1 L Mar-4-05 SMM009, Exam Review Overview Working problems - - PDF document

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Mar-4-05 SMM009, Exam Review 1 L

INSTITUTIONEN FÖR SYSTEMTEKNIK

LULEÅ TEKNISKA UNIVERSITET

Exam Review

David Carr Virtual Environments, Fundamentals Spring 2004

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Overview

• Working problems
• Summary of the course

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INSTITUTIONEN FÖR SYSTEMTEKNIK

LULEÅ TEKNISKA UNIVERSITET

Summary

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Elements of Image Formation

• Objects
• Viewer
• Light source(s)
• Attributes that govern how light interacts with the

materials in the scene

• Note the independence of the objects, viewer, and light

source(s)

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Three-Color Theory

• Human visual system has two types of sensors
• Rods: monochromatic, night vision
• Cones

+ Color sensitive + Three types of cone + Only three values (the tri stimulus values) are sent to the brain

• Need only match these three values
• Need only three primary colors

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Pinhole Camera & Synthetic Camera Model

xp= -x/z/d yp= -y/z/d Use trigonometry to find projection of a point These are equations of simple perspective zp= d center of projection image plane projector p projection of p

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Following the Pipeline: Transformations

• Much of the work in the pipeline is in converting object

representations from one coordinate system to another

• World coordinates
• Camera coordinates
• Screen coordinates
• Every change of coordinates is equivalent to a matrix

transformation

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Clipping

• Just as a real camera cannot “see” the whole world,

the virtual camera can only see part of the world space

• Objects that are not within this volume are said to be clipped
• ut of the scene

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Projection

• Must carry out the process that combines the 3D

viewer with the 3D objects to produce the 2D image

• Perspective projections: all projectors meet at the center of

projection

• Parallel projection: projectors are parallel, center of

projection is replaced by a direction of projection

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Rasterization

• If an object is visible in the image, the appropriate

pixels in the frame buffer must be assigned colors

• Vertices assembled into objects
• Effects of lights and materials must be determined
• Polygons filled with interior colors/shades
• Must have also determine which objects are in front (hidden

surface removal)

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Geometry and Representation

• Geometry
• Relationships among objects in an n-dimensional space
• In computer graphics, objects exist in 3 dimensions
• Minimum set of primitives for building more sophisticated objects

+ Scalars, Vectors, Points

• Representations
• Dimension and basis
• Coordinate systems for representing vectors spaces
• Frames for representing affine spaces
• Change of frames and basis
• Homogeneous coordinates

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Affine Spaces

• Point + a vector space
• Operations
• Vector-vector addition → vector
• Scalar-vector multiplication → vector
• Point-vector addition → point
• Scalar-scalar operations → scalar
• For any point P and the scalars 0,1 define
• 1 • P = P
• 0 • P = 0 (zero vector)

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Convexity

• An object is convex if and only if for any two points in

the object all points on the line segment between these points are also in the object

P Q Q P

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P(α,β)=R+αu+βv P(α,β)=R+α(Q-R)+β(P-Q)

Planes

• A plane be determined by a point and two vectors or by

three points

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Normals

• Every plane has a vector n normal (perpendicular,
• rthogonal) to it
• From point-two vector form P(α,β) = R + αu + βv, we

know we can use the cross product to find n = u × v and the equivalent form (P(α)-P) × n = 0

u v P

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Frames

• Coordinate System is insufficient to present points
• If we work in an affine space we can add a single point,

the origin, to the basis vectors to form a frame

• Frame determined by (P0, v1, v2, v3)
• Within this frame, every vector can be written as

v = α1v1 + α2v2 + … + αnvn

• Every point can be written as

P = P0 + β1v1 + β2v2 + … + βnvn

P0 v1 v2 v3

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A Single Representation

• If we define 0•P = 0 and 1•P = P then we can write

v=α1v1+ α2v2 +α3v3 = [α1 α2 α3 0] [v1 v2 v3 P0]T P = P0 + β1v1+ β2v2 +β3v3

= [β1 β2 β3 1] [v1 v2 v3 P0] T

• Thus we obtain the four-dimensional homogeneous

coordinate representation v = [α1 α2 α3 0] T p = [β1 β2 β3 1] T

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Homogeneous Coordinates

• The general form of 4 dimensional homogeneous coordinates is

p=[x y x w]T

• We return to a three dimensional point (for w ≠ 0) by

x←x/w y←y/w z←z/w

• If w=0, the representation is that of a vector
• Note that homogeneous coordinates replaces points in three

dimensions by lines through the origin in four dimensions

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Working with Representations

