raytracing: why is my sphere rendered as an oval?

Question

I am to write a raytracer, however I already seem to hit my first big problem. For whatever reason, my sphere (which - since I only begin - I simply color white when a ray hits) is rendered as an oval.

Furthermore, it seems that the distortion is getting worse, the farther I am moving the sphere's center away from x = 0 and y = 0

Here's the intersection and main-loop code:

double const Sphere::getIntersection(Ray const& ray) const
{
  double t;
  double A = 1;
  double B = 2*( ray.dir[0]*(ray.origin[0] - center_[0]) + ray.dir[1] * (ray.origin[1] - center_[1]) + ray.dir[2] * (ray.origin[2] - center_[2]));
  double C = pow(ray.origin[0]-center_[0], 2) + pow(ray.origin[1]-center_[1], 2) + pow(ray.origin[2] - center_[2], 2) - radius_pow2_;
  double discr = B*B - 4*C;

  if(discr > 0)
  {
    t = (-B - sqrt(discr))/2;
    if(t <= 0)
    {
      t = (-B + sqrt(discr))/2;
    }
  }
  else t = 0;

  return t;
}

Sphere blub = Sphere(math3d::point(300., 300., -500.), 200.);
Ray mu = Ray();
// for all pixels of window
for (std::size_t y = 0; y < window.height(); ++y) {
  for (std::size_t x = 0; x < window.width(); ++x) {
    Pixel p(x, y);
    mu = Ray(math3d::point(0., 0., 0.), math3d::vector(float(x), float(y), -300.));

    if (blub.getIntersection(mu) == 0. ) {
      p.color = Color(0.0, 0.0, 0.0);
    } else {
      p.color = Color(1., 1., 1.);
    }
  }
}

What I also do not understand is why my "oval" isn't centered on the picture. I have a window of 600 x 600 pixels, so putting the sphere's center at 300 x 300 should afaik put the sphere in the center of the window as well.

---

my specific solution

(Thanks to Thomas for pushing me to the right direction!)

As Thomas rightly said, my questions where two distinct problems. Considering projecting the sphere in the center, I did as he suggested and changed the origin and projections of the rays.

To get the perspective right, I did not realize I already had to calculate the focal length from the dimensions.

focal_length = sqrt(width^2 + height^2) / ( 2*tan( 45/2 ) )

The result:

Answer 1

This is normal for linear perspective projections, and exacerbated by wide camera angles; see http://en.wikipedia.org/wiki/Perspective_projection_distortion. Most games use something like 90 degrees in the horizontal direction, 45 on either side. But by casting rays up across 600 pixels in the x direction but 300 in the z direction, yours is significantly wider, 126 degrees to be precise.

---

The reason why your sphere doesn't appear centered is that you're casting rays from the bottom left corner of the screen:

mu = Ray(math3d::point(0.,0.,0.),math3d::vector(float(x),float(y),-300.));

That should be something like:

mu = Ray(math3d::point(width/2,height/2,0.),math3d::vector(float(x-width/2),float(y-height/2),-300.));