http://www.ugrad.cs.ubc.ca/~cs314/Vjan2005

1
Prog 2, Kangaroo Hall of Fame, Midterm Review,
(Interpolation if time)Week 6, Wed Feb 9

• http//www.ugrad.cs.ubc.ca/cs314/Vjan2005

2
Program 2 Corrections/Clarifications

• handin 314 proj2 (not 414)
• f not s to toggle flat/smooth shading
• s already in use for camera
• add t to toggle between randomly colored and
  grey terrain
• makes it easier to check if lighting correct
• add u to replace terrain with new randomly
  generated geometry
• consider adding for a bit of extra credit
• /- toggle to increment/decrement abs cam speed

3
Program 2 Corrections/Clarifications

• roll/yaw confusion
• first para. correct, second para. wrong
• left horiz drag yaw, right horiz drag roll
• image flying expertise courtesy of Matt Baumann

4
Program 2 Quick Demo
5
Review Midpoint Algorithm

• moving incrementally along x direction
• draw at current y value, or move up to y1?
• check if midpoint between two possible pixel
  centers above or below line
• candidates
• top pixel (x1,y1),
• bottom pixel (x1, y)
• midpoint (x1, y.5)
• check if midpoint above or below line
• below top pixel
• above bottom pixel
• assume , slope

6
Review Bresenham Algorithm

• yy0 e0
• for (xx0 x lt x1 x)
• draw(x,y)
• if (2(edy) lt dx)
• e edy
• else
• yy1
• eedy-dx

• all integer arithmetic
• cumulative error function

yy0 eps0 for ( int x x0 x lt x1 x )
draw(x,y)eps dy if ( (eps ltlt 1)
gt dx ) y eps - dx
7
Review Flood Fill

• draw polygon edges, seed point, recursively set
  all neighbors until boundary is hit to fill
  interior
• drawbacks visit pixels up to 4x, per-pixel
  memory storage needed

P
8
Review Scanline Algorithms

• set pixels inside polygon boundary along
  horizontal lines one pixel apart
• use bounding box to speed up

9
Review Edge Walking

• basic idea
• draw edges vertically
• interpolate colors down edges
• fill in horizontal spans for each scanline
• at each scanline, interpolate edge colors across
  span

10
Review General Polygon Rasterization

• idea use a parity test
• for each scanline
• edgeCnt 0
• for each pixel on scanline (l to r)
• if (oldpixel-gtnewpixel crosses edge)
• edgeCnt
• // draw the pixel if edgeCnt odd
• if (edgeCnt 2)
• setPixel(pixel)

11
Hall of Fame
12
But Wait, Theres More!

• nice comment )
• this project is fun, only one of the very few
  that I actually enjoyed. Too bad there's only one
  CG course in UBC (
• two fourth year CG courses await you!
• 424 Geometric Modelling
• 426 Animation

13
Midterm Review
14
Midterm Exam

• Friday Feb 11 10am-1050am
• you may use one handwritten 8.5x11 sheet
• one side of page
• no other notes, no books
• nonprogrammable calculators OK
• arrive on time!
• sit every other seat, ID out in front of you
• coats and bags in front of room

15
Whats Covered

• transformations
• viewing and projections
• coordinate systems of rendering pipeline
• lighting and shading
• not scan conversion

16
Reading

• FCS book, Red book
• see web page for details
• you can be tested on material in book but not
  covered in lecture
• you can be tested on material covered in lecture
  but not covered in book

17
Old Exams Posted

• see course web page

18
The Rendering Pipeline

• pros and cons of pipeline approach

19
Transformations
20
Homogeneous Coordinates

w
w1
y
x
21
Composing Transformations
Ta Tb Tb Ta, but Ra Rb ! Rb Ra and Ta Rb !
Rb Ta
22
Composing Transformations

• example rotation around arbitrary center

23
Composing Transformations

• example rotation around arbitrary center
• step 1 translate coordinate system to rotation
  center

24
Composing Transformations

• example rotation around arbitrary center
• step 2 perform rotation

25
Composing Transformations

• example rotation around arbitrary center
• step 3 back to original coordinate system

26
Composing Transformations

• rotation about a fixed point
• p TRT-1p
• rotation around an arbitrary axis
• considering frame vs. object
• p DCBAp

OpenGL D C B A draw p
frame
object
27
Transformation Hierarchies

• hierarchies dont fall apart when changed
• transforms apply to graph nodes beneath

28
Matrix Stacks

• push and pop matrix stack
• avoid computing inverses or incremental xforms
• avoid numerical error

29
Matrix Stacks
glPushMatrix()
glPopMatrix()
DrawSquare()
glPushMatrix()
glScale3f(2,2,2)
glTranslate3f(1,0,0)
DrawSquare()
glPopMatrix()
30
Transformation Hierarchies

• example

glTranslate3f(x,y,0) glRotatef(
,0,0,1) DrawBody() glPushMatrix()
glTranslate3f(0,7,0) DrawHead() glPopMatrix()
glPushMatrix() glTranslate(2.5,5.5,0)
glRotatef( ,0,0,1) DrawUArm()
glTranslate(0,-3.5,0) glRotatef( ,0,0,1)
DrawLArm() glPopMatrix() ... (draw other
arm)
y
x
31
Display Lists