Within the two frames any point or vector has a representation of the same form a=[α1 α2 α3 α4 ] in the first frame b=[β1 β2 β3 β4 ] in the second frame where α4 = β4 = 1 for points and α4 = β4 = 0 for vectors and The matrix M is 4 x 4 and specifies an affine transformation in homogeneous coordinates

a=MTb

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Affine Transformations

• Every linear transformation is equivalent to a change in

frames

• Every affine transformation preserves lines
• However, an affine transformation has only 12 degrees
• f freedom because 4 of the elements in the matrix are

fixed and are a subset of all possible 4 x 4 linear transformations

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• 1

d 1 d 1 d 1

z y x

Matrix Representation in 4D Homogeneous Coordinates

• This form is better for implementation because:
• All affine transformations can be expressed this way
• Multiple transformations can be concatenated together
• Translation

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• Rotation about z axis in three dimensions leaves all

points with the same z

• Equivalent to rotation in 2 dimensions in planes of constant z
• x- and y-axes similar

x’ = x cos θ – y sin θ y’ = x sin θ + y cos θ z’ = z

Rotation About the Z-Axis

• 1

1 cos sin sin cos

Rz(θ) =

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• 1

z y x

s s s

x’=sxx y’=syy z’=szz p’=Sp S = S(sx, sy, sz) =

Scaling

• Expand or contract along each axis (fixed point of
• rigin)
• Reflections are negative scaling

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Shear

• Equivalent to pulling faces in
• pposite directions
• On the x-axis

x’ = x + y cot θ y’ = y z’ = z

• 1

1 1 cot 1 H(θ) =

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Inverses

• Could compute inverse matrices by general

formulas,

• But, simple geometric observations are easier
• Translation: T-1(dx, dy, dz) = T(-dx, -dy, -dz)
• Rotation: R -1(θ) = R(-θ)

+Holds for any rotation matrix +Note that since cos(-θ) = cos(θ) and sin(-θ)=-sin(θ), R -1(θ) = R T(θ)

• Scaling: S-1(sx, sy, sz) = S(1/sx, 1/sy, 1/sz)

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Concatenation

• Form arbitrary affine transformation matrices by multiplying
• Rotation, translation, and scaling matrices
• Same transformation is applied to many vertices:
• Cost of forming a matrix M=ABCD is not significant compared to the

cost of computing Mp for many vertices p

• Difficulty is forming a desired transformation
• Order of Transformations
• Note that matrix on the right is the first applied
• Mathematically, the following are equivalent

p’ = ABCp = A(B(Cp))

• Note many references use column matrices to present points. In terms of

column matrices pT’ = pTCTBTAT

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Projection

Perspective Parallel

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Three Aspects of Computer Viewing

• Positioning the camera
• Setting the model-view matrix
• Selecting a lens
• Setting the projection matrix
• Clipping
• Setting the view volume
• All are implemented in the

pipeline

• Default OpenGL projection is
• rthogonal

Orthogonal Projection

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xp = x yp = y zp = 0

Projections and Normalization

• The default projection in the eye (camera) frame is
• rthogonal
• For points within the default view volume
• Most graphics systems use view normalization
• All other views are converted to the default view by

transformations that determine the projection matrix

• Allows use of the same pipeline for all views

M =

• 1

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Simple Perspective

• Center of projection at the origin
• Projection plane z = d, d < 0
• Consider top

and side views

xp =

d z x /

yp =

d z y /

zp = d

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M =

• /

1 1 1 1 d

• 1

z y x

• d

z z y x /

p = ⇒ q =

Homogeneous Coordinate Form

consider q = Mp where

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xp =

d z x /

yp =

d z y /

zp = d

Perspective Division

• However w ≠ 1, so we must divide by w to return from

homogeneous coordinates

• This perspective division yields
• The desired perspective equations

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OpenGL Viewing

near and far measured from camera Orthogonal Projection Perspective Projection

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Pipeline View of Projection Matrices

Model-View Transformation Projection Transformation Perspective Division Clipping Projection nonsingular 4D → 3D Against Default Cube 3D → 2D

