# Model gets distorted when rotating the camera

I'm currently developing my own 3d graphics engine and I'm having a hard time figuring out why my 3D models gets distorted when rotating the camera around.

This is my projection matrix. I'm following the OpenGL's Model.

def get_projection_mat(aspect_ratio, camera):
fov = camera.fov
z_near = camera.z_near
z_far = camera.z_far
top = math.tan(fov * 0.5) * z_near
bottom = -top
right = top * aspect_ratio
left = -right
projection_mat = np.identity(4, dtype=float)
projection_mat[0, 0] = 2 * z_near / (right - left)
projection_mat[0, 2] = (right + left) / (right - left)
projection_mat[1, 1] = 2 * z_near / (top - bottom)
projection_mat[1, 2] = (top + bottom) / (top - bottom)
projection_mat[2, 2] = -(z_far + z_near) / (z_far - z_near)
projection_mat[2, 3] = -2 * z_far * z_near / (z_far - z_near)
projection_mat[3, 2] = -1
projection_mat[3, 3] = 0

return projection_mat



This is my view matrix:

def camera_matrix(self):
camera_matrix = np.identity(4, dtype=float)
camera_matrix[:3, 0] = self.left[:]
camera_matrix[:3, 1] = self.up[:]
camera_matrix[:3, 2] = self.forward[:]
camera_matrix[:3, 3] = self.pos[:]

return camera_matrix



This is how I get the ViewProjection matrix:

    projection_mat = get_projection_mat(self.aspect_ratio, self.camera)
view_mat = self.camera.camera_matrix()
view_mat = np.linalg.inv(view_mat)
self.view_projection_mat = np.dot(projection_mat, view_mat)


And, finally, this is the ViewProjectionModel matrix:

    view_projection_model_mat = np.dot(model.transform_mat.T, self.view_projection_mat)


This is the method that I use to rotate the camera:

def rotate(self, yaw, pitch, degress=True):
rotation_mat_y = helper.rotate_matrix_y(yaw, degrees)
rotation_mat_x = helper.rotate_matrix_x(pitch, degrees)
rotation_mat = np.dot(rotation_mat_y, rotation_mat_x)
self.forward = np.dot(np.insert(self.forward, 3, 1), rotation_mat)
self.forward = helper.normalized((self.forward / self.forward)[:3])
self.up = np.dot(np.insert(self.up, 3, 1), rotation_mat)
self.up = helper.normalized((self.up / self.up)[:3])
self.left = np.dot(np.insert(self.left, 3, 1), rotation_mat)
self.left = helper.normalized((self.left / self.left)[:3])



A video demo, showing the problem

To summarize, the order of multiplication is ProjectionMat X ViewMat x ModelMat x ModelFaces. If it isn't clear by now, I'm using Python for this project. Any help would be greatly appreciated.

With best regards, João Pedro

EDIT: Like this?

    directions, r = np.linalg.qr(np.array([self.forward, self.up, self.left]))
self.forward = directions
self.up = directions
self.left = directions


I see two potential errors here:

1. You're transforming directions like they're points.

np.insert(self.forward, 3, 1) says "Add a homogenouse coordinate of 1 to this direction vector so that matrix translation affects it" - but you absolutely do not want translation to affect your directions! Turning 30 degrees is turning 30 degrees, no matter whether I do it at the origin or 50 km away from the origin.

Your translation is already tracked by the pos variable, so don't let it leak into your directions this way. Instead, your 4th component / "w" of a direction vector should be zero, to ensure you factor in only the rotation of the vector.

This also makes the division by w unnecessary, though you were already obliterating its result by normalizing it anyway.

2. You're allowing rounding errors to accumulate

forward, up, and left are used as inputs to compute their own successors, meaning that any error in the input gets replicated in the output, along with any new rounding errors from the latest multiplication / normalization. So each small incremental error adds up and gradually snowballs up to a perceptible scale.

You're re-normalizing the vectors, so you're correcting any errors in their lengths, but not their directions. So as the errors build up, the three basis vectors drift from being fully orthogonal, causing the skewing artifacts you're seeing.

To fix this, you'll want to orthonormalise your basis to correct both magnitude and directional errors and stop them from building up. One way to do this is using the Gram-Schmidt process. You can compute the left vector from the cross product of up and forward to get a mutually-perpendicular unit vector by construction with a bit less work.

• Thank you for your answer, I already fixed the 1º point. But something that i didn't grasp is, where I'm supposed to do the orthonormalization? To the camera's rotation matrix? To the vecto's itself? I've already tried making a column matrix out of my directions,orthonormalizating it and extracting the camera's directions, but things got a little too funky. I'm quite frustated at this point :P. – Joao Pedro Braz Sep 7 at 1:32
• Do it when you update forward, up, and left, ie. in your rotate method. – DMGregory Sep 7 at 1:33
• I edited my question, please take a look. – Joao Pedro Braz Sep 7 at 1:41
• I think that's heavier-weight than you need, given that you're discarding r anyway, but if it works for you then it works for you. – DMGregory Sep 7 at 1:46