Graphics Pipeline: Viewspace & Back face culling

I'm attempting to model a simple graphics pipeline - using Matlab at the moment as a modelling tool to get the transformations correct. I appreciate there are software tools that would make this easier - but I'm looking to understand the maths behind it and hence am looking to mostly use simple functions & matrices that I'm learning from a book (very retro!).

I've successfully got through the stages of defining simple objects and converting them into a universal world space - but have come unstuck with the mathematics required to convert an object into view space and the back face culling.

I believe my view space transformation is correct because when I plot the composite vectors they appear correct - but - when I do the back-face culling, it seems to fail to remove the correct triangles. Given that it depends only on two things, the line-of-sight vector and face normals, I can't work out what I'm doing wrong.

When defining the triangles in local definition space, I did it so that all the normals pointed outwards. I've put my results into the image below

My questions are:

- Have I gone wrong, or are my expectations incorrect? - If so where, and how do I fix?

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## Progress

I've plotted the normal's of all the shapes when in view space. They have all been inverted and now point inwards. Is this a property of the transformation and could it be responsible - or should this have no effect because all the polygons are affected the same?

(Have changed the code to show this)

    clc; clear all; close all;
    %============Initial verticies & Faces of the shape===========
    [s1_vtx,s1_fcs] = Pyramid();
    [s2_vtx,s2_fcs] = Cube();
    %==============Transform Shape 1 ======================
    Tx = 0; Ty = 0; Tz = 0; %Translation vectors for x,y,z axes
    Sx = 2; Sy = 2; Sz = 2;%Scaling factors in x,y,z dimensions
    Rx = pi/2; Ry = pi/4; Rz = pi/4; %Rotating factors in x,y,z dimensions
    transform = scale(Sx,Sy,Sz)*rotate(Rx,0,0)*translate(Tx,Ty,Tz); %Merge transforms together
    s1_vtx = transform*vertcat(s1_vtx,(ones(1,length(s1_vtx)))); %Add row of ones to end for multiplication
    s1_vtx = s1_vtx(1:3,:); %And remove afterwards
    %==============Transform Shape 2 ======================
    Tx = 0.5; Ty = 0; Tz = 0.5;
    transform = scale(1,2,1)*translate(Tx,Ty,Tz);
    s2_vtx = transform*vertcat(s2_vtx,(ones(1,length(s2_vtx))));
    s2_vtx = s2_vtx(1:3,:);
    %======Create World Space ===========
    ws_vtx = horzcat(s1_vtx(1:3,:), s2_vtx(1:3,:)); %remove homogenous column for patching
    ws_fcs = horzcat(s1_fcs,(s2_fcs+(length(s1_vtx))));
    %======Plot World Space ===========
    grid on; hold on;
    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',16, 'Color','r', 'FontWeight','b');
    end
    points = zeros(3,3); %Contains 1 triangle
    for i = 1:length(ws_fcs); %For each triangle
        points(:,1:3) = ws_vtx(:,ws_fcs(1:3,i));
        U = points(:,2) - points(:,1); %Get two non-parallel vectors
        V = points(:,3) - points(:,1);
        average = [0,0,0];
        for j = 1:length(points)
            average(j) = (points(j,1) + points(j,2) + points(j,3))/3;
        end
        N = cross(U,V)/norm(cross(U,V)); %Normal, normalised to mag 1
        scatter3(average(1),average(2),average(3));
        plot3([average(1), average(1)+N(1)],[average(2), average(2)+N(2)],[average(3), average(3)+N(3)]);
    end
    %==================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
    T = viewMat(focus, [Cx,Cy,Cz],Vspec); %Create viewspace transform matrix
    p = norm(focus - [Cx,Cy,Cz]);
    U = T(1,1:3); V = T(2,1:3); N = T(3,1:3); %New Up, Right & View direction vectors
    %============Plot the camera vectors=================
    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)]);
    plot3([Cx, Cx+V(1)],[Cy, Cy+V(2)],[Cz, Cz+V(3)]);
    plot3([Cx, Cx+U(1)],[Cy, Cy+U(2)],[Cz, Cz+U(3)]);
    %==================Transform into View Space===================
    ws_vtx = T*vertcat(ws_vtx,(ones(1, length(ws_vtx)))); %Transform matrix
    ws_vtx = ws_vtx(1:3,:); %Remove homogenous dimension
    origin = T*[Cx;Cy;Cz;1]; %Transform origin
    Cx = origin(1); Cy = origin(2); Cz = origin(3); %remove homogenous dimension
    focus = (T*horzcat(focus,1)')';%Transform focus point
    focus = focus(:,1:3);%remove homogenous dimension
    %==================Plot View Space=================
    figure(); hold on; grid on;
    patch('Faces',ws_fcs','Vertices',ws_vtx','Facecolor', 'none');
    scatter3(Cx, Cy, Cz, 's');
    scatter3(focus(1), focus(2), focus(3), 's');
    plot3([Cx, focus(1)],[Cy, focus(2)],[Cz,focus(3)], 'g');
    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',16, 'Color','r', 'FontWeight','b');
    end
    %================Plot normals of world space==============
    for i = 1:length(ws_fcs); %For each triangle
        points(:,1:3) = ws_vtx(:,ws_fcs(1:3,i));

        U = points(:,2) - points(:,1); %Get two non-parallel vectors
        V = points(:,3) - points(:,1);

        average = [0,0,0];
        for j = 1:length(points)
            average(j) = (points(j,1) + points(j,2) + points(j,3))/3;
        end
        N = cross(U,V)/norm(cross(U,V)); %Normal, normalised to mag 1
        scatter3(average(1),average(2),average(3));
        plot3([average(1), average(1)+N(1)],[average(2), average(2)+N(2)],[average(3), average(3)+N(3)]);
    end

Answer 1

[Mike Garrity]

Are you sure your triangle orientations are consistent?

For the calculations you're using for normals and backfaces, they must be. This means that if you apply the right-hand rule to the vertices of each face, then your thumb should be pointing out of the solid.

[David Hood]

Is this the same as having the normal of each triangle pointing out of the solid? (as in my picture)...if it is then yes, if it isn't - then I guess I need to go through and reassign the triangles

[Mike Garrity]

If you're computing those normals from the points in a consistent way, then yes, all the normals pointing out means that the triangles are consistently oriented.

But I asked because I don't see the normals in your picture of the arrow. Only in the picture of the cube. The fact that the triangles are consistently in one model doesn't imply anything about the other model. Have you done the same plot for the arrow?