How does ray tracing differ from rasterisation?

Rasterisation maps 3-D geometry onto a 2-D screen triangle-by-triangle — fast, but physically approximate. Ray tracing fires rays from the camera through every pixel, finding which object each ray hits and following secondary rays for shadows, reflections and refractions. The result is physically accurate light transport, at the cost of many more calculations per pixel.

What is path tracing?

Path tracing is a Monte Carlo extension of ray tracing. Instead of evaluating all light paths analytically, the renderer importance-samples a random path through the scene for each pixel sample and averages the results over many samples. With enough samples-per-pixel the image converges to a reference-quality solution with physically correct global illumination, caustics and colour bleeding.

What is the Fresnel effect?

The Fresnel equations describe how much light is reflected versus transmitted at a dielectric boundary. At grazing angles almost all light reflects; at normal incidence most transmits. The Schlick approximation r0 + (1–r0)(1–cos?)5 is used in real-time graphics for glass, water and car paint.

Why does the image look noisy at first?

Path tracing estimates each pixel with random samples. Early samples are few, so variance (noise) is high. As the sample count — shown as 'spp' (samples per pixel) — increases, the Monte Carlo estimate converges and noise decreases proportional to 1/vspp. This progressive refinement is exactly how professional offline renderers like Blender Cycles work.

Ray Tracing Simulator – Interactive Path Tracer

What is Ray Tracing?

Ray tracing is a rendering algorithm that simulates the physical behaviour of light by following rays from the camera through each pixel and computing how they interact with scene geometry. Unlike rasterisation — which projects triangles onto a screen — ray tracing produces global illumination effects such as reflections, refractions, shadows and colour bleeding naturally from the underlying physics.

A path tracer extends this further: instead of evaluating all possible light paths, it importance-samples random paths through the scene using Monte Carlo integration. The rendering equation

L_o(x,?_o) = L_e(x,?_o) + ?_O f_r(x,?_i,?_o) L_i(x,?_i) (n^·?_i) d?_i

is the Kajiya rendering equation (1986). The left side is outgoing radiance; the integral sums all incoming light weighted by the BRDF f_r and cosine foreshortening. Path tracing estimates this integral stochastically — each ray bounce samples one random incoming direction.

Ray–Sphere Intersection

A ray is defined as P(t) = O + tD. Substituting into the sphere equation |P - C|² = r² gives a quadratic in t:

|O + tD - C|² = r²
t² (D·D) + 2t (D·(O-C)) + (O-C)·(O-C) - r² = 0
discriminant ? = b² - c ? t = -b ± v?

If ? < 0 the ray misses the sphere. The smallest positive root gives the closest hit.

Reflection

R = D - 2(D·N^)N^

The reflected direction is simply the incident direction with the normal component negated. Mirror surfaces bounce the ray without any colour or angle randomness.

Refraction (Snell's Law)

n1 sin ?1 = n2 sin ?2 ? T = (n1/n2) I + ((n1/n2) cos ?1 - cos ?2) N^

Glass objects use Snell's law for refraction and the Schlick approximation for Fresnel blending between reflection and transmission:

F(?) ˜ r0 + (1 - r0)(1 - cos ?)5 where r0 = ((n1-n2)/(n1+n2))²

At grazing incidence (? ? 90°) almost all light reflects; at normal incidence (? = 0°) most transmits.

Monte Carlo Path Tracing

The rendering equation cannot be solved analytically for complex scenes. Monte Carlo integration approximates it by drawing N random samples from the hemisphere above each hit point:

L ˜ (1/N) S_i=1..N f(?_i) L(?_i) (n^·?_i) / p(?_i)

where p(?_i) is the sampling PDF. Using cosine-weighted hemisphere sampling (p = n^·? / p) cancels the cosine factor, leaving:

L ˜ p/N S f(?_i) L(?_i)

Error decreases as 1/vN — doubling quality requires four times the samples. Progressive refinement accumulates samples over multiple frames, so the image automatically improves while you watch.

Russian Roulette

Recursion is terminated stochastically: at each bounce a ray is killed with probability (1 - albedo) and surviving rays are scaled up by 1/albedo to keep the estimator unbiased. This prevents infinite loops while correctly handling diffuse global illumination.

Tone Mapping

Real-world luminance ranges far exceed a monitor's [0, 1] range. Reinhard tone mapping compresses HDR radiance values:

L_display = L / (1 + L) then gamma ˜ L^1/2.2

Gamma correction converts linear light values to the perceptual sRGB colour space of your display.

Scene Presets — Phenomena Demonstrated

Preset	Key Phenomena	Materials	Educational Highlight
Cornell Box	Colour bleeding, area shadows, global illumination	Diffuse, mirror, glass	Classic GI benchmark scene (Cornell 1984)
Mirrors	Infinite reflections, light tunnelling	Mirror walls, diffuse spheres	Eigenvalue recursion depth limit
Glass Spheres	Fresnel reflection/refraction, caustics	Glass (IOR 1.33–1.6), diffuse floor	Index of refraction table: water/glass/diamond
Soft Shadows	Penumbra, umbra, two-light interference	Area lights, diffuse/mirror spheres	Shadow hardness vs light source size
Night City	Neon colour bleeding, glossy reflections	Emissive neon, mirror, glass, diffuse	Multiple light sources, additive colour
Colourful	Interreflections between coloured spheres	Diffuse, mirror, glass mix	Colour bleeding — light carries surface colour

Material Models

Material	BRDF Type	Parameters	Real-World Examples
Diffuse (Lambertian)	f_r = ?/p (constant)	Albedo colour	Chalk, plaster, matte paint
Mirror (Specular)	Delta BRDF (Dirac peak)	Reflectance tint	Polished silver, perfect metal
Glass (Dielectric)	Fresnel blend — reflect + transmit	IOR n (1.33–1.9), tint	Water (1.33), glass (1.5), diamond (2.42)
Emissive (Area Light)	L_e = constant emission	Emission colour + intensity	Light panels, neon tubes, LEDs

Curriculum Connections

Topic	Qualification	Concepts Covered
Optics — reflection & refraction	GCSE / A-Level Physics	Snell's law, total internal reflection, critical angle
Wave & ray optics	A-Level Physics / IB	Huygen's principle, Fermat's principle of least time
Computer graphics pipeline	A-Level / BTEC CS	Rasterisation vs ray tracing, z-buffer, shading models
Numerical methods	A-Level Maths / Further Maths	Monte Carlo integration, random sampling, convergence 1/vN
Linear algebra	Further Maths / University	Vector dot/cross products, ONB construction, matrix transforms
Probability & statistics	A-Level Maths	Variance, standard error, importance sampling, PDF/CDF

Ray Tracing Simulator