/// <summary>
/// Transforms a vector by a matrix.
/// </summary>
public static Vector Transform(Matrix mat, Vector vec)
{
return vec.X * mat.U + vec.Y * mat.V + vec.Z * mat.W;
}
/// <summary>
/// Creates a translation matrix.
/// </summary>
public static Matrix Translation(Vector vec)
{
return new Matrix(new Vector(1, 0, 0),
new Vector(0, 1, 0),
new Vector(0, 0, 1),
vec);
}
/// <summary>
/// Creates a possibly non-uniform scaling matrix.
/// </summary>
public static Matrix Scaling(Vector scale)
{
return new Matrix(new Vector(scale.X, 0, 0),
new Vector(0, scale.Y, 0),
new Vector(0, 0, scale.Z),
Vector.Zero);
}
/// <summary>
/// Constructs a ray with an origin and a direction.
/// </summary>
public Ray(Point origin, Vector direction)
{
this.origin = origin;
this.direction = direction.Normalize();
}
/// <summary>
/// Constructs a matrix from four column vectors.
/// </summary>
private Matrix(Vector u, Vector v, Vector w, Vector t)
{
this.u = u;
this.v = v;
this.w = w;
this.t = t;
}
/// <summary>
/// Renders the scene into a pixel buffer.
/// </summary>
public void Render(PixelBuffer pixbuf)
{
float dx = 1.0f / pixbuf.Width, dy = 1.0f / pixbuf.Height;
camera.AspectRatio = (float)pixbuf.Width / pixbuf.Height;
// Free parallelism, why not! Note a Parallel.For loop
// over each row is slightly faster but less readable.
Parallel.ForEach(pixbuf, (pixel) =>
{
var color = Vector.Zero;
float u = pixel.X * dx;
float v = pixel.Y * dy;
var rays = new[]
{
camera.Trace(2 * (u - 0.25f * dx) - 1, 2 * (v - 0.25f * dy) - 1),
camera.Trace(2 * (u + 0.25f * dx) - 1, 2 * (v - 0.25f * dy) - 1),
camera.Trace(2 * (u - 0.25f * dx) - 1, 2 * (v + 0.25f * dy) - 1),
camera.Trace(2 * (u + 0.25f * dx) - 1, 2 * (v + 0.25f * dy) - 1),
};
// Trace a packet of 4 coherent AA rays
var packet = scene.Intersects4(rays);
// Convert the packet to a set of usable ray-geometry intersections
Intersection<Model>[] hits = packet.ToIntersection<Model>(scene);
for (int t = 0; t < 4; ++t)
{
if (hits[t].HasHit)
{
color += new Vector(0.1f, 0.1f, 0.1f);
var ray = rays[t];
var model = hits[t].Instance;
// Parse the surface normal returned and then process it manually
var rawNormal = new Vector(hits[t].NX, hits[t].NY, hits[t].NZ);
var normal = model.CorrectNormal(rawNormal); // Important!
// Calculate the new ray towards the light source
var hitPoint = ray.PointAt(hits[t].Distance);
var toLight = lightPosition - hitPoint; // from A to B = B - A
var lightRay = new Ray(hitPoint + normal * Constants.Epsilon, toLight);
// Is the light source occluded? If so, no point calculating any lighting
if (!scene.Occludes(lightRay, 0, toLight.Length()))
{
// Compute the Lambertian cosine term (rendering equation)
float cosLight = Vector.Dot(normal, toLight.Normalize());
// Calculate the total light attenuation (inverse square law + cosine law)
var attenuation = lightIntensity * cosLight / Vector.Dot(toLight, toLight);
color += model.Material(hits[t].Mesh).BRDF(toLight.Normalize(), ray.Direction, normal) * attenuation;
}
}
}
// Average the 4 per-pixel samples
pixbuf.SetColor(pixel, color / 4);
});
}
/// <summary>
/// Returns the length of a vector.