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Orthogonal Normalization

• Two steps
• Move center to origin
• Scale to have sides of length 2
• Final Projection, set z =0

2 right left right + left right left 2 top bottom top + bottom top bottom 2 near far far + near far near 1

• P = ST =

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Simple Perspective

• Consider a simple perspective with the COP at the
• rigin, the near clipping plane at z = -1, and a 90

degree field of view determined by the planes

• x = ±z, y = ±z

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Generalized Projection Matrix

1 1

• 1
• N =

after perspective division, the point (x, y, z, 1) goes to x’’ = -x/z y’’ = -y/z z’’ = -(α+β/z) which projects orthogonally to the desired point regardless of α and β

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If we pick α = β =

near + far far near 2near far near far

the near plane is mapped to z = -1 the far plane is mapped to z =1 and the sides are mapped to x = ± 1, y = ± 1 Hence the new clipping volume is the default clipping volume

Picking α and β

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Phong Shading Model

• Has three components
• Diffuse, Specular, Ambient
• Uses four vectors
• To source, To viewer, Normal,
• Ideal reflector
• For each light source and each color component, the Phong

model can be written (without the distance terms) as I =kd Id l · n + ks Is (v · r )α + ka Ia

• For each color component we add contributions from all sources

φ

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r = 2 (l · n ) n - l

Components

• Specular surfaces, ideal reflector
• Normal is determined by local orientation
• Angle of incidence = angle of reflection
• The three vectors must be coplanar
• Ir ~ ks I cosαφ
• Shininess coeffient, metal 100-200, plastic 5-10
• Lambertian Surface
• Perfectly diffuse reflector, light scattered equally in all directions
• Amount of light reflected is proportional to the vertical component of

incoming light + reflected light ~cos θi, cos θi = l · n if vectors normalized + 3 coefficients, kr, kb, kg show how much of each color is reflected

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Distance Terms

• The light from a point source that reaches a surface is

inversely proportional to the square of the distance between them

• We can add a factor of the

form 1/(a + bd +cd2) to the diffuse and specular terms

• The constant and linear terms soften

the effect of the point source

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Object Oriented Models

• Hierarchical models, trees, and DAGs
• Scene graphs
• Other graphical trees
• Constructive Solid Geometry

+ Solid primitives, set operations ∪, ∩, & - + Construct expression trees

• Shade trees

+ Expressions for custom rendering

• BSP trees

+ Depth ordering

• Quadtrees and Octrees

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Scene Graphs, Design Goals

• Node-Link Structure
• Links can be implicit
• Different types of nodes for:
• Transformations
• Object aggregation
• Material properties
• Cameras
• Want:
• Properties to propagate down the

graph

• Coordinates to be local to each

group

• Transformations to apply to all

children

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Scene Graphs, Basic Class Structure

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Cameras in a Scene Graph

• Views of the scene graph
• Separate from the scene
• Can also be interior to a scene graph
• Provide view of part of the scene
• For example, set below an InvisibleGroupNode

+ View of the hidden for “x-ray” vision

• Need to provide settings such as:
• Location, orientation, view angle, …

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Events in Scene Graphs

• We need a way to interact
• Nodes should generate events when accessed, selected, …
• Applications register for event notification

+ Listeners attached to scene graph nodes + Multiple listeners allowed

• May be associated with any scene graph node
• Two kinds
• Input, mouse events, etc.
• Object, result from modifications of the scene graph
• Events passed up the tree to ancestors by default
• The application can stop this.

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Discrete Methods

• XOR writing mode and buffers
• Pixel pipeline
• Texture Mapping
• Environment mapping, box maps
• Aliasing/anti-aliasing
• RGBA, composition, and blending
• Translucency, fog, anti-aliasing

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Renderer Implementation

• 4 Tasks
• Modeling
• Geometric processing
• Rasterization
• Display
• Clipping
• Line

+ Cohen-Sutherland + Liang-Barsky + Cohen-Sutherland in 3D

• Polygon

+ Sutherland-Hodgman

• Hidden surface removal
• Back-face removal
• Depth sorting
• Painter’s algorithm
• Z-buffer algorithm
• Octree methods
• Scan conversion
• Digital difference analyzers
• Bresenham’s algorithm
• Circles
• Flood fill
• Antialiasing

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Questions?