1. Computer Graphics CC416
Week 14
3D Graphics

2. 3D Viewing

3. 3D Viewing Viewing: virtual camera Projection Depth
Visible lines and surfaces
Surface rendering

4. 3D Viewing pipeline 1 Similar to making a photo
Position and point virtuele camera, press button;
Projection plane aka
Viewing plane
Pipeline has +/ same structure as in 2D

5. Synthetic Camera Metaphor for creating 3D scenes: Coordinate system:
Camera: u, v, n
Object: x, y, z

6. 3D Viewing pipeline 2 MC: Modeling Coordinates WC: World Coordinates
VC: Viewing Coordinates
PC: Projection Coordinates
NC: Normalized Coordinates
DC: Device Coordinates
Apply model transformations
To camera coordinates
Project
To standard coordinates
Clip and convert to pixels

7. Right hand coordinate system
Coordinate Systems Y +z x x +z
Left hand coordinate system
Not used in this class and
Not in OpenGL
Right hand coordinate system

8. 3D viewing coordinates 1
zvp
Specification of projection: P0 : View or eye point Pref : Center or look-at point V: View-up vector (projection along vertical axis) zvp : positie view plane yw P0 N V
Pref xw zw
P0, Pref , V: define viewing coordinate system
Several variants possible

9. 3D viewing coordinates 2
xview
yview
zview yw P0 N V
Pref xw zw
P0, Pref , V: define viewing coordinate system
Several variants possible

10. 3D view coordinates 3 n
xview
yview u v
zview yw P0 N V
Pref xw zw

11. Transform the left-handed eye coordinate system defined by v, u and VPN into the right-handed world coordinate system
The origin of the eye coordinate system, COP is translated to the origin (0, 0, 0,) of the world coordinate system.
Rotate so that
The axis in the u direction is parallel to the world coordinate x-axis.
The axis in the v direction is parallel to the world coordinate y-axis.
The VPN is parallel to the negative z-axis, that is, going into the display screen.
Make the negative z-axis the positive direction so that the positive z direction goes into the screen, i.e., make the eye coordinate system left-handed.

12. The matrices required to accomplish this transformation are given next.
The viewing coordinate origin is at a world position P0 (x0, y0, z0), the translation matrix to translate the viewing origin to the world origin is: x0
T = y0 z0
The rotation matrix that super imposes the viewing axes onto the world frame, using the u, v, n unit vectors, is
ux uy uz 0
R = vx vy vz 0
nx ny nz 0

13. The coordinate transformation matrix is obtained by product R.T
MWC, VC = R.T
ux uy uz -u.P0
= vx vy vz -v.P0
nx ny nz -n.P0
Where the translation unit vectors are the dot product:
u.P0 = -x0ux – y0uy – z0uz
v.P0 = -x0vx – y0vy – z0vz
n.P0 = -x0nx – y0ny – z0nz

14. Steps in 3D Viewing Projection type specification
Why projection?
Objects 3D, device 2D
Two most important
Perspective
Parallel orthographic
Viewing parameter specification
Viewing plane
Viewing (eye) coordinate system
Scene coordinate system

15. Projections Projection Our concern Terminology
Transformation from n-D coordinate system to m-D coordinate system, where m < n
Our concern
n = 3 and m = 2
 projection from 3D to 2D
Terminology
Projectors: Straight projection rays
Center of projection: Where the projectors emanated from
Projection plane: Where the projection forms

16. Projections Projection from 3D to 2D defined by
Projectors emanate from center of projection (COP), pass through each point of the object, and intersect the projection plane
Planar geometric projections
Projection is onto a plane
Referred to as "projections" here

17. Projections Two Basic Classes Perspective Parallel
Distance between projection plane and COP is finite
Visual effect similar to human visual system
Perspective foreshortening:
distance from COP longer, size smaller
Exact shape, measurement, parallelism not reserved

18. Perspective Projection
Given a vertex in the eye coordinate system, v = (xe, ye, ze), the projected screen coordinates, (xs, ys) are computed as:
Where d is the distance from COP to the view plane. These formulas are easily derived by considering the projection onto the x-z and y-z planes

19. Projections
Parallel
Distance between projection plane and COP is infinite
Less realistic view
Exact measurement and parallelism preserved
Perspective
Parallel

20. Classification of Planar Geometric Projections

21. Specification of an Arbitrary 3D View
View Plane
Projection Plane
Defined by
VRP (View Reference Point)
look at point in OpenGL
VPN (View Plane Normal)
(eye - look) in OpenGL

