,(ones(1,length(ws_vtx)))); %Transform worldspace
ws_vtx = ws_vtx(1:3,:); %Remove homogenous column
position = T*[Cx;Cy;Cz;1]; focus = T*horzcat(focus,1)'; %Transform set points
position = position(1:3,:); focus = focus(1:3,:); %Remove homogenous columns
%================Plot new viewspace===============================
figure(); grid on; hold on; xlabel('x'); ylabel('y'); zlabel('z');
scatter3(position(1),position(2),position(3),'s'); %Plot new camera vectors
scatter3([position(1); position(1)+focus(1)],...
    [position(2); position(2)+focus(2)],[position(3); position(3)+focus(3)],'s');
plot3([position(1), focus(1)],[position(2), focus(2)],[position(3), focus(3)],'g');
patch('Faces',ws_fcs','Vertices',ws_vtx','Facecolor', 'none');

Answer 1

Your transform looks correct. To transform from world to eye coordinates, I I always use a "lookat" transform, defined by 3 vectors: $\bf{e}$, $\bf{a}$ and $\bf{u}$; in english, the eye position, the point it's looking at, and an up vector, which must not be in the same direction as $\bf{a} - \bf{e}$ (more specifically, not a multiple of it).

The space is defined using $\bf{z} = {{(\bf{e} - \bf{a})}\over{|\bf{e} - \bf{a}}|}$, which means the negative z axis points in the direction of what I'm looking at (this works well for OpenGL); $\bf{x} = {{\bf{u} \times \bf{z}}\over{|\bf{u} \times \bf{z}|}}$ and $\bf{y} = \bf{z} \times \bf{x}$, which, again, for OpenGL, makes $\bf{x}$ rightward on the screen and $\bf{y}$ upward.

To transform into eye coordinates is a matter of subtracting they eye's position, then projecting into the space defined by the vectors above:

$$\begin{bmatrix} \bf{x}_x & \bf{x}_y & \bf{x}_z & 0 \\ \bf{y}_x & \bf{y}_y & \bf{y}_z & 0 \\ \bf{z}_x & \bf{z}_y & \bf{z}_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} 1 & 0 & 0 & -\bf{e}_x \\ 0 & 1 & 0 & -\bf{e}_y \\ 0 & 0 & 1 & -\bf{e}_z \\ 0 & 0 & 0 & 1 \end{bmatrix} $$

Which is exactly what you have to begin with.

Answer 2

I've done some small changes to how I usually construct my view matrix, here is what I've modified:

// put it after N = ...
U = cross(N, Vspec);
U = U / sqrt(dot(U,U)); % normalize
V = cross(U, N);

Then I also set N vector as -N to R matrix, like this:

R= [U(1),U(2),U(3),0; % U is direction of camera space X axis
    V(1),V(2),V(3),0; % V is direction of camera space Y axis
    -N(1),-N(2),-N(3),0;
    0   ,   0,   0,1];

as shown e.g. in the code for lookAt matrix for OpenGL

Now you can orient your first graph, as if you look through your camera, this will give you a reference of how it should look on the second graph.

Then orient second graph so that you look through -Z axis, Y looks up, and X from left to right. Because your camera view vector is -Z, up is Y and right is X. Once you orient your second graph like this, you'll see that arrow looks identically to the first graph, ray on graph 2 will be drawn as point, because it is directed along -Z (your view direction), and in camera space end and start points will fall on a same pixels (another hint that it is correct).

I've also modified a bit camera pos, view direction and axes (just to make it easier to observe for myself).

Here are the values, that I've used: focus = [1.2,0,1.5]; %The point we're looking at Cx = 3; Cy = -3; Cz = 3; %Position of camera

And for the axes: axis([-5,5,-6,6,-7,7]);

I can send you whole code if you need.