1. Last Time Clipping Homework 4, due Nov 2 in class
Why we care
Sutherland-Hodgman
Cohen-Sutherland
Intuition for Liang-Barsky
Homework 4, due Nov 2 in class
Midterm, Nov 2 in class
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2. Midterm Info November 2 in class
All material listed in online “to-know” list
Allowed:
Pencil/pen, ruler
One sheet of letter (standard) sized paper, with anything on it, both sides
Nothing to increase the surface area
Not allowed:
Calculator, anything else
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3. This Week Liang-Barsky Details Weiler-Atherton clipping algorithm
Drawing points and lines
Visibility
Z-Buffer and transparency
A-buffer
Area subdivision
BSP Trees
Exact Cell-Portal
Lighting and Shading – Part 1
Project 2 is posted and due at Nov 18
10/27/09
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4. General Liang-Barsky Liang-Barsky works for any convex clip region
E.g. Perspective view volume in world or view coordinates
Require a way to perform steps 1 and 2
Compute intersection t for all clip lines/planes
Label them as entering or exiting
Far
Near
Left
Right
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5. In View Space
For Project 2, you need to clip edges to a view frustum in world space
Situation is:
xleft
xright
frustum x1 x2
eye, e
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6. First Step Compute inside/outside for endpoints of the line segment
Determine which side of each clip plane the segment endpoints lie
Use the cross product
What do we know if (x1 - e)  (xleft - e) > 0 ?
Other cross products give other information
What can we say if both segment endpoints are outside one clip plane?
Stop here if we can, otherwise…
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7. Finding Parametric Intersection
Left clip edge: x = e + (xleft - e) t
Line: x = x1 + (x2 - x1) s
Solve simultaneous equations in t and s:
Use endpoint inside/outside information to label as entering or leaving
Now we have general Liang-Barsky case
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8. General Clipping
Liang-Barsky can be generalized to clip line segments to arbitrary polygonal clip regions
Consider clip edges as non-infinite segments
Look at all intersecting ts between 0 and 1
Clipping general polygons against general clip regions is quite hard: Weiler-Atherton algorithm
Start with polygons as lists of vertices
Replace crossing points with vertices
Double all edges and form linked lists of edges
Adjust links at crossing vertices
Enumerate polygon patches
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9. Weiler-Atherton – Form Lists
Original Polygons
Double Linked Lists -
outside and inside lists
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10. Weiler-Atherton – Find Crossings
Crossing vertices added –
links re-written
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11. Weiler-Atherton – Enumerate
“Clip and Poly”
“Poly not Clip”
“Not clip not poly”
“Clip not Poly”
Every link used once
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12. Where We Stand At this point we know how to: Next thing:
Convert points from local to screen coordinates
Clip polygons and lines to the view volume
Next thing:
Determine which pixels to fill for any given point, line or polygon Convert points from local to window coordinates
Determine which pixels are covered by any given line or polygon
Anti-alias
Put the slides in the extra lecture note but we won’t discuss it because it is too low level.
Determine which polygon is in front
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13. Visibility
Given a set of polygons, which is visible at each pixel? (in front, etc.). Also called hidden surface removal
Very large number of different algorithms known. Two main classes:
Object precision: computations that operate on primitives
Image precision: computations at the pixel level
All the spaces in the viewing pipeline maintain depth, so we can work in any space
World, View and Canonical Screen spaces might be used
Depth can be updated on a per-pixel basis as we scan convert polygons or lines
Actually, run Bresenham-like algorithm on z and w before perspective divide
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14. Visibility Issues
Efficiency – it is slow to overwrite pixels, or rasterize things that cannot be seen
Accuracy - answer should be right, and behave well when the viewpoint moves
Must have technology that handles large, complex rendering databases
In many complex environments, few things are visible
How much of the real world can you see at any moment?