22. Specification of an Arbitrary 3D View
Window
Similar to the window in 2D
Contents mapped to the viewport
Projection on the view plane outside the window not shown
Specification needs the following
Minimum and maximum window coordinates
Two orthogonal axes
Can be defined by view volume

23. Specification of an Arbitrary 3D View
PRP (Projection Reference Point)
Parallel: Direction of Projection (DOP) = from PRP to CW
Perspective: COP
“Eye” in OpenGL
View volume
Clipping and projection
Perspective
Semi-infinite pyramid with apex at PRP and edges passing through window corners
Parallel
Infinite parallelepiped with sides parallel to DOP

24. Specification of an Arbitrary 3D View
View volume (cont’d)
Finite view volume
Front clipping plane
Parallel to VRP
Specified by F (front distance) = distance(FCP - VRP)
Back clipping plane
Specified by B (back distance) = distance(BCP - VRP)
In OpenGL, “near” and “far” represents “front” and “back”, respectively, in the camera coordinate

25. Finite View Volumes
Parallel
Perspective

26. Specification of an Arbitrary 3D View
Mapping from view volume to 2D display
View volume --> NPC (Normalized Projection Coordinates), i.e., standard cube
3D viewport specified in NPC
z=1 face of NPC cube mapped to display
If wire-frame, z coordinate discarded
If surface, hidden-surface removal

27. OpenGL 3D Viewing 1 3D Viewing in OpenGL: Position camera;
Specify projection.

28. OpenGL 3D Viewing 2
View volume y x z
Camera always in origin, in direction of negative z-axis.
Convenient for 2D, but not for 3D.

29. OpenGL 3D Viewing 3
Solution for view transform: Transform your model such that you look at it in a convenient way.
Approach 1: Do it yourself. Apply rotations, translations, scaling, etc., before rendering the model. Surprisingly difficult and error-prone.
Approach 2: Use gluLookAt();

30. OpenGL 3D Viewing 4
MatrixMode(GL_MODELVIEW);
gluLookAt(x0,y0,z0, xref,yref,zref, Vx,Vy,Vz);
x0,y0,z0: P0, viewpoint, location of camera;
xref,yref,zref: Pref, centerpoint;
Vx,Vy,Vz: V, view-up vector.
Default: P0 = (0, 0, 0); Pref = (0, 0, 1); V=(0, 1, 0).

31. OpenGL 3D Viewing 5 Orthogonal projection: y x z
MatrixMode(GL_PROJECTION);
glOrtho(xwmin, xwmax, ywmin, ywmax, dnear, dfar);
xwmin, xwmax, ywmin,ywmax:
specification window
dnear: distance to near clipping plane
dfar : distance to far clipping plane
dnear
dfar y
ywmax x
Select dnear and dfar right:
dnear < dfar,
model fits between clipping planes. z
xwmax
ywmin
xwmin

32. OpenGL 3D Viewing 6 Perspective projection: y x z
MatrixMode(GL_PROJECTION);
glFrustrum(xwmin, xwmax, ywmin, ywmax, dnear, dfar);
xwmin, xwmax, ywmin,ywmax:
specification window
dnear: distance to near clipping plane
dfar : distance to far clipping plane
dfar
dnear y x
ywmax
Select dnear and dfar right:
0 < dnear < dfar,
model fits between clipping planes. z
ywmin
xwmax
xwmin
Standard projection: xwmin = -xwmax,
ywmin = -ywmax

33. OpenGL 3D Viewing 7 Finally, specify the viewport (just like in 2D):
glViewport(xvmin, yvmin, vpWidth, vpHeight);
xvmin, yvmin: coordinates lower left corner (in pixel coordinates);
vpWidth, vpHeight: width and height (in pixel coordinates);
vpHeight
vpWidth
(xvmin, yvmin)

34. OpenGL 2D Viewing 8 In short: To prevent distortion, make sure that:
glMatrixMode(GL_PROJECTION);
glFrustrum(xwmin, xwmax, ywmin, ywmax, dnear, dfar);
glViewport(xvmin, yvmin, vpWidth, vpHeight);
glMatrixMode(GL_MODELVIEW);
gluLookAt(x0,y0,z0, xref,yref,zref, Vx,Vy,Vz);
To prevent distortion, make sure that:
(ywmax – ywmin)/(xwmax – xwmin) = vpWidth/vpHeight
Make sure that you can deal with resize/reshape of the (OS) window.

35. 3D NDC to 2D Image (Near) Plane
Resulting image on the near plane
Chapter 14