42_miniRT

Ray equation

$ray(t) = o + d * t$

• t: scalar parameter
• d: direction (normalized)
• o: origin point (camera)

Plane equation

The vector formed from any two points of a plane and its normal vector are perpendicular.

$(x - p) · n = 0$

• x: any point on the plane
• p: known point on the plane
• n: normal vector

Intersection of a ray with a plane

Let's substitute the ray in the plane equation and solve for t.

$((o + d * t) - p) · n = 0$
$(d * t + o - p) · n = 0$
$(d * t) · n + (o - p) · n = 0$
$(d * t) · n = (p - o) · n$
$t = \frac{(p - o) · n}{d · n}$

Edge cases:

• The denominator d · n == 0, the ray and the plane are parallel.
  • If the numerator (p - o) · n != 0, the ray is outside the plane, there is no intersection.
  • If the numerator (p - o) · n == 0, the ray is in the plane, there are infinite intersections.
• t < 0, the intersection is behind the camera, it is not considered.
• t >= 0, t indicates the distance from the origin of the ray to the intersection.

Sphere equation

Every point on the surface of the sphere are at a constant distance from the center.

$|x - c| = r$

• x: any point on the sphere
• c: center of the sphere
• r: radius of the sphere

Intersection of a ray with a sphere

Let's substitute the ray in the sphere equation and solve for t.
We will not solve using the magnitude formula because it has a square root.
We will substitute k = o - c for clarity.
We will use the fact that the |d·d| == 1, since d is normalized.

$|(o + d * t) - c| = r$
$|d * t + o - c| = r$
$|d * t + k| = r$
$(d * t + k) · (d * t + k) = r²$
$(d · d) t²+ 2(k · d)t + k · k = r²$
$t²+ 2(k · d)t + (k · k - r²) = 0$
$t = \frac{-2d · k \pm \sqrt{4(d · k)² - 4(k · k - r²)}}{2}$
$t = \frac{-2d · k \pm 2 \sqrt{(d · k)² - k · k + r²}}{2}$
$t = -d · k \pm \sqrt{(d · k)² - k · k + r²}$

Edge cases:

• The discriminant < 0, there's no solutions so the ray doesn't intersect the sphere.
• The discriminant == 0, there's one solutions so the ray is tangent to the sphere.
  • If t < 0, the intersection is behind the camera, it is not considered.
  • If t >= 0, t indicates the distance from the origin of the ray to the intersection.
• The discriminant > 0, there's two solutions so the ray intersects the sphere twice.
  • If t_1 < t_2 < 0, the sphere is behind the camera, it is not considered.
  • If 0 < t_1 < t_2, t_1 indicates the distance from the origin of the ray to the intersection.
  • If t_1 < 0 < t_2, the origin of the ray is inside the sphere.

Cylinder equation

Every point on the surface of the cylinder body is at a constant distance from its projection onto the cylinder's axis.

$|(x - c) - ((x - c) · a)a| = r$
$(x - c) · a &lt;= h$

• x: any point on the cylinder body
• c: center of the cylinder
• a: axis of the cylinder
• r: radius of the cylinder
• h: half height of the cylinder

Intersection of a ray with a cylinder

Let's substitute the ray in the cylinder body equation and solve for t.
We will substitute k = o - c, D = (d - (d · a)a) and K = (k - (k · a)a) for clarity.

$|((o + d * t) - c) - (((o + d * t) - c) · a)a| = r$
$|(k + d * t) - ((k + d * t) · a)a| = r$
$|k + d * t - (k · a)a - (d * t · a)a| = r$
$|t (d - (d · a)a) + (k - (k · a)a)| = r$
$|t * D + K| = r$
$(D · D)t² + 2(D · K)t + (K·K - r²) = 0$
$t = \frac{-2(D · K) \pm \sqrt{4(D· K)² - 4(D·D)(K · K - r²)}}{2(D · D)}$
$t = \frac{-(D · K) \pm \sqrt{(D· K)² - (D·D)(K · K - r²)}}{(D · D)}$