CS 430/585 Computer Graphics I 3D Clipping Backface Culling, Z-buffering, Ray Casting, Ray Tracing Week 7, Lecture 14

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CS 430/585Computer Graphics I3D
ClippingBackface Culling,Z-buffering,Ray
Casting, Ray TracingWeek 7, Lecture 14

• David Breen, William Regli and Maxim Peysakhov
• Geometric and Intelligent Computing Laboratory
• Department of Computer Science
• Drexel University
• http//gicl.cs.drexel.edu

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Overview

• 3D Clipping
• Advanced topics
• Z-buffering
• Back-Face Culling
• Ray Tracing (Ray Casting)

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Simple Mesh Format (SMF)

• Michael Garland http//graphics.cs.uiuc.edu/garla
  nd/
• Triangle data
• Vertex indices begin at 1

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3D Clipping

• Cohen-Sutherland and Cyrus-Beck can be trivially
  extended to 3D
• We will cover
• Cohen-Sutherland for 3D, (parallel projection)
• Cohen-Sutherland for 3D, (perspective projection)

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Recall Cohen-Sutherland

• Line is completely visible iff both code values
  of endpoints are 0, i.e.

• If line segments are completely outside the
  window, then

Pics/Math courtesy of Dave Mount _at_ UMD-CP
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Cohen-Sutherland for 3D,Parallel Projection

• Use 6 bits
• Trivially accept if both end-codes are 0
• Trivially reject if bit-by-bit AND of end-codes
  is not 0
• Up to 6 intersections may have to be computed

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Cohen-Sutherland for 3D computing intersection
points.

• Use parametric representation of the line to
  compute intersections
• So for y1 replace y with 1 and solve for t
• If 1 ? t ? 0 use it to find x and z
• Test if x and z are in valid range
• Repeat for planes y-1, x1, x-1, z-1, z0

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Cohen-Sutherland for 3D, Perspective Projection

• Use 6 bits identical to parallel view volume
  clipping
• Conditions on the codes are different
• Trivially accept/reject lines using same roles
• Intersection points computed differently

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Cohen-Sutherland for 3D computing intersection
points.

• Intersections with planes z-1, zzmin is the
  same.
• Calculating intersections with a sloping plane
  
• For plane yz these two equations are equal
• Repeat for planes y-z, xz, x-z

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Clipping in Homogeneous Coordinates

• Same algorithm for clipping parallel and
  perspective view volumes
• Avoid problems of clipping rational parametric
  splines
• Transform canonical perspective view volume to
  canonical parallel view volume

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Application of Matrix M

• If using Nper we can always clip against the
  parallel projection View Volume
• Nper M Sper SHpar T(-PRP) R T(-VRP)

Before
After
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Homogeneous Clipping

• 3D parallel view-volume defined by
• -1 ? x ? 1, -1 ? y ? 1, -1 ? z ? 0
• -1 ? X/W ? 1, -1 ? Y/W ? 1, -1 ? Z/W ? 0
• We must change the inequalities for W lt0
• W gt0 - W ? X ? W,
  - W ? Y ? W,
  - W ? Z/W ? 0
• W lt0 - W ? X ? W,
  - W ? Y ? W,
  - W ? Z/W ? 0

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Homogeneous Clipping

• Clip once for W gt0 and once for W lt0
• Too much work
• For each point P with W lt0 find symmetric 1P
  and clip it

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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More Efficient Alternative?

• Use Cohen-Sutherland to do trivial accept/reject
• Project remaining edges onto view plane
• Clip lines in 2D

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end clipping
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Mesh/Faceted Model
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Back-Face Culling

• Assumptions
• Object approximated as closed polyhedron
• Polyhedron interior is not exposed by the front
  cutting plane
• Eye-point not inside object
• Polygons not facing the viewer called Back-Facing
• Right-hand vertex ordering defines outward normal
• Back-Face Culling is a technique for eliminating
  these polygons
• On average eliminates half of the polygons

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Back-Face Culling

• After projection, examine normal Nk (xk , yk, zk)
  to the face.
• If zk lt 0, face is a Back-Face - dont draw it
• More general test looks at NkV
• V - View vector
• The only test necessary for a single convex
  polyhedron

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Back-Face Culled Wire-Frame
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Z-buffering

• Z-buffering (depth-buffering) is a visible
  surface detection algorithm
• Implementable in hardware and software
• Requires data structure (z-buffer) in addition to
  frame buffer.
• Z-buffer stores values 0 .. ZMAX corresponding
  to depth of each point.
• If the point is closer than one in the buffers,
  it will replace the buffered values

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Z-buffering




1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Z-buffering

• for (y 0 y lt YMAX y)
• for (x 0 x lt XMAX x)
• Fxy BACKGROUND_VALUE
• Zxy ZMIN
• for (each polygon)
• for (each pixel in polygons projection)
• pz polygons z-value at pixel coordinates
  (x,y)
• if (pz gt Zxy) / New point is closer /
  
• Zxy pz
• Fxy polygons color at pixel coordinates
  (x,y)

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Z-buffering

• We can simplify the calculation of z by
  exploiting the fact that triangle is planar.
• