Exam Topics and Notes

midterm
final

NOTES:

Write down answers that convince me you know what you're talking about. Some questions will require short discussions - I'm looking for the reasons that you gave an answer, not so much the answer itself.
For transformation matrices, you don't need to multiply the matrices out. If you set up the transformation matrices correctly, that will get you credit.
I expect you to know the simple equations that we've covered in class, e.g.,
- computing a normal vector from a corner of a polygin (v1, v2, v3).
- diffuse illumination: dot_product(N,L)
- computing the reflection vector
I don't expect you to memorize the more complex equations e.g.,
- summation method of normal vector calculation,
- the full rendering equation
but you should be able to
- explain an equation that we covered in class if I give it to you, and
- tell me in words what the equation does and how it does it if I don't give it to you but just ask you about it.
Be able to intelligently discuss issues. When there are various ways to do things you should know the alternatives and be able to compare advantages and disadvantages of each, for example:
- for computing ray-object intersections, inverse transforming ray to object space
- in specular reflection, comparing angle made by E+L to N v. angle made by R and L
- directional v. point light sources
- shadowing by transparent objects
- in transparency, blending v. adding colors
- what normal to use for inside intersections of transparent objects

Midterm

Closed book, closed notes
questions will be from material covered in class, most of which is on the slides

Midterm Topics

This is a representative, not exhaustive, list of topics

Computer Graphics - general terms and concepts include:
- vector graphics
- raster graphics
- frame buffer
- scan conversion
3D Graphics - general terms and concepts include:
- Perspective, orthogonal projection
- z-buffer
- scanline algorithms
- viewing frustum
- ray tracing
Vectors and Transformations - use to compute quantities of interest
- normalizing vectors
- dot product
- cross product
- parametric equation of a line
- modeling and viewing transformations
- Affine transformations
- Composing transformations
- Homogeneous coordinates
- scale - uniform and non-uniform
- translation
- rotation - x, y, z axis rotation
- shearing, reflection
- coordinate frames, right v. left hand coordinate system
- transforming data v. transforming coordinate system
- transforming vectors
- transforming normal vectors
- inverse transforming back into object space
- compute normal vector from polygon vertex list (2 methods)
- compute reflection and refraction ray
Ray Tracing Geometry
- Forming camera coordinate system
  - view direction vector
  - up vector
  - third coordinate axis ('over' vector)
- Forming rays through pixels
  - positioning the virtual pixel screen
  - finding the upper left-hand corner of the screen
  - looping through pixel centers
Basic illumination and shading
- Calculating normal vectors
- Ambient: what is it, how is it calculated
- Diffuse: what is it, how is it calculated
- Specular: what is it, how is it calculated
- Shadows: how are they calculated
- Full illumination equation with multiple light sources - be able to explain it
- Light source modeling: point, directional, directed & hooded (Warn)
Recursive Ray Tracing
- Shadow calculation
- Calculating the reflection vector
- Calculating the refraction vector
  - Snell's Law
  - index of refraction
  - Critical angle
  - Wavelength dependence
Intersection Calculations
- Implicit v. explicit forms of equations
- Solve implicit plus parametric ray
- Ray & sphere (algebraic, geometric) intersection
- Ray & infinite plane intersection
- Ray & axis-aligned cube intersection
- Normal vector calculation: first corner v. summation method
- Ray & convex-polyhedron intersection
- Ray & concave-polyhedron (convex/concave polygons) intersection
- Point inside a convex/concave polygon test
- Transforming rays back to object space and intersecting
Solid Texturing
- texturing - the general idea
  - complexity on demand
  - can texture may any display attribute
- solid texturing - the general idea
  - objects sits in 3D texture space
  - F(x,y,z) = <some display attribute>
- checkerboard - modulo arithmetic to alternate between values
- rings - as in checkerboard but use radial distance
- noise
  - random values at integer lattice points
  - interpolate random values
  - index into 3D table using scaled pixel centers
- turbulence - summing scaled multiples of base frequency
- marble - add more variation by using sine function, undulation function, etc.
2D Texturing
- Mapping surface to (u,v) space
- Mapping (u,v) space to indice space for table of values
- Value from table: Closest value, bilinear interpolation
- Pixel size considerations: Mip-mapping

Bounding Elements

ease of computation, efficiency of testing, tightness of fit
object-based bounding volumes
- sphere
- AABB
- slabs
- convex hull
screen space bounding axis aligned rectangles
world space spatial subdivision
- spatial enumeration
- octtree
- BSP
hierarchical bounding elements
object space v. world space

Approximate Point Distribution

Topic	Percent
General Graphics	10
Math, Geometry Vectors and Transformations Ray Tracing Geometry Intersection Calculations	35
Recursive ray tracing	25
Basic Illumination	10
Texturing	10
Bounding elements	10