Complexity - object precision visibility may generate many small pieces of polygon
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15. Painters Algorithm (Image Precision)
Choose an order for the polygons based on some choice (e.g. depth to a point on the polygon)
Render the polygons in that order, deepest one first
This renders nearer polygons over further
Difficulty:
works for some important geometries (2.5D - e.g. VLSI)
doesn’t work in this form for most geometries - need at least better ways of determining ordering
Fails zs
Which point for choosing ordering? xs
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16. The Z-buffer (1) (Image Precision)
For each pixel on screen, have at least two buffers
Color buffer stores the current color of each pixel
The thing to ultimately display
Z-buffer stores at each pixel the depth of the nearest thing seen so far
Also called the depth buffer
Initialize this buffer to a value corresponding to the furthest point (z=1.0 for canonical and window space)
As a polygon is filled in, compute the depth value of each pixel that is to be filled
if depth < z-buffer depth, fill in pixel color and new depth
else disregard
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17. The Z-buffer (2) Advantages: Disadvantages:
Simple and now ubiquitous in hardware
A z-buffer is part of what makes a graphics card “3D”
Computing the required depth values is simple
Disadvantages:
Over-renders – rasterizes polygons even if they are not visible
Depth quantization errors can be annoying
Can’t easily do transparency or filter-based anti-aliasing (Requires keeping information about partially covered polygons)
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18. Visibility Recap
You are given a set of polygons to draw and you need to figure out which one is visible at every pixel
Issues include:
Efficiency – it is slow to overwrite pixels, or scan convert things that cannot be seen
Accuracy – answer should be right, and behave well when the viewpoint moves
Complexity – object precision visibility may generate many small pieces of polygon
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19. Z-Buffer and Transparency
Say you want to render transparent surfaces (alpha not 1) with a z-buffer
Must render in back to front order
Otherwise, would have to store at least the first opaque polygon behind transparent one
Front
Partially transparent
3rd
1st or 2nd
3rd: Need depth of 1st and 2nd
Opaque
2nd
Opaque
1st
1st or 2nd
Must recall this color and depth
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20. OpenGL Depth Buffer
OpenGL defines a depth buffer as its visibility algorithm
The enable depth testing: glEnable(GL_DEPTH_TEST)
To clear the depth buffer: glClear(GL_DEPTH_BUFFER_BIT)
To clear color and depth: glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT)
The number of bits used for the depth values can be specified (windowing system dependent, and hardware may impose limits based on available memory)
The comparison function can be specified: glDepthFunc(…)
Sometimes want to draw furthest thing, or equal to depth in buffer
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21. The A-buffer (Image Precision)
Handles transparent surfaces and filter anti-aliasing
At each pixel, maintain a pointer to a list of polygons sorted by depth, and a sub-pixel coverage mask for each polygon
Coverage mask: Matrix of bits saying which parts of the pixel are covered
Algorithm: Drawing pass (do not directly display the result)
if polygon is opaque and covers pixel, insert into list, removing all polygons farther away
if polygon is transparent or only partially covers pixel, insert into list, but don’t remove farther polygons
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22. The A-buffer (2) = Algorithm: Rendering pass Advantage: Disadvantages:
At each pixel, traverse buffer using polygon colors and coverage masks to composite:
Advantage:
Can do more than Z-buffer
Coverage mask idea can be used in other visibility algorithms
Disadvantages:
Not in hardware, and slow in software
Still at heart a z-buffer: Over-rendering and depth quantization problems
But, used in high quality rendering tools
over =
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23. Area Subdivision
Exploits area coherence: Small areas of an image are likely to be covered by only one polygon
The practical truth of this assertion varies over the years (it’s currently going from mostly false to more true)
Three easy cases for determining what’s in front in a given region:
a polygon is completely in front of everything else in that region
no surfaces project to the region
only one surface is completely inside the region, overlaps the region, or surrounds the region
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24. Warnock’s Area Subdivision (Image Precision)
Start with whole image
If one of the easy cases is satisfied (previous slide), draw what’s in front
Otherwise, subdivide the region and recurse
If region is single pixel, choose surface with smallest depth
Advantages:
No over-rendering
Anti-aliases well - just recurse deeper to get sub-pixel information
Disadvantage:
Tests are quite complex and slow
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25. Warnock’s Algorithm Regions labeled with case used to classify them:
One polygon in front
Empty
One polygon inside, surrounding or intersecting
Small regions not labeled
Note it’s a rendering algorithm and a HSR algorithm at the same time
Assuming you can draw squares 2 3 2 2 3 3 3 2 3 3 3 1 3 1 1 1 1 3 3 3 3 3 2 3 3 3 2 2 2 2
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26. BSP-Trees (Object Precision)
Construct a binary space partition tree
Tree gives a rendering order
A list-priority algorithm
Tree splits 3D world with planes
The world is broken into convex cells
Each cell is the intersection of all the half-spaces of splitting planes on tree path to the cell
Also used to model the shape of objects, and in other visibility algorithms
BSP visibility in games does not necessarily refer to this algorithm
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27. BSP-Tree Example A A C 4 3 - + B C - B + - + 1 3 2 4 1 2 10/27/09
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28. Building BSP-Trees Choose polygon (arbitrary)
Split its cell using plane on which polygon lies
May have to chop polygons in two (Clipping!)
