1. Joshua Barczak CMSC 435 UMBC
Viewing
Joshua Barczak
CMSC 435
UMBC

2. Rendering Image-Order(Raytracing) Object-Order(Rasterization)
For each pixel:
What is under it?
What color is it?
Object-Order(Rasterization)
For each object:
What pixels does it cover?
What color are they?

3. Object-Order Rendering
Object-Space
Primitives
World-space Primitives
Camera-Space Primitives
Clip-Space Primitives
Clip
Modeling
XForm
Viewing
XForm
Projection
XForm
Last time
Today
Clip-Space Primitives
Rasterize
Window
XForm
Viewport
XForm
Displayed Pixels
Pixels
Raster-Space Primitives

4. Coordinate Systems Model/Object/Local space World Space
Coordinates model is authored in
Typically with Origin at object center
World Space
“The” coordinate system
Model Space
World Space

5. Coordinate Systems Camera/Eye/View Space Origin at Viewer
Image plane slightly in front…
X/Y axes screen aligned
Z axis points in/out of screen
Camera Space

6. Coordinate Systems Camera Space Handedness Right handed Left handed
OGL convention
Left handed
D3D convention
Half the world likes one, and half the other… Z X Y Z X

7. Coordinate Systems Clip Space NDC (-1,1) on x/y (-1,1) on Z (OpenGL)
(0,1) on Z (D3D)
NDC
Normalized Device Coordinates
0,0
1,0
0,1
There is much disagreement about what ‘NDC’ space actually is, the name isn’t very well standardized…

8. Coordinate Systems Raster Space Screen space
Width,0
0,0
Raster Space
Continuous pixel coordinates
1 unit 1 pixel
NDC scaled by resolution
Screen space
Raster space translated to window location
0,Height
There is much disagreement about what ‘Screen’ space actually is, the name isn’t very well standardized…

9. The Major Transforms Projection Matrix Model-World Matrix View Matrix
How OpenGL sets it up…
The right way to set it up…
Changes for every object…
Changes once for a given camera…

10. Viewing Transform The job of the viewing transform is:
Get us from these axes
Onto these axes

11. Camera Coordinate System
From, At, Up
Derive an orthonormal basis for the camera
Right handed, to match OGL
Make sure everything is normalized… Up Z Y
From X
Book calls these U,V,W At

12. Viewing Transform Two ways to think of this: Take 1: Moving the world
Take 2: Changing the coordinate system

13. The “Look-At” Transform
Take 1: Rotations and translations
Translate to origin
Subtract eye coords
Put the Z axis on From-At
Want matrix to turn this into 0,0,1

14. The “Look-At” Transform
Take 1: Rotations and translations
Translate to origin
Subtract eye coords
Put Z axis on From-At
Inverse of:

15. The “Look-At” Transform
Take 1: Rotations and translations
Translate to origin
Subtract eye coords
Put Z axis on From-At
Don’t forget to spend 3 hours flipping signs and axes… 

16. The “Look-At” Transform
Take 2: Projecting onto new basis
Use cross products to get a basis
Subtract eye position
Project onto new basis
From At Up X Y Z
IMO this one is a lot harder to screw up….

17. The “Look-At” Transform
From At Up X Y
Vector form: Z
Matrix form:
IMO this one is a lot harder to screw up….

18. Camera Controls Orbit/Arcball camera Fixed target point
User pivots in a sphere around model
ex = r*cos(azimuth)*sin(zenith); ey = r*sin(azimuth)*sin(zenith); ez = r*cos(zenith);
gluLookAt( ex, ey, ez, 0,0,0, 0,0,1 );

19. Camera Controls Orbit/Arcball camera
Use mouse to rotate around the unit sphere
Use mouse wheel/keys to adjust distance
//On mouse drag:
// record differences from last position
float dx = x - oldX;
float dy = y - oldY;
oldX = x; // update stored position
oldY = y;
// rotation angle, scaled so across the
// window = one rotation
view->azimuth += 2*M_PI * dx / view->width;
view->zenith -= M_PI * dy / view->height ;
// constrain the polar angle so we don't
// suddenly flip around
float EPS = f;
if( view->zenith < EPS )
view->zenith = EPS;
else if( view->zenith > M_PI-EPS )
view->zenith = M_PI-EPS;

20. Camera Controls First Person Camera
Track Zenith/Azimuth as with orbit camera, construct Z axis
Derive other axes using cross products
Good idea to smooth it
Mousepos = s * Mousepos + (1-s) * last_Mousepos
From At Up X Y

