rotation once

%Notes: R=v*v'+cos(alpha)*(identity-v*v')+sin(alpha)*cross(v,v);
%Where v is the vector we want to rotate around, not the camera axis itself

%World coordinate system.
ey = [0 0 0]'; %Camera is at world origin.
look_at = [0 0 -1]'; %Bunny is centered around 0,0,-1
up = [0 1 0]'; %1 on the y axis is up. Makes sense

%Get the camera coordinate system.
n = normc(ey-look_at); %z-axis
u = cross(up', n)'; %x-axis
v = cross(n,u); % y-axis

%convert degrees to radians
alpha_deg = 25;
alpha = (alpha_deg*pi)/180;

%Develop the view matrix to go from world to camera coordinate.
%trans = [-(e'*u),-(e'*v),-(e'*n),1]';
trans = [0,-1,-2,1]';
left = vertcat(u',v',n',[0 0 0]);
V = horzcat(left,trans);

%Find the new e aka camera position
new_e = V*vertcat(ey,1);
%Find the new look_at
new_look_at = V*vertcat(look_at,1);
%Find the new up
new_up = V*vertcat(up,1);

%Trim off the fourth for now
new_e = new_e(1:3);
new_look_at = new_look_at(1:3);
new_up = new_up(1:3);

%We now have our rotation matrix, rotating about v, our y-axis
R=v*v'+cos(alpha)*(eye(3)-v*v')+sin(alpha)*cross(v,v);

%Calculate the new u,v,n.
new_n = n'*R;
new_u = u'*R; %Our new x-axis about y.
new_v = v'*R;

%Calculate the rotated e,look_at, and up
new_e = vertcat((new_e'*R)',0);
new_look_at = vertcat((new_look_at'*R)',0);
new_up = vertcat((new_up'*R)',0);
%We need to translate back to the world for new_e,new_look_at, and new_up
%Get the transpose of our V (same as inverse)
V = inv(V);

%Convert back to world coordinates
new_e = V*new_e;
new_look_at = V*new_look_at;
new_up = V*new_up;

%Chop off the fourth for the final results
new_e = new_e(1:3)'
new_look_at = new_look_at(1:3)'
new_up = new_up(1:3)'