Raytracing in one weekend in C++

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Welcome to my adventure following the raytracing in one weekend series in C++.

Part 3 (Time for Maths)

Now to add shapes

Let's start with spheres

First we have to talk about Ray-Sphere Intersection

A sphere is the easiest thing to test for an intersection with a ray.

The equation for a sphere of radius r is :

X² + Y² + Z² = r²

This equation is true for the surface of a sphere at the center of the world.

That means that any point for which :

X² + Y² + Z² < r²

is true is inside, and outside for the opposite (>)

Displacing the sphere

If the sphere is not placed at the center of the world and its center is at a C_X, C_Y, C_Z then the equation becomes :

(X - C_x)² + (X - C_X)² + (X - C_X)² = r²

If we simplify this with vectors C the center of the sphere and P the point we want to check

The vector from the center to the point is (P - C)

DOT Product

We can then obtain the formula by doing:

(P - C) * (P - C) = (X - C_X)² + (Y - C_Y)² + (Z - C_Z)²

Any point P that satisfies this equation is on the sphere.

Do we hit tho ?

We want to know if our ray P(t) = O + td ever hits the sphere anywhere.

That would mean that there exists some t for which P(t) satisfies the equation

(P(t) - C) * (P(t) - C) = r²

which can be found by replacing P(t) by its expanded :

((O + td) - C) * ((O + td) - C = r²

Resolving - Developing

((O + td) - C) * ((O + td) - C) = r²

Remember that we only care for the t related components of the resolution.

(td + (O - C)) * ( td + (O - C)) = r²

t²d * d + 2td * (O - C) + (O - C) * (O - C) = r²

t²d * d + 2td * (O - C) + (O - C) * (O - C) - r² = 0

Resolving - Quadratic Solve ;)

t2d * d + 2td * (O - C) + (O - C) * (O - C) - r² = 0

We know the radius and the direction of the ray. The vectors are now scalars thanks to the dot product.

Our only unknown is now t and we have t².

We can solve for a quadratic equation the question :

What are the times t for which the ray touches the surface of the sphere ?

Quadratic Solve

$$\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

Where :

- a = d * d

- b = 2d * (O - C)

- c = (O - C) * (O - C) - r²

Now to actually Raytrace

The discriminant is :

0 : 2 real solutions

= 0 : 1 real solution

< 0 : 2 irrational solutions

We will test our code by hard coding the formula

bool HitSphere(const Position& rCenter, double radius, const Ray& rRay)
{
   Vector3 oC = rRay.GetOrigin() - rCenter;
   double a = Dot(rRay.GetDirection(), rRay.GetDirection());
   double b = 2.0 * Dot(oC, rRay.GetDirection());
   double c = Dot(oC, oC) - radius * radius;

   double discriminant = b * b - 4 * a * c;
   return (discriminant >= 0);
}

Color RayColor(const Ray& rRay)
{
   if(HitSphere(Position(0, 0, -1), 0.5, rRay))
       return Color(1, 0, 0);
   Vector3 unitDirection = Unit(rRay.GetDirection());
   double blue = 0.5 * (unitDirection.y + 1.0);
   return (1.0 - blue) * Color(1.0, 1.0, 1.0) + blue * Color (0, 0, 1.0);

}

You should now see a sphere emerge from the background.

It has no shading yet, but it’s a start !

Be aware though that this solution we have for now allows negative t values (behind the camera) but we will fix it later !

We will see shading in the next one

see ya