03 - Ray Tracing Flashcards
What’s the “Parametric Representation”
- independent variables
- equations for all variables
- plugging in all permissible parameter values yields structure
e.g.:
x = r * cos(t); y = r * sin(t); t = [0,,2PI)
What is the “Explicit Representation”
- structure represented by a function
e.g.:
y = f(x)
What is the “Implicit Representation”
- structure represented as solution set of system of equations
e.g.:
G = { (x, y) | ax + by + c = 0 ∧ a, b, c ∈ ℝ, a ≠ 0 ∨ b ≠ 0}
What’s the principle of a “Pinhole Camera”? What is based on this principle?
- small hole in wall
- outside light generates mirror-inverted image on inside
-> base for “Virtual Camera”
What defines a “Virtual Camera” (3)
- position and viewing direction
- orientation of vertical axis
- image plane (width, height, distance in front of camera)
What’s “Object-Order Rendering”
- “Rasterization”
- sequence of all objects in scene
- determine pixels, affected by object
- determine pixel color (opacity of objects)
What’s “Image-Order Rendering”
- “Ray Tracing”
- sequence of pixels
- determine objects visible at pixel
- determine pixel color (opacity of objects)
What’s the principle of “Ray Tracing”? Of what parts is “Ray Tracing” made off (3)
- one pixel after another
- view ray from camera through pixel
- first object to intersect is sought
- color for pixel is computed
-> ray generation
-> ray intersection
-> shading
What are “Barycentric Coordinates”
- given k points P1, …, Pk ∈ ℝ^n | (k ≤ n+1)
- Q = 𝜆1P1+𝜆2P2+…+𝜆kPk
-> 𝜆1, …, 𝜆k = “Barycentric Coordinates” of Q w.r.t. points P1, …, Pk
What’s a “Simplex”
- region of points Q where 0 ≤ 𝜆i ≤ 1 holds for the Barycentric Coordinates
How do you calculate the “Barycentric Coordinates” for a triangle
- 𝜆1 = A⊿(Q, P2, P3) / A⊿(P1, P2, P3)
- 𝜆2 = A⊿(P1, Q, P3) / A⊿(P1, P2, P3)
- 𝜆3 = A⊿(P1, P2, Q) / A⊿(P1, P2, P3)
How to check if a point is inside a triangle
- if “Barycentric Coordinates” add up to 1 and are all ≥0
What is the “Barycentric Interpolation”
- colors c1, c2, c3 vertices of triangle
- color cQ at Q with Barycentric Coordinates
How is a “Virtual Camera” defined for “Ray Generation”? How is the image plane defined? How is the “View Ray” defined?
Virtual Camera:
- e: position (= projection center)
- z: aim point
- up: up vector
- w: negative view
- u = up x w
- v = u x w
Image Plane:
- d: distance to camera
- l: left border
- r: right border
- b: bottom border
- t: top border
- s: aim point (between borders of image plane)
View Ray:
r(t) = e + td | d = (s - e) / ||(s - e)||
How to calculate intersection point(s) of ray and sphere
- parametric ray representation
-> r(t) = e + td - implicit representation of sphere
-> ||x - c||^2 - r^2 = 0 - plug in ray for x
- solve for t
What are the two extrema of “Illumination”
- specular reflection (angle of incidence = angle of reflection)
- diffuse reflection (equal reflection in all directions)
What is the “Bidirectional Reflectance Distribution Function”? What are its inputs?
- radiometric concept to model reflection at a surface point
- proportion of light from incoming direction that gets reflected in outgoing direction at point x
- 𝝎i: direction of incident irradiance
- x: point of reflection on surface
- 𝝎r: direction of outgoing radiance
directions 𝝎 can also be described by two angles θ and ϕ
How is a “Bidirectional Reflectance Distribution Function” measured
- Gonioreflectometer
What are the parts of the “Phong Shading Model” (3)
- ambient: indirect illumination, light from other surfaces
- diffuse
- specular: imperfect mirroring
What’s the “Lambertian Reflection”
- perfectly diffuse model
- determine intensity
- incoming intensity * diffuse reflection coefficient (wavelength dependent) * cos 𝜃
=> incoming intensity * diffuse reflection coefficient (wavelength dependent) * (N * L)
What’s “Specular Reflection”? Where does it occur?
- perfect mirroring
- in direction of R = (N * L) * 2N - L
What are “Highlights”? How are they calculated?
- non perfect reflections
- dependent on “Specular Reflection” and angle to it
- incoming intensity * specular reflection coefficient * cos^n 𝛼
=incoming intensity * specular reflection coefficient * (R * V)^n - with R being perfect mirror direction and V being view direction
- n is called “Phong exponent” or “Shininess”
What are the parts of the “Phong Reflection Model” (3)
- Ia: ambient intensity
- Id: diffuse intensity
- Is: specular intensity
=> I = Ia + Id + Is
=> I = ka * IL + kd * IL * (N * L) + ks * IL * (R * V)^n
How does “Flat Shading” work
- face normal used for illumination
How does “Phong Shading” work
- illusion of smooth curved surface
- interpolation of “Vertex-Normals”
- compute “Face Normals” for triangles
- each vertex -> sum face normals (opt.: scale by area)
- “Normalize” vertex normals
* introduce point Q * - compute Q = 𝜆1P1 + 𝜆2P2 + 𝜆2P2
- calculate nQ = 𝜆1n1 + 𝜆2n2 + 𝜆2n2
- normalize nQ
How are shadows handled / calculated? What problem can occur here and why? What’s a solution for this?
- send “Shadow Ray” from object to light source
- if it intersects another object -> shadow
problem:
- finite accuracy (floating point numbers) -> ray might intersect surface itself
solution:
- test if object has been intersected once more (for convex objects)
- start ray a bit further away from surface
What types of “Primary and Secondary Rays” are there? How are they used
P: primary ray (from camera to object)
R: reflection ray
S: shadow ray
T: transmission ray
-> recursive calculation of rays (with set recursion depth)
-> weight rays by their reflection coefficient
What’s “Shell’s Law of Reflection”?
- describes change of direction of light when transmitting form one medium to another
-> different refraction indices needed
What is “Distributed Ray Tracing”? What are “Soft Shadows”?
- shoot multiple rays into one direction
-e.g. shadow rays: multiple rays onto light source - same principle for mirroring and transmissions
-> image looks softer / not too immaculate
What are the parts of the “Rendering Equation”
L(x, 𝝎) = E(x, 𝝎) + ∫Ω+ f(x, 𝝎i -> 𝝎) * Lin(x, 𝝎i) * cosθi d𝝎i
L: radiance
E: emission term
f: BRDF
Ω+ and cosθi => Integration over positive hemisphere (arriving light)
f(x, 𝝎i -> 𝝎) => bidirectional distribution function (how much light gets reflected at certain angles)
Lin(x, 𝝎i) => incoming radiance
How can you create “Motion Blur”
- calculate ray tracing for times in time interval
- average out all values
How can you create a “Depth of Field”
- create virtual lens
- choose point in lens -> rays get focused in focal plane