Continue until each cell contains only one polygon fragment
Splitting planes could be chosen in other ways, but there is no efficient optimal algorithm for building BSP trees
Optimal means minimum number of polygon fragments in a balanced tree
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29. Building Example
We will build a BSP tree, in 2D, for a 3 room building
Ignoring doors
Splitting edge order is shown
“Back” side of edge is side with the number 5 2 3 4 1 6
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30. Building Example (Done)1 - + 2 3a - + + 3b 2 4b 4a 3b 4b + + + 6 5b 5a 1 4a 3a 6
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31. BSP-Tree Rendering
Observation: Things on the opposite side of a splitting plane from the viewpoint cannot obscure things on the same side as the viewpoint
Rendering algorithm is recursive descent of the BSP Tree
At each node (for back to front rendering):
Recurse down the side of the sub-tree that does not contain the viewpoint
Test viewpoint against the split plane to decide which tree
Draw the polygon in the splitting plane
Paint over whatever has already been drawn
Recurse down the side of the tree containing the viewpoint
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32. Using a BSP-Tree
Observation: Things on the opposite side of a splitting plane from the viewpoint cannot obscure things on the same side as the viewpoint
A statement about rays – a ray must hit something on this side of the split plane before it hits the split plane and before it hits anything on the back side
NOT a statement about distance – things on the far side of the plane can be closer than things on the near side
Gives a relative ordering of the polygons, not absolute in terms of depth or any other quantity
Split plane
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33. Rendering Example 1 5b 9 5a 6 - + 2 Eye 3a - 3b 2 4b + + 8 7 5 4a 3b
Back-to-front rendering order is 3a,4a,6,1,4b,5a,2,3b,5b
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34. BSP-Tree Rendering (2) Advantages: Disadvantages:
One tree works for any viewing point
Filter anti-aliasing and transparency work
Have back to front ordering for compositing
Can also render front to back, and avoid drawing back polygons that cannot contribute to the view
Uses two trees – an extra one that subdivides the window
Major innovation in Quake
Disadvantages:
Can be many small pieces of polygon
Over-rendering
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35. Exact Visibility
An exact visibility algorithm tells you what is visible and only what is visible
No over-rendering
Warnock’s algorithm is an example
Difficult to achieve efficiently in practice
Small detail objects in an environment make it particularly difficult
But, in mazes and other simple environments, exact visibility is extremely efficient
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36. Cells and Portals Assume the world can be broken into cells
Simple shapes
Rooms in a building, for instance
Define portals to be the transparent boundaries between cells
Doorways between rooms, windows, etc
In a world like this, can determine exactly which parts of which rooms are visible
Then render visible rooms plus contents
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37. Cell-Portal Example (1)
View
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38. Cell and Portal Visibility
Start in the cell containing the viewer, with the full viewing frustum
Render the walls of that room and its contents
Recursively clip the viewing frustum to each portal out of the cell, and call the algorithm on the cell beyond the portal
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39. Cell-Portal Example (2)
View
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40. Cell-Portal Example (3)
View
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41. Cell-Portal Example (4)
View
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42. Cell-Portal Example (5)
View
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43. Cell-Portal Example (6)
View
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44. Cell-Portal Operations
Must clip polygons to the current view frustum (not the original one)
Can be done with additional hardware clipping planes, if you have them
Must clip the view frustum to the portal
Easiest to clip portal to frustum, then set frustum to exactly contain clipped portal
In Project 2, you implement these things in software, for a 2.5d environment
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45. Cell-Portal Properties
Advantages
Extremely efficient - only looks at cells that are actually visible: visibility culling
Easy to modify for approximate visibility - render all of partially visible cells, let depth buffer clean up
Can handle mirrors as well - flip world about the mirror and pretend mirror is a portal
Disadvantages
Restricted to environments with good cell/portal structure
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46. Project 2 Intro X (cx,cy,cz) You are given the following:
Rooms, defined in 2D by the edges that surround the room
The height of the ceiling
Each edge is marked opaque or clear
For each clear edge, there is a pointer to the thing on the other side
You know where the viewer is and what the field of view is
The viewer is given as (cx,cy,cz) position
The view frustum is given as a viewing angle and an angle for the field of view X
(cx,cy,cz)
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47. Project 2 (2) Represent the frustum as a left and right clipping line
You don’t have to worry about the top and bottom
Each clip line starts at the viewer’s position and goes to infinity in the viewing direction
Write a procedure that clips an edge to the view frustum
This takes a frustum and returns the endpoints of the clipped edge, or a flag to indicate that the edge is not visible
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48. Project 2 (3)
Write a procedure that takes a room and a frustum, and draws the room
Clip each edge to the frustum
If the edge is visible, draw the wall that the edge represents
Create the 3D wall from the 2d piece of edge
Project the vertices
Draw the polygon in 2D
If the edge is clear, recurse into neighboring polygon
Draw the floor and ceiling first, because they will be behind everything
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49. Where We Stand So far we know how to: Next Transform between spaces
Draw polygons
Decide what’s in front
Next
Deciding a pixel’s intensity and color
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50. Normal Vectors
The intensity of a surface depends on its orientation with respect to the light and the viewer
The surface normal vector describes the orientation of the surface at a point
Mathematically: Vector that is perpendicular to the tangent plane of the surface
What’s the problem with this definition?
