1. Viewing and Projection
Images were taken from the book: “Interactive Computer Graphics” by Angel and Shreiner

2. Graphics Pipeline Consider our program as a pipeline
At each step, we manipulate the data
Our goal is to output the data in screen coordinates
חלק מהשלבים יכולים להיות טריוויאליים – כלומר מטריצת היחידה

3. Graphics Pipeline Object coordinates Camera coordinates Clip
Vertices 𝑃
Projection transformation
𝑇 𝑚
𝑇 𝑤
𝑇 𝑐 −1
Model-view transformation

4. Positioning the Camera
Initially, the object and camera frames are the same (the identity matrix)
The camera is located at the origin and points in the negative 𝑧 direction
We place the camera through the Model-view matrix
המצלמה והאובייקט נמצאים שניהם בראשית מערכת הצירים, ולכן צריך להזיז את המצלמה/האובייקט כדאי שנראה משהו

5. The LookAt Function eye: The position of the camera
mat4 LookAt(vec4& eye, vec4& at, vec4& up)
eye: The position of the camera
at: The position the camera looks at
Up: The upside (𝑦) direction of the camera

6. The LookAt Function Translate the camera to eye and change frame
mat4 LookAt(vec4& eye, vec4& at, vec4& up ) {
vec4 n = normalize(eye - at);
vec4 u = normalize(cross(up,n));
vec4 v = normalize(cross(n,u));
vec4 t = vec4(0.0, 0.0, 0.0, 1.0);
mat4 c = mat4(u, v, n, t);
return c * Translate( -eye ); }

7. Viewing Perspective vs. Orthographic DOP COP
אפשר להסתכל על הטלה אורתוגרפית כמקרה הגבול של הטלה פרספקטיבית – מרכז ההטלה עובר לאינסוף

8. Orthogonal Projections
How can we construct 𝑃?
𝑥 𝑝 =𝑥, 𝑦 𝑝 =𝑦, 𝑧 𝑝 =0
In matrix form:
𝑥 𝑝 , 𝑦 𝑝 , 𝑧 𝑝 ,1 𝑇 =𝑀I 𝑥,𝑦,𝑧,1 𝑇

9. Orthogonal Projections
How can we construct 𝑃=𝑀I?
𝑥 𝑝 , 𝑦 𝑝 , 𝑧 𝑝 ,1 𝑇 =𝑀I 𝑥,𝑦,𝑧,1 𝑇
Where I is the identity matrix and 𝑀=

10. Orthogonal Projections
Take a finite portion of the projection plane
The default view volume is a cube with sides of length 2 centered at the origin
Namely, the planes 𝑥=±1, 𝑦=±1, 𝑧=±1, determine the clipping volume

11. Orthogonal Viewing We can specify a different view volume
mat4 Ortho(left, right, bottom, top, near, far)
Values are in
Camera coordinates

12. Projection Normalization
The graphics card has a fixed viewing volume
Thus, we transform our viewing volume into the canonical one

13. Projection Normalization
How to normalize the viewing volume?
Translate: 𝑇 − 𝑟𝑖𝑔ℎ𝑡+𝑙𝑒𝑓𝑡 2 ,− 𝑏𝑜𝑡𝑡𝑜𝑚+𝑡𝑜𝑝 2 , 𝑛𝑒𝑎𝑟+𝑓𝑎𝑟 2
Scale: 𝑆 2 𝑟𝑖𝑔ℎ𝑡−𝑙𝑒𝑓𝑡 , 2 𝑡𝑜𝑝−𝑏𝑜𝑡𝑡𝑜𝑚 , 2 𝑛𝑒𝑎𝑟−𝑓𝑎𝑟
In matrix form:
לשים לב לסימנים!
𝑆𝑇= 2 𝑟𝑖𝑔ℎ𝑡−𝑙𝑒𝑓𝑡 𝑡𝑜𝑝−𝑏𝑜𝑡𝑡𝑜𝑚 − 2 𝑓𝑎𝑟−𝑛𝑒𝑎𝑟 0 − 𝑙𝑒𝑓𝑡+𝑟𝑖𝑔ℎ𝑡 𝑟𝑖𝑔ℎ𝑡−𝑙𝑒𝑓𝑡 − 𝑡𝑜𝑝+𝑏𝑜𝑡𝑡𝑜𝑚 𝑡𝑜𝑝−𝑏𝑜𝑡𝑡𝑜𝑚 − 𝑓𝑎𝑟+𝑛𝑒𝑎𝑟 𝑓𝑎𝑟−𝑛𝑒𝑎𝑟 1

14. Summary: Orthogonal Projection𝑃
Translate
Scale
Normalize 𝑇 𝑆 𝑀
Orthographic projection
Project
Projection transformation

15. Perspective Projections
How can we construct 𝑃 for 𝑑=−1?
𝑥 𝑝 = 𝑥 𝑧/−1 , 𝑦 𝑝 = 𝑦 𝑧/−1 , 𝑧 𝑝 =−1

16. Perspective Projections
How can we construct 𝑃?
𝑃= −
Perspective
division
לחשב על הלוח, להראות שמקבלים מה שרוצים אחרי חלוקה בקומפוננטה הרביעית

17. Perspective Projections
𝑃 corresponds to a frustum with COP at the origin, 90 degrees FOV and bounded by the planes 𝑥=±𝑧, 𝑦=±𝑧.

18. Perspective Viewing We can specify a different viewing frustum
mat4 Frustum(left, right, bottom, top, near, far)

19. Perspective Viewing We can specify a different viewing frustum
mat4 Perspective(fovy, aspect, near, far)
aspect = 𝑤/ℎ

20. Perspective Normalization
𝑃 is independent of the far clipping plane
We take
𝑃= 𝛼 − 𝛽 0
Thus, 𝑥,𝑦,𝑧,1 T goes (after division) to
− 𝑥 𝑧 ,− 𝑦 𝑧 ,−𝛼− 𝛽 𝑧 T

21. Perspective Normalization
We choose
𝛼=− 𝑛𝑒𝑎𝑟+𝑓𝑎𝑟 𝑛𝑒𝑎𝑟−𝑓𝑎𝑟 ,
𝛽=− 2∗𝑛𝑒𝑎𝑟∗𝑓𝑎𝑟 𝑛𝑒𝑎𝑟−𝑓𝑎𝑟
Then, our frustum is transformed into the default viewing volume (i.e., bounded by the planes 𝑥=±1, 𝑦=±1, 𝑧=±1)

22. Viewport Transformation
How to transform into screen coordinates?
𝑟= 𝑤 2 𝑥+1 , 𝑠= ℎ 2 𝑦+1 .

23. Complete Graphics Pipeline
Object
coordinates
Camera
coordinates
Clip
coordinates
Normalized device
coordinates
Screen
coordinates
Model-view
Projection
Perspective division
Viewport 𝑉 𝑉

24. Suggested Readings
Interactive Computer Graphics, Chapter 4