/// </summary>
public static float Length(Vector u)
{
return (float)Math.Sqrt(Dot(u, u));
}
/// <summary>
/// Reflects this vector about a normal.
/// </summary>
public Vector Reflect(Vector n)
{
return Reflect(this, n);
}
/// <summary>
/// Returns the dot product of this vector with another.
/// </summary>
public float Dot(Vector v)
{
return Dot(this, v);
}
/// <summary>
/// Returns the dot product of two vectors.
/// </summary>
public static float Dot(Vector u, Vector v)
{
return u.X * v.X + u.Y * v.Y + u.Z * v.Z;
}
/// <summary>
/// Returns the cross product of two vectors.
/// </summary>
public static Vector Cross(Vector u, Vector v)
{
return new Vector(u.Y * v.Z - u.Z * v.Y,
u.Z * v.X - u.X * v.Z,
u.X * v.Y - u.Y * v.X);
}
/// <summary>
/// Returns the azimuth of a vector.
/// </summary>
/// <remarks>
/// Horizontal angle, zero towards the z-axis.
/// </remarks>
public static float Azimuth(Vector u)
{
var len = Length(u);
if (len > 0) return (float)Math.Atan2(u.Z, u.X);
else throw new InvalidOperationException("Vector has no direction");
}
/// <summary>
/// Returns the inclination of a vector.
/// </summary>
/// <remarks>
/// Vertical angle, zero on the xz-plane.
/// </remarks>
public static float Inclination(Vector u)
{
var len = Length(u);
if (len > 0) return (float)(Math.Acos(u.Y / len) - Math.PI / 2);
else throw new InvalidOperationException("Vector has no direction");
}
/// <summary>
/// Normalizes a vector to unit length.
/// </summary>
public static Vector Normalize(Vector u)
{
var len = Length(u);
if (len > 0) return u / len;
else throw new InvalidOperationException("Vector has no direction");
}
/// <summary>
/// Transforms a vector by this matrix.
/// </summary>
public Vector Transform(Vector vec)
{
return Transform(this, vec);
}
/// <summary>
/// Returns the cross product of this vector with another.
/// </summary>
public Vector Cross(Vector v)
{
return Cross(this, v);
}
/// <summary>
/// Returns the inverse of a matrix.
/// </summary>
public static Matrix Invert(Matrix mat)
{
// Work out determinant of the 3x3 submatrix.
var det = mat.U.X * (mat.V.Y * mat.W.Z - mat.W.Y * mat.V.Z)
- mat.V.X * (mat.W.Z * mat.U.Y - mat.W.Y * mat.U.Z)
+ mat.W.X * (mat.U.Y * mat.V.Z - mat.V.Y * mat.U.Z);
if (Math.Abs(det) < 0)
throw new ArgumentException("Matrix is not invertible");
// Compute inv(submatrix) = transpose(submatrix) / det.
var inv_u = new Vector(mat.U.X, mat.V.X, mat.W.X) / det;
var inv_v = new Vector(mat.U.Y, mat.V.Y, mat.W.Y) / det;
var inv_w = new Vector(mat.U.Z, mat.V.Z, mat.W.Z) / det;
// Transform the translation column by this inverse matrix.
var inv_t = -(mat.T.X * inv_u + mat.T.Y * inv_v + mat.T.Z * inv_w);
return new Matrix(inv_u, inv_v, inv_w, inv_t);
}
/// <summary>
/// Converts a vector into a point.
/// </summary>
/// <remarks>
/// This is meaningless mathematically.
/// </remarks>
public static Point ToPoint(Vector u)
{
return Point.Zero + u;
}
/// <summary>
/// Corrects an Embree.NET normal, which is unnormalized
/// and in object space, to a world space normal vector.
/// </summary>
public Vector CorrectNormal(Vector normal)
{
return (inverseTranspose * normal).Normalize();
}
/// <summary>
/// Reflects a vector about a normal.
/// </summary>
public static Vector Reflect(Vector i, Vector n)
{
return i - 2 * n * Dot(i, n);
}