# What causes this distortion in my perspective projection at steep view angles?

For various reasons – partly educational, partly technical – I'm writing a 3D game engine from scratch, using OpenGL 3.0; aiming for Core Profile, eventually, so of course I'm using a custom matrix stack and shaders.  I've got nearly all of that working, but I'm getting strange results and not sure where I'm making a mistake.

The problem:  using a perspective projection matrix (based on the formulas from the OpenGL 2.1 Reference), everything looks great… until I aim the camera up or down at a steep angle, when it gets horribly distorted, as shown in the screenshots below.

I've checked out the answers here (though the video is unavailable, so I'm not sure if it's the same sort of distortion), and double-checked my code against my references.  So far as I can tell the math is correct and I don't see any glaring typos, but I'm hoping a fresh pair of eyes can help me find the cause – thanks in advance.

### Code

If there's anything else relevant you need to look at, just ask.

Vector3f Util::normalize(const Vector3f &v)
{
float d = std::sqrt(dot(v, v));
return d ? (v / d) : v;
}

Vector4f Util::buildQuaternion(const Angle &angle, const Vector3f &axis)
{
Angle halfAngle = 0.5f * angle;
float c = halfAngle.cos(), s = halfAngle.sin();
return Vector4f(s * axis.x, s * axis.y, s * axis.z, c);
}

Vector3f Util::rotate(const Vector3f &p, const Vector4f &q)
{
Vector4f r = qmul(qmul(q, Vector4f(p.x, p.y, p.z, 0.0f)), ~q);
return Vector3f(r.x, r.y, r.z);
}

void Util::orbit(Vector3f &planet, const Vector3f &sun, const Vector3f &up, const Angle &yaw, const Angle &pitch)
{
Vector3f f = planet - sun;
Vector3f s = cross(up, f);
Vector4f qx = buildQuaternion(pitch, normalize(s));
Vector4f qy = buildQuaternion(yaw, normalize(up));
Vector3f r = rotate(f, qmul(qx, qy));
planet = r + sun;
}

Transform &Transform::makeRotate(const Vector4f &q)
{
float xx = q.x * q.x;
float xy = q.x * q.y;
float xz = q.x * q.z;
float xw = q.x * q.w;
float yy = q.y * q.y;
float yz = q.y * q.z;
float yw = q.y * q.w;
float zz = q.z * q.z;
float zw = q.z * q.w;
float ww = q.w * q.w;

float *m = m_matrix;
m[ 0] = 1-2*(yy+zz); m[ 4] =   2*(xy-zw); m[ 8] =   2*(xz+yw); m[12] = 0.0;
m[ 1] =   2*(xy+zw); m[ 5] = 1-2*(xx+zz); m[ 9] =   2*(yz-xw); m[13] = 0.0;
m[ 2] =   2*(xz-yw); m[ 6] =   2*(yz+xw); m[10] = 1-2*(xx+yy); m[14] = 0.0;
m[ 3] = 0.0;         m[ 7] = 0.0;         m[11] = 0.0;         m[15] = 1.0;
return *this;
}

Transform &Transform::makeRotate(const Angle &angle, const Vector3f &axis)
{
return makeRotate(Util::buildQuaternion(angle, axis));
}

Transform &Transform::makePerspective(const Angle &fov, float aspect, float zNear, float zFar)
{
float f = (fov * 0.5f).cot();
float e = f / aspect;
float g = (zFar + zNear) / (zNear - zFar);
float h = 2.0f * zFar * zNear / (zNear - zFar);

float *m = m_matrix;
m[ 0] =  e ; m[ 4] = 0.0; m[ 8] =  0.0; m[12] = 0.0;
m[ 1] = 0.0; m[ 5] =  f ; m[ 9] =  0.0; m[13] = 0.0;
m[ 2] = 0.0; m[ 6] = 0.0; m[10] =   g ; m[14] =  h ;
m[ 3] = 0.0; m[ 7] = 0.0; m[11] = -1.0; m[15] = 0.0;
return *this;
}

Transform &Transform::makeLookAt(const Vector3f &eye, const Vector3f &center, const Vector3f &up)
{
Vector3f f = Util::normalize(center - eye);
Vector3f u = Util::normalize(up);
Vector3f s = cross(f, u);
u = cross(Util::normalize(s), f);

float *m = m_matrix;
m[ 0] =  s.x; m[ 4] =  s.y; m[ 8] =  s.z; m[12] = 0.0;
m[ 1] =  u.x; m[ 5] =  u.y; m[ 9] =  u.z; m[13] = 0.0;
m[ 2] = -f.x; m[ 6] = -f.y; m[10] = -f.z; m[14] = 0.0;
m[ 3] =  0.0; m[ 7] =  0.0; m[11] =  0.0; m[15] = 1.0;
return translate(-eye);
}

void Camera::orbit3P(const Angle &yaw, const Angle &pitch)
{
/// 3rd-person orbit - move camera around subject
Util::orbit(m_eye, m_center, m_up, yaw, pitch);
m_dirty = true;
}

void Camera::orbit1P(const Angle &yaw, const Angle &pitch)
{
/// 1st-person orbit - change camera direction
Util::orbit(m_center, m_eye, m_up, -yaw, pitch);
m_dirty = true;
}


### Screenshots

Horizontal - all is well

About 45° - distortion begins to appear

Almost 90° - what is this I don't even

(The ortho overlay was to help visualize the camera eye and center locations.)

• Well, it's hard to debug this code, the first thing I would check is the matrix multiplication order, it is very easy to miss this one. Check you multiplication order, I would also try to set identity matrices and debug the others. Jan 28 '15 at 14:31
• Sure, it's some 10 kloc, but not all in one file. I actually think it's rather well organized. :) Anyway, currently I'm only using identity matrices, except for some that are part of the model data and known to be correct (it's from a published game that has no such issues), and the projection matrix. I will check out the ordering, though. Jan 28 '15 at 14:56
• OK, didn't find any problems with the multiplication order: it's strictly hierarchical, resulting in Camera*Scene*Actor*Model (in fact Scene and Actor are always identity for now). I did discover that tweaking the camera's up-vector changes where the distortion happens, so maybe it's not a problem with the projection, but in makeLookAt or orbit. Jan 28 '15 at 17:26

Vector3f f = Util::normalize(center - eye);
Vector3f u = Util::normalize(up);
Vector3f s = cross(f, u);
u = cross(Util::normalize(s), f);


s is not normalized.

It should be:

Vector3f f = Util::normalize(center - eye);
Vector3f u = Util::normalize(up);
Vector3f s = Util::normalize(cross(f, u));
u = cross(s, f); // no norm needed because s,f both unit length and orthogonal


If up and f (forward) are perpendicular, then s is already (approximately) unit length. That is why it works for looking horizontally but not vertically.

• Believe it or not, after comparing my version to the Mesa library, I came to the same conclusion and successfully tested it – just moments before finding your answer. I had this system working in an older version of the engine that used gluLookAt, without the distortion, so I knew it had to be possible. This was definitely the solution, thanks! Jan 29 '15 at 2:51