Basic Ray Tracing
Rendering: Reality
Eye acts as pinhole camera
Photons from light
hit objects
Rendering: Reality
Eye acts as pinhole camera
Photons from light
hit objects
Bounce everywhere
Extremely few
hit eye, form image
ec
019 photons/s
ulb = 1
one lightb
Synthetic Pinhole Camera
Useful abstraction: virtual image plane
Rendering: Ray Tracing
Reverse of reality
• Shoot rays through image plane
• See what they hit
• Secondary rays for:
• Reflections
• Shadows
• Embarrassingly parallel
Local Illumination
Simplifying assumptions:
• Ignore everything except eye, light, and
object
• No shadows, reflections, etc
“Ray Tracing is Slow”
Very true in the past;
still true today
Ray tracing already
used within the
“raster” pipeline
Real-time, fully ray-
traced scenes are [Nvidia OptiX]
here for older
games
Big Hero 6 (2014)
Control (2019)
Fully Path Traced Portal and Quake 2 (2022)
Side Note: RTX
Side Note: DLSS
Why is Ray-Tracing Slow?
Why Slow?
Naïve algorithm: O(NR)
• R: number of rays
• N: number of objects
But rays can be cast in parallel
• each ray O(N)
• even faster with good culling
Why Slow?
Despite being parallel:
1. Poor cache coherence
• Nearby rays can hit different geometry
2. Unpredictable
• Must shade pixels whose rays hit object
• May require tracing rays recursively
Basic Algorithm
For each pixel:
• Shoot ray from camera through pixel
• Find first object it hits
• If it hits something
• Shade that pixel
• Shoot secondary rays
Shoot Rays From Camera
Ray has origin and direction
Points on ray are the positive span
How to create a ray?
Shoot Rays From Camera
Creating a ray:
• Origin is eye
• Pick direction to pierce center of pixel
Whitted-style Ray Tracing
• Turner Whitted introduced ray tracing to
graphics in 1980
• Combines eye ray tracing + rays to light
and recursive tracing
• Algorithm:
1. For each pixel, trace primary ray in
direction V to the first visible surface.
2. For each intersection trace secondary
rays:
• Shadow in direction L to light sources
• Reflected in direction R
• Refracted (transmitted) in direction T
3. Calculate shading of pixel based on light
attenuation
Find First Object Hit By Ray
Collision detection: find all values of t
where ray hits object boundary
Take smallest positive value of t
When Did We Hit an Object?
How do we know?
How can we calculate this efficiently?
Efficient Approximations
Multiple approximate checks eliminate
candidates more efficiently than a single,
accurate check
Checks (in order):
• Ray-Plane intersection
• Ray-Triangle intersection
• Position of intersection on triangle
Ray-Plane Collision Detection
Plane specified by:
• Point on plane
• Plane normal
In-class Activity:
Use the plane equation to determine
where point Q is based on the ray origin P
and direction d ⃗ assuming we also know at
least one other point on this plane
N⋅Q+d=0
N ⋅ (P + d ⃗t) + d = 0
N ⋅ P + N ⋅ d ⃗t = − d
N ⋅ d ⃗t = − (d + N ⋅ P)
N⋅P+d
t=−
N ⋅ d⃗
Q = P + d ⃗t
Ray-Triangle Collision Detection
• Intersect ray with triangle’s supporting
plane:
N = (A - C)x(B - C)
• Check if inside triangle
How to Check if Inside?
• Using triangle edges
• Using barycentric coordinates
• Using projections
Ray-Triangle Collision Detection
Normal:
Ray-Triangle Collision Detection
Normal:
Idea: if P inside, must be left
of line AB
How can we determine if point Q is to
the left or right of a triangle edge?
Intuition
Cross product will point in opposite
direction if point Q is to the right
Therefore dot product will now be negative
(cosϴ < 0 if ϴ > 90°)
Ray-Triangle Collision Detection
Normal:
Idea: if P inside, must be left
of line AB
Inside-Outside Test
Check that point Q is to the left of all edges:
[(B-A)x(Q-A)]⋅n >= 0
[(C-B)x(Q-B)]⋅n >= 0
[(A-C)x(Q-C)]⋅n >= 0
If it passes all three tests, it is inside the triangle
Barycentric Coordinates
Affine frame defined by origin
(t = c) and vectors from c (v
= a-c, w = b-c)
Point can be represented
using area coordinates , β,
(ratio between sub-area
and total triangle area):
Q = a + βb + c
𝛾
𝛼
𝛾
𝛼
Barycentric Coordinates
What does these area
coordinates tell us?