Sample Questions

terms: raster, frame buffer, scanline, vector graphics, scan conversion, etc.
Given: a normal vector,N, and another vector, A. Assume the vector A is coming into a surface at a point, p, with normal N. Calculate the vector of reflection. Neither N nor A are normalized.
Set up the transformation matrices to (in order) rotate an object around the x-axis, scale it by 2 along the y-axis, and translate it to (3,-5,7).
Set up the transformation matrices to roll an airplane along it's longitudinal axis if
- it's tail is at (3,4,5) and its nose is at (3,4,8).
- it's tail is at (0,0,0) and its nose is at (1,1,0).
Given the diagram for the geometric ray-sphere intersection procedure, how do know when the ray misses the sphere?
In the algebraic ray-intersection procedure, how do you know when the ray misses the sphere?
How do you calculate the intersection of a ray and a plane?
How do you calculate the intersection of a ray and a triangle?
How could you calculate the intersection of a ray defined by source point S=(Sx,Sy,Sz) and direction vector C=(Cx,Cy,Cz) with a rectangular solid which is a copy of an axis-aligned cubed transformed by an affine 4x4 transformation matrix M?
In diffuse illumination, what does N.L model?
How is a refractive color combined with other colors to form the final color?
What is the color of diffuse reflection and why?
What is the color of specular reflection and why?
What is the constant ambient term an approximation for?
Under what object-to-world space transformations can illumination calculations be done in objects space by inverse-transforming the appropriate information, such as light, camera, and intersection point, back to object space.
What is Snell's Law?
How is the refractive ray calculated from the incoming ray and normal vector?
How do you tell if a ray is inside of an object?
When a ray of refraction has passed through an object and has hit a surface ready to exit an object, what normal vector do you use? how is that surface illuminated by the light source?
How would you make a "ring" solid texture that has one color of stripe twice as wide as the other color stripe?

Final Exam

Format

Closed book, closed notes
The final exam will be about two hours long.
Review what you have programmed in the labs. There will be questions concerning specific things required by the lab assignments.
The final is comprehensive with an emphasis on the material not covered in the midterm.
There will be approximately sixteen questions which will require short descriptions of algorithms, pseudo code, setting up matrices, explaining terms in equations, etc. There may be a few other questions that require shorter answers covering the miscellaneous topics we covered toward the end of the quater (photorealistic rendering, sampling, CSG, volume visualization, etc.)
I'm looking for answers that convince me you know what you're talking about. If I ask a yes/no question, a simple 'Yes' or 'No' will receive little value - you always need to explain your answer. Sometimes I ask questions for which I'll accept either a 'Yes' or a 'No' if accompanied by appropriate explanations.
I give partial credit. It's never a good idea to leave an answer blank. Tell me what you know about the problem/answer to get at least partial credit.
I don't expect you to memorize complex equations, but if I give an equation to you, you should be able to explain its terms, describe what it does, be able to modify it appropriately, etc.
You should be able to describe all of the various methods at a high level.
You should read the relevant sections in the book to reinforce what we've covered in class.
You are responsible for what's been covered in class.

Material

Terms: know these terms and similar terms and how they relate to Ray Tracing

Pinhole camera
Ambient, diffuse, specular
Solid texture, noise, turbulence, marble
Surface texture mapping
Recursive ray tracing
Shadows
Reflection
Transparency, Snell's Law, index of refraction, critical angle
Bounding sphere, axis-aligned bounding box, slabs, convex hull
Anti-aliasing, convolution, sampling, comb function, reconstruction
Distributed ray tracing: jittering, Poison distribution, subsamples
Motion blur
Depth of field, lens, focal length, aperature
Gloss
Translucency
Soft shadows: umbra, penumbra
Environmental and bump mapping
CSG
Implicit surufaces, blobs, metaballs
Volume visualization, ray casting
Path notation
Photorealistic rendering, radiosity, path tracing, light tracing, photon mapping, bidirectional path tracing
Monte Carlo integration and sampling, expected value, variance, stratified sampling, importance sampling

Topics (the number of '*' indicate emphasis on the final)