Just “the normal vector” or “the normal”
Will use n or N to denote
Normals are either supplied by the user or automatically computed
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51. Transforming Normal Vectors
Normal vectors are directions
To transform a normal, multiply it by the inverse transpose of the transformation matrix
Recall, rotation matrices are their own inverse transpose
Don’t include the translation! Use (nx,ny,nz,0) for homogeneous coordinates
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52. Local Shading Models
Local shading models provide a way to determine the intensity and color of a point on a surface
The models are local because they don’t consider other objects
We use them because they are fast and simple to compute
They do not require knowledge of the entire scene, only the current piece of surface. Why is this good for hardware?
For the moment, assume:
We are applying these computations at a particular point on a surface
We have a normal vector for that point
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53. Local Shading Models What they capture: What they don’t do:
Direct illumination from light sources
Diffuse and Specular reflections
(Very) Approximate effects of global lighting
What they don’t do:
Shadows
Mirrors
Refraction
Lots of other stuff …
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54. “Standard” Lighting Model
Consists of three terms linearly combined:
Diffuse component for the amount of incoming light from a point source reflected equally in all directions
Specular component for the amount of light from a point source reflected in a mirror-like fashion
Ambient term to approximate light arriving via other surfaces
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55. Diffuse Illumination
Incoming light, Ii, from direction L, is reflected equally in all directions
No dependence on viewing direction
Amount of light reflected depends on:
Angle of surface with respect to light source
Actually, determines how much light is collected by the surface, to then be reflected
Diffuse reflectance coefficient of the surface, kd
Don’t want to illuminate back side. Use
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56. Diffuse Example Where is the light?
Which point is brightest (how is the normal at the brightest point related to the light)?
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57. Illustrating Shading Models
Show the polar graph of the amount of light leaving for a given incoming direction:
Show the intensity of each point on a surface for a given light position or direction
Diffuse?
Diffuse?
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58. Specular Reflection (Phong Reflectance Model)V R
Incoming light is reflected primarily in the mirror direction, R
Perceived intensity depends on the relationship between the viewing direction, V, and the mirror direction
Bright spot is called a specularity
Intensity controlled by:
The specular reflectance coefficient, ks
The Phong Exponent, p, controls the apparent size of the specularity
Higher n, smaller highlight
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59. Specular Example
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60. Illustrating Shading Models
Show the polar graph of the amount of light leaving for a given incoming direction:
Show the intensity of each point on a surface for a given light position or direction
Specular?
Specular?
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61. Specular Reflection ImprovementH N V
Compute based on normal vector and “halfway” vector, H
Always positive when the light and eye are above the tangent plane
Not quite the same result as the other formulation (need 2H)
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62. Putting It Together Global ambient intensity, Ia:
Gross approximation to light bouncing around of all other surfaces
Modulated by ambient reflectance ka
Just sum all the terms
If there are multiple lights, sum contributions from each light
Several variations, and approximations …
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63. Color Do everything for three colors, r, g and b
Note that some terms (the expensive ones) are constant
Using only three colors is an approximation, but few graphics practitioners realize it
k terms depend on wavelength, should compute for continuous spectrum
Aliasing in color space
Better results use 9 color samples
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64. Approximations for Speed
The viewer direction, V, and the light direction, L, depend on the surface position being considered, x
Distant light approximation:
Assume L is constant for all x
Good approximation if light is distant, such as sun
Distant viewer approximation
Assume V is constant for all x
Rarely good, but only affects specularities
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