14. A Complete Scene and the Cost of Seeing It

A completed picture is not the end of the renderer. It is the first point at which all of its responsibilities can be examined together. The camera generates samples, geometry supplies candidates, closest-hit traversal selects visibility, materials create local and secondary contributions, and the pixel averages the results. If any one of those stages is hidden behind the final scene, the course has produced an image without producing the understanding it was intended to develop.

We shall therefore finish by making an account. The laboratory separates primary, shadow, reflection and transmission rays, then counts primitive and bounding tests. It also compares linear sphere traversal with two simple bounded groups. The grouped version is not a full production bounding volume hierarchy. It is the smallest experiment that demonstrates why an acceleration structure can reduce work without changing the required closest hit.

The complete route

This table is a debugging route. A wrong pixel can be followed backwards to its sample positions, rays, hit records and material decisions. The stages are not independent in execution, but they have distinct criteria. Keeping those criteria visible prevents a plausible final colour from excusing an unexplained intermediate state.

Linear traversal has a clear cost

With R rays and N primitives, testing every primitive gives up to R × N primitive tests. Early any-hit exits and shrinking closest-hit intervals may reduce actual work, but the traversal still attempts many objects that are nowhere near the ray.

Adding samples multiplies primary routes. Shadows and recursive materials add secondary routes according to what those samples hit. The lesson counters therefore distinguish ray categories before reporting primitive tests. Render time alone is not enough: it varies with browser, device, background work and warm-up. The operation counts describe the algorithmic workload of this particular scene more directly.

Reject a group using an axis-aligned box

An axis-aligned bounding box stores minimum and maximum coordinates on X, Y and Z. A ray can intersect the allowed parameter interval for each pair of slabs. If those three intervals do not overlap, the ray cannot reach anything wholly contained by the box.

function hitBounds(ray, box, tMin, tMax) {
    for (let axis = 0; axis < 3; ++axis) {
        const direction = ray.direction[axis];
        if (Math.abs(direction) < 1e-12) {
            const outside = ray.origin[axis] < box.minimum[axis]
                         || ray.origin[axis] > box.maximum[axis];
            if (outside) return false;
            continue;
        }

        const inverse = 1 / direction;
        let near = (box.minimum[axis] - ray.origin[axis]) * inverse;
        let far  = (box.maximum[axis] - ray.origin[axis]) * inverse;

        if (inverse < 0) {
            [near, far] = [far, near];
        }

        tMin = Math.max(tMin, near);
        tMax = Math.min(tMax, far);
        if (tMax < tMin) {
            return false;
        }
    }
    return true;
}

The laboratory puts spheres into two groups and computes a bound around each. A ray tests both bounds. Only a group whose box is hit has its member spheres tested. Planes remain outside because an infinite plane does not fit a finite world-space box. Triangles could be grouped in the same way, but the small demonstration keeps the before-and-after count easy to inspect.

From two groups to a hierarchy

A bounding volume hierarchy places bounds in a tree. The root encloses a large set. Child bounds divide it, and leaves reference small primitive groups. A traversal tests the root, descends only through intersected children and can order children by near distance to find a close hit earlier. A good build tries to balance the cost of more bound tests against the primitive tests they are likely to reject.

Our two groups demonstrate the rejection mechanism but not a general BVH builder, surface-area heuristic, dynamic update policy or robust traversal stack. Calling it a complete BVH would overstate the result. It is a bounded-group precursor whose counters let us see the operative idea before the data structure becomes elaborate.

Correctness remains the first gate

Switch between linear and bounded traversal. The rendered image should remain the same for the same sample positions and scene. Primitive-test counts should change; visibility should not. If the accelerated image differs, the lower count is not an optimisation result. It is evidence that a required candidate was incorrectly rejected or visited with the wrong interval.

This is the necessary order of judgement. First establish agreement with the simple traversal across deliberately difficult rays, including box boundaries, zero direction components and objects touching bounds. Then measure whether work or time improves. Speed obtained by skipping visible geometry is merely a faster wrong answer.

Read the manual renderer as source

The live course uses one cumulative source file so every lesson stage can be compared without loading a hidden service or external library. Open the browser ray tracer source and locate the vector operations, camera construction, three primitive tests, scene traversal, local shading, secondary-ray creation, sampling loop and metrics. The code is intentionally compact. Compactness is useful only while each responsibility can still be identified.

Canvas receives an ImageData buffer. It does not intersect a sphere, choose a triangle, cast a shadow or refract a ray. That boundary was stated in Lesson 1 and remains true in the completed scene. The browser API displays our answer; it has not answered the ray-tracing questions for us.

You have now constructed a ray traced scene by hand. The important result is not the arrangement of spheres. It is that a pixel can be challenged: which sample created it, which ray carried that sample, which interval admitted the hit, which surface record survived, which secondary routes contributed and what work was spent. When those questions have answers, the image is no longer a convincing black box.