Basics of Computer Graphics *
I won't ask definition questions, but you need to know these to understand questions about ray tracing.
- 2D v. 2 1/2 D v. 3D
- Orthographic v. perspective projections
- scan conversion v. ray tracing
- Raster graphics v. Vector graphics
- Z-buffer v. ray tracing
Vectors, Transformations and comuptations *
There will probably not be any questions specifically asking about vectors and transformations, but you will have to use this information in answering questions about ray tracing.
- Normalizing vectors
- Dot product
- Cross product
- Parametric equation of a line
- Modeling and viewing transformations
- Affine transformations
- Composing transformations
- Homogeneous coordinates
- Scale - uniform and non-uniform
- Translation
- Rotation - x, y, z axis rotation
- Shearing, reflection
- Coordinate frames, right v. left hand coordinate system
- Transforming data v. transforming coordinate system
- Transforming vectors
- Transforming normal vectors
- Barycentric coordinates
Ray Tracing Geometry *
- Forming camera coordinate system
- Forming rays through pixels
- Calculating reflection vectors
- Calculating the refraction vector
  - Snell's Law
  - Index of refraction
  - Critical angle
  - Wavelength dependence
- Forming ray from surface to light source
Basic illumination and shading **
- Ambient: what is it, how is it calculated
- Diffuse: what is it, how is it calculated
- Specular: what is it, how is it calculated
- Shadows: how are they calculated
- Full illumination equation with multiple light sources
Recursive Ray Tracing ***
- Recursively shading a ray
- Blending surface, reflection, transparancy colors
Intersection Calculations **
- Implicit v. explicit forms
- Solve implicit plus parametric ray
- Ray-object intersections in object space
- Transforming ray to object space and intersection info to world space
- Ray-sphere (algebraic, geometric)
- Ray-infinite-plane
- Ray-oriented-square
- Normal vector calculation: e.g., first corner v. summation method
- Ray-convex-polyhedron
- Ray-concave-polyhedron (convex/concave polygons)
- Point inside a polygon test
Texturing: Solid and Surface **
- Texturing - the general idea
- Surface
  - Parameterization of sphere, quadrilateral, triangle, polyhedral mesh
  - function v. table look-up
  - Mip-mapping
- Solid texturing - the general idea
  - functional v. table look-up
  - checkerboard
  - Rings
  - Noise, turbulence, marble
- Environmental mapping
  - preprocessing computations
  - execution-time computations
  - hemi-sphere, hemi-cube
  - limitations & advantagess
- Bump mapping
  - Height field
  - Normal vector perturbation
  - limitations & advantages
Smooth Shading *
- Calculation of vertex normals
- Interpolation of u,v values in a triangle or quadrilateral
- Interpolation of u,v values in an arbitrary polygon in a scanline algorithm
- Gouraud smooth shading
- Phong smooth shading
Speed-ups **
- Bounding volumes
  - Spheres
  - Axis-aligned bounding box
  - Slabs
  - Convex hull
  - Hierarchical organization
- Bounding rectangle of projected image
- Spacial partitioning
Anti-aliasing ***
- Fourier transform: spatial domain & frequency domain
- Low pass filter, high pass filter
- Aliasing: manifestations, definition, in frequency domain
- Convolution theorem: convolution in one is multiplication in the other
- Ideal anti-aliasing given signal and sampling frequency
- Sampling (in both domains): ideal & practical
- Reconstruction (in both domains): ideal & practical
Distributed Ray Tracing ***
- Supersample, adaptive supersampling
- Poison distribution
- Jittering
- Temporal sampling: motion blur
- Gloss
- Tranlucency
- Soft shadows
- Depth of field
- motion blur
CSG models *
- boolean operations of primitive objects
- tree structure & traversal
- determining the normal
Implicit surfaces *
- blob, metaball
- density functions
- threshold value, threshold radius
- radius of influence
- weighted sum of density functions
- distance-based implicit functions
- distance-based elements: point, line, polygon, polygonal mesh, polyhedron
- searching along the ray
- determining the normal
- bulge problem: use of max v. sum
Volumes: density grid *
- attenuation integral
- illumination, self-shadowing
- contributions to light leaving cell
Photorealistic rendering *
- Rendering equation, irradiance, radiance
- Monte Carlo integration
- Path notation
- Probability density function (pdf)
- Expected value
- Variance & standard deviation
- Non-uniform sampling
- Non-uniform distribution
- Importance sampling
- Stratification
- Law of Diminishing returns
- Radiosity
- Path tracing
- Light tracing
- Photon mapping
- Bi-directional path tracing

Approximate Point Distribution

Topic	Percent
General Graphics	5
Forming rays, calculating intersections	10
Recursive ray tracing incl. Illumination & texturing, combining shading effects	20
Speed-ups	10
Anti-aliasing	20
Distributed ray tracing	25
Other topics (alt. texture maps, vol. viz., CSG, impl. surf., 782 topics,)	10

Sample Questions

Given: a normal vector,N, and another vector, A. Assume the vector A is coming into a surface at a point, p, with normal N. Calculate the vector of reflection. Neither N nor A are normalized.
What is Snell's Law?
Given the diagram for the geometric ray-sphere intersection procedure, how do know when the ray misses the sphere?
In the algebraic ray-intersection procedure, how do you know when the ray misses the sphere?
In diffuse illumination, what does N.L model?
How is a refractive color combined with other colors to form the final color?
What is vector graphics?
How do you caculate the intersection of a ray and a plane?
How would you model a coffee cup using a CSG model?
How is sampling in the spatial domain represented in the frequency domain?
What is aliasing? In your answer, show sampling of a sign wave that exhibits aliasing.
Explain why you can't transform a vector the same way you transform a point.
How is motion blur calculated?
What is turbulence and how is it related to noise?
What is the convex hull of an object? How could it be used in a ray tracer?
Discuss the possibilities/problems of using hierarchical bounding boxes together with spatial subdivision. Would the combination work? If not, why not.
Could adaptive anti-aliasing work with jittered sampling? Discuss the possibilities.
Discuss the limitations of environmental mapping.
What are the 3 contributions to the light leaving a cell in volume visualization?
Give an example of a type of illumination that ray tracing does not handle. Use Path notation for your answer.

Last updated 3/13/11