21. Camera Controls First person camera Forward Motion:
Translate along Z
Sideways (Strafe) Motion:
Translate along X/Y
From At Up X Y

22. Camera Controls First person camera Motion Up Y From X At velocity=0;
if( w )
velocity -= Z; // right handed
if( s )
velocity += Z;
if( a )
velocity -= X;
if( d )
velocity += X;
velocity *= SPEED/length(velocity);
position += velocity * elapsed_time;

23. Projection Transform The job of the projection transform is:
Cram the visible universe into a unit box
Project

24. Canonical View Volume (GL)
(-1,1,1)
(1,1,1)
Top Edge of Screen
(0,1,-1)
Center of screen
Left Edge of Screen
(0,0,-1)
(-1,0,-1)
(1,0,-1)
(1,-1,1)
Points outside the cube are not visible…
(1,-1,-1)
(0,-1,-1)
D3D uses z=0. I like theirs better…
Bottom Edge of Screen

25. Types of Projection Orthographic Perspective
Parallel lines stay parallel
Often used by modellers
Perspective
Parallel Lines converge
Used for first-person views

26. The View Frustum
Figure: Olano
Orthographic
Perspective

27. Orthographic Projection
Parameters:
l,b,t,f: Corners of viewing rectangle
n,f: distances to near/far planes
I’m using left handed, because its less confusing…
(-1,-1,-1)
(1,1,1)
(r,t,f)
(l,b,n)

28. Orthographic Projection
X,Y simply scales
Z gets n:f range mapped to -1:1
XLate(-n)
Scale(2/f-n)
XLate(-1)
(-1,-1,-1)
(1,1,1)
(r,t,f)
(l,b,n)

29. Orthographic Projection
(-1,-1,-1)
(1,1,1)
(r,t,f)
(l,b,n)

30. Orthographic Projection
The book’s version does not do what they say it does!
Z is backwards!
(-1,-1,-1)
(1,1,1)
(r,t,f)
(l,b,n)

31. Orthographic Projection
Typical, Symmetric Case… (l=-r, b=-t)
General case (off center)
(-1,-1,-1)
(1,1,1)
(r,t,f)
(l,b,n)

32. Perspective Raytracing: Given y’, find y
Perspective projection is the opposite: Given y, find y’ y’ y

33. Similar Triangles y y’ n z

34. Division by Z
Black  Up close
WhiteFar away

35. Division by Z
Black  Up close
WhiteFar away

36. Perspective Projection
Our desired transform…
Q: How do we trick a matrix into dividing?
Matrices don’t work this way…

37. Perspective Projection
A: Use a projective transform
Homogeneous coordinates again…

38. Perspective Projection
Each homogeneous point projects to a w=1 point
“Equivalent” points, different coordinates -w 1 w

39. Perspective Projection
Perspective Transform
After homogeneous divide…
Z isn’t quite right…

40. Perspective Projection
At near plane, z=n
At far plane, z=f
In between, z is non-linear…
Fortunately, at least, its monotonically increasing……

41. Perspective Projection
Perspective matrix “squishes” geometry along Z
Turns frustum into rectangular prism
Add an ortho transform afterwards…

42. Perspective Projection

43. Perspective Projection

44. Perspective with FOV Set dimensions based on FOV/Aspect
Just like in raytracing
Much more common… h 1
FOV

45. GL Transforms glPushMatrix/PopMatrix
Matrix stack
glRotate/glTranslate/glScale, etc…
glLoadIdentity
glMatrixMode( mode )
Set current stack
GL_MODELVIEW
Modeling + Viewing Transforms
GL_PROJECTION

46. GL Transforms gluLookAt( eye, at, up )
Apply a viewing transform
glOrtho(left,right,bottom,top,near,far)
Apply an orthographic projection
glFrustum(left,right,bottom,top,near,far)
Apply a perspective projection
gluPerspective(fov,aspect,near,far)
Perspective with FOV

47. OpenGL Ortho Our ortho: glOrtho:
-Z is reversed for right-handed camera space
-They want absolute distances for n,f

48. OpenGL Perspective Ours: glFrustum: Differences: Right handed
Supports off-center projection

49. OpenGL Perspective FOV
Ours:
gluPerspective:
Differences:
Right handed

50. D3D Ortho Our ortho: D3DXMatrixOrthoLH: Differences:
Maps Z to 0:1 not -1:1

51. D3D Perspective Ours: D3DXMatrixPerspectiveLH: Differences:
Maps Z to 0:1 not -1:1