Barycentric Coordinates
If point Q’s
, β, >= 0
and
+β+ =1
then Q is within the
triangle!
𝛼
𝛼
𝛾
𝛾
Barycentric Coordinates
Proportional to lengths of crossproducts:
Aa = ||((C-B)x(Q-B))||/2
Ab = ||((A-C)x(Q-C))||/2
Ac = ||((B-A)x(Q-A))||/2
Beyond Triangle Intersections…
• Barycentric coordinates can interpolate
• Vertex properties
• Material properties
• Texture coordinates
• Normals
kd (Q) = α kd (A) + β kd (B) + γ kd (C)
• Used everywhere!
Barycentric Coordinates in 2D
Project down into 2D and compute
barycentric coordinates
Möller-Trumbore Triangle Intersect
• Introduced as an optimized triangle-ray intersection test
• Based on the barycentric parameterization
• Direction of ray intersection from ray origin becomes
3rd axis (uw are barycentric axes)
• Still commonly used
Full details here:
https://www.scratchapixel.com/lessons/3d-basic-
rendering/ray-tracing-rendering-a-triangle/moller-
trumbore-ray-triangle-intersection
Other Common Intersects
• Sphere
• Box
• Cylinder
Ray Tracing: Shading
• Shading colors the pixels
• Color depends on:
• Object material
• Incoming lights
• Angle of viewer
Object Materials
Different materials can behave very
differently
• opaque vs translucent vs transparent
• shiny vs dull
We classify different responses to light
into “types”
Emissive Lighting
Light generated within material
Diffuse Reflection
Light comes in, bounces out randomly (Lambertian)
Typical for “rough” unpolished materials
View angle doesn’t matter
Specular Reflection
Light reflects perfectly
Typical for smooth, “polished” surfaces
General Opaque Materials
Diffuse-specular spectrum:
What About Translucent?
Subsurface Scattering
What About Translucent?
Subsurface Scattering
Refraction
What About Translucent?
Subsurface Scattering
Refraction
Structural Color
…
Not today.
Phong Shading Model
We’ll talk about the specific math behind
shading models later. For now, let’s
focus on the “ray-tracing” aspect of
shading…
Ray Tracing: Shading
Let I(P, d) be the intensity along ray P + td
I(P, d) = Idirect + Ireflected + Itransmitted
• Idirect computed from Phong model
• Ireflected = krI(Q, R)
• Itransmitted = ktI(Q, T)
Reflection and Transmission
Law of reflection:
θi = θr
Snell’s law of refraction:
ηisinθi = ηtsinθt
(η is index of refraction)
What is this effect?
Total Internal Reflection
• Occurs if:
• ηi > ηt (index of refraction of current
medium > index of refraction of other
medium)
• ϴi > ϴc (angle of incidence > critical
angle)
• Critical angle is an angle of incidence that
provides an angle of refraction of 90°
• No transmission occurs — only reflection
Critical Angle in TIR
• If θt = 90°, light moves along boundary
surface
• If θt > 90°, light is reflected within current
medium
Light and Shadow Attenuation
Light attenuation:
• Light farther from the source contributes
less intensity
Shadow attenuation:
• If light source is blocked from point on an
object, object is in shadow
• Attenuation is 0 if completely blocked
• Some attenuation for translucent objects
Light Attenuation
Real light attenuation: inverse square law
Tends to look bad: too dim
or washed out
So, we cheat:
d is light-to-point distance
Tweak constant & linear terms to taste:
1
fatten(d) =
a + bd + cd 2
Equation used in raytracing assignment
Shooting Shadow Rays
Local Illumination Redux
Simplifying assumptions:
• ignore everything except eye, light, and
object
• no shadows, reflections, etc
• only point lights
• only simple (diffuse & specular)
materials
Beyond Local Shading