• reuse block of OpenGL code
• more efficient than immediate mode
• code reuse, driver optimization
• good for static objects redrawn often
• cant change contents
• not just for multiple instances
• interactive graphics objects redrawn every frame
• nest when possible for efficiency

32
Double Buffering

• two buffers, front and back
• while front is on display, draw into back
• when drawing finished, swap the two
• avoid flicker

33
Projective Rendering Pipeline
glVertex3f(x,y,z)
viewing/camera
object
world
alter w
WCS
VCS
OCS
glFrustum(...)
projection transformation
clipping
glTranslatef(x,y,z) glRotatef(th,x,y,z) ....
gluLookAt(...)
/ w
CCS
perspective division
normalized device

• OCS - object coordinate system
• WCS - world coordinate system
• VCS - viewing coordinate system
• CCS - clipping coordinate system
• NDCS - normalized device coordinate system
• DCS - device coordinate system

glutInitWindowSize(w,h) glViewport(x,y,a,b)
NDCS
device
DCS
34
Projection

• theoretical pinhole camera

eye point
image plane

• image inverted, more convenient equivalent

eye point
image plane
35
Projection Taxonomy
planar projections
parallel
perspective 1,2,3-point
orthographic
oblique
cavalier
cabinet
axonometric isometric dimetric trimetric
top, front, side
36
Projective Transformations

• transformation of space
• center of projection moves to infinity
• viewing frustum transformed into a parallelpiped

x
x
Frustum
-z
-z
37
Normalized Device Coordinates

• left/right x /- 1, top/bottom y /- 1,
  near/far z /- 1

NDC
Camera coordinates
x
x
x1
right
Frustum
-z
z
left
x -1
z1
z -1
z-n
z-f
38
Projection Normalization

• distort such that orthographic projection of
  distorted objects is desired persp projection

39
Transforming View Volumes
NDCS
y
(1,1,1)
z
(-1,-1,-1)
x
40
Basic Perspective Projection
P(x,y,z)
y
similar triangles
P(x,y,d)
z
zd
but
also

• nonuniform foreshortening
• not affine

41
Basic Perspective Projection

• can express as homogenous 4x4 matrix!

42
Projective Transformations

• determining the matrix representation
• need to observe 5 points in general position,
  e.g.
• left,0,0,1T?-1,0,0,1T
• 0,top,0,1T?0,1,0,1T
• 0,0,-f,1T?0,0,1,1T
• 0,0,-n,1T?0,0,-1,1T
• leftf/n,topf/n,-f,1T?-1,1,1,1T
• solve resulting equation system to obtain matrix

43
OpenGL Orthographic Matrix

• scale, translate, reflect for new coord sys
• understand derivation from VCS!

44
OpenGL Perspective Matrix

• shear, scale, reflect for new coord sys
• understand derivation from VCS!

45
Viewport Transformation
onscreen pixels map from -1,1 to 0,
displaywidth
DCS

• confusion on hw
• pixel locations should not be negative or huge!

46
Light Sources

• directional/parallel lights
• point at infinity (x,y,z,0)T
• point lights
• finite position (x,y,z,1)T
• spotlights
• position, direction, angle
• ambient lights

47
Reflectance

• specular perfect mirror with no scattering
• gloss mixed, partial specularity
• diffuse all directions with equal energy
• 
  
• specular glossy diffuse
• reflectance distribution

48
Review Reflection Equations

• Idiffuse kd Ilight (n l)

2 ( N (N L)) L R
49
Review Reflection Equations 2

• Blinn improvement
• full Phong lighting model
• combine ambient, diffuse, specular components

50
Lighting vs. Shading

• lighting
• simulating the interaction of light with surface
• shading
• deciding pixel color
• continuum of realism when do we do lighting
  calculation?

51
Shading Models

• flat shading
• compute Phong lighting once for entire polygon
• Gouraud shading
• compute Phong lighting at the vertices and
  interpolate lighting values across polygon
• Phong shading
• compute averaged vertex normals
• interpolate normals across polygon and perform
  Phong lighting across polygon

52
Transforming Normals

• apply nonuniform scale stretch along x by 2
• cant transform normal
• by modelling matrix
• solution

normal to any surface transformed by inverse
transpose of modelling transformation
53
Interpolation
54
Scan Conversion

• done
• how to determine pixels covered by a primitive
• next
• how to assign pixel colors
• interpolation of colors across triangles
• interpolation of other properties

55
Interpolation During Scan Conversion

• interpolate values between vertices
• z values
• r,g,b colour components
• use for Gouraud shading
• u,v texture coordinates
• surface normals
• equivalent methods (for triangles)
• bilinear interpolation
• barycentric coordinates

56
Bilinear Interpolation

• interpolate quantity along L and R edges, as a
  function of y
• then interpolate quantity as a function of x

P1
P3
P(x,y)
PL
PR
y
P2
57
3. Barycentric Coordinates

• weighted combination of vertices

(1,0,0)
(0,0,1)
(0,1,0)
58
Computing Barycentric Coordinates

• for point P on scanline

P1
P3
PL
P
PR
d2 d1
P2
59
Computing Barycentric Coords

• similarly

P1
P3
PL
P
PR
b1 b2
d2 d1
P2
60
Computing Barycentric Coords

• combining
• gives

P1
P3
PL
P
PR
b1 b2
c1 c2
d2 d1
P2
61
Computing Barycentric Coords

• thus

62
Computing Barycentric Coords

• can verify barycentric properties