Announcements. Tutorial this week Life of the polygon A1 theory questions

Transcription

1 Announcements Assignment programming (due Frida) submission directories are ied use (submit -N Ab cscd88 a_solution.tgz) theor will be returned (Wednesda) Midterm Will cover all o the materials so ar including toda's lecture Lecture notes lecture slides readings assignment are all air game Practice midterms are on-line (no solutions will be given) Tutorial this week Lie o the polgon A theor questions Oice Hours I will have oice hours toda - pm Ale will have oice hours later in the week I will also have oice hours on Tuesda 4-5pm

2 Last week s review Cameras (theor) Pinhole Camera Thin Lens model Virtual pinhole camera Perspective and orthographic projections Cameras (practice) Location o camera in space Transormation o geometr rom camera to world coordinate rame and (vice versa) Homogeneous Perspective Projection (how do we represent perspective using a single 44 matri) Homogeneous Prospective Projection with Pseudodepth

3 Projecting Triangle Lets review steps in the rendering hierarch Triangle is given in the object-based coordinate rame as three vertices o p o p z o p

4 Projecting Triangle Lets review steps in the rendering hierarch Triangle is given in the object-based coordinate rame as three vertices w o Transorm to world coordinated p i = Mowpi w p w p z w p z

5 Projecting Triangle Lets review steps in the rendering hierarch Triangle is given in the object-based coordinate rame as three vertices w o Transorm to world coordinated p i = Mowpi c w Transorm rom world to camera coordinates p = M p i wc i e c p v r w r c p z c p u r z

6 Projecting Triangle Lets review steps in the rendering hierarch Triangle is given in the object-based coordinate rame as three vertices w o Transorm to world coordinated p i = Mowpi c w Transorm rom world to camera coordinates p = M p Appl homogeneous perspective Divide b last component e v r u r p p p w r z p = M i p p c i c p z i wc c p c p i

7 Projecting Triangle Lets review steps in the rendering hierarch Triangle is given in the object-based coordinate rame as three vertices w o Transorm to world coordinated p i = Mowpi c w Transorm rom world to camera coordinates p = M p Appl homogeneous perspective Divide b last component e v r u r p p p w r z p = M i p p c i c p z i wc c p c p i

8 Visibilit Computer Graphics CSCD8 Fall 8 Instructor: Leonid Sigal

9 Clipping Idea: Remove points and parts o objects outside view volume Sounds simple but consider i we have an object on a boundar

10 View Volume Consider what we can actuall see T e α R Near Plane B L Far Plane Optical ais (z-ais)

11 Side note: Field o View T e α B / (T B) tan( α) = T e α R Near Plane B L Far Plane

12 View Volume What does homogeneous perspective projection do to our view volume? + = / F F F F M p parallepiped R L B T R L B T -

13 Canonical View Volume Can we alter homogeneous perspective projection to help us clip? = / F F F F B T B T B T L R L R L R M p () cube R L B T (---)

14 Back-ace Removal Idea: Remove surace patches that point awa rom the camera (like backside o the object as it viewed rom the ront) Consider a cube Back Faces 4 Back Faces 5 Back Faces Top Back Back Back Let Let Right Let Right Bottom Bottom Bottom We onl need to render at most hal o the sides depending on the view

15 Back-ace Removal How do we know i the patch (triangle) points awa rom the camera? Consider a normal o the patch (triangle) e p ( p e) n r r I ( p e) n > then triangle is part o the back-ace and needs to be removed r I ( p e) n < then triangle ma be visible

16 Back-ace Removal Does it matter which point we consider on the patch? Not i this is a planar patch Consider a normal o the patch (triangle) e p ( p e) n r r I ( p e) n > then triangle is part o the back-ace and needs to be removed r I ( p e) n < then triangle ma be visible

17 Back-ace Removal Does it matter which point we consider on the patch? Not i this is a planar patch How do we compute I p p p are patch vertices in CCW order n r p e p ( p e) n r p p r I ( p e) n > then triangle is part o the back-ace and needs to be removed r I ( p e) n < then triangle ma be visible

18 Back-ace Removal Does it matter which point we consider on the patch? Not i this is a planar patch r ( p n = ( p How do we compute p p ) ( p ) ( p p p ) ) p e p ( p e) n r p p r I ( p e) n > then triangle is part o the back-ace and needs to be removed r I ( p e) n < then triangle ma be visible

19 Z-Buer (a.k.a Depth Buer) We have a rame-buer (this is where an image that we see on the screen is stored) We also have a z-buer that keeps track o the z coordinate or ever piel in the rame-buer To draw point in the world with color c that projects to ( z) we can eecute the ollowing algorithm i z < z-buer( ) then rame-buer( ) = c z-buer( ) = z end

20 Z-Buer (a.k.a Depth Buer) We need to initialize the z-buer with some value. What is the good value to initialize with? I we are using canonical view volume then would work To draw point in the world with color c that projects to ( z) we can eecute the ollowing algorithm i z < z-buer( ) then rame-buer( ) = c z-buer( ) = z end

21 Z-Buer (a.k.a Depth Buer) Advantages o Z-buering Simple and accurate Independent o the order the polgons are drawn Disadvantages o Z-buering Memor or a Z-buer (small consideration) Wasted computation in drawing distant points irst (this potentiall can be a large drawback)

22 Z-Buer (a.k.a Depth Buer) We represent a patch using vertices How do we get a pseudodeph and proper rendering everwhere else? ) ( z ) ( z ) ( z

23 Z-Buer (a.k.a Depth Buer) We represent a patch using vertices How do we get a pseudodeph and proper rendering everwhere else? ) ( z ) ( z ) ( z Linearl interpolate along a scan line z

24 Painter s Algorithm Idea: Order the patches and draw them in the order o depth (with most distant patches irst) This is an alternative to Z-buering ) ( z ) ( z ) ( z

25 Painter s Algorithm How do we deal with intersecting patches? Break patches into smaller patches ) ( z ) ( z ) ( z

26 BSP Trees Binar space partition tree (BSP tree) is an algorithm or making back-to-ront ordering o polgons eicient and to break polgons to avoid intersections ) ( z ) ( z ) ( z

27 BSP Tree I e and T on the same side o T (let) then draw T irst then T I e and T are on dierent sides o T (right) then draw T irst then T How do we know i points are on the same side? r ( n ) = ( p) ( ) = ( ) > ( ) < on the plane " outside" " inside" n r T T T n r n r T e outside-acing normals n r e

28 BSP Tree Eample Let s tr building a BSP tree or this scene e The tree will be the same regardless o the camera placement

29 BSP Tree Eample Let s tr building a BSP tree or this scene inside outside

30 BSP Tree Eample Let s tr building a BSP tree or this scene inside outside

31 BSP Tree Eample Let s tr building a BSP tree or this scene inside outside

32 BSP Tree Eample Let s tr building a BSP tree or this scene inside outside

33 BSP Tree Eample Let s tr building a BSP tree or this scene inside outside b 9a

34 BSP Tree Traversal Tree traversal algorithm i ee in the outside hal-space o the root Draw aces on inside sub-tree o the root Draw the root Draw aces is the outside o sub-tree o the root else Draw aces is the outside o sub-tree o the root Draw the root Draw aces on inside sub-tree o the root end inside outside Eas to modi to do backace removal 4 7 9b 9a

35 BSP Tree Advantages Can easil discard portions o the scene behind the camera Artiacts o z-buer quantization are not seen Tree construction ied or the static scenes Disadvantages How can we handle dnamic scenes? This is what is tpicall done in games because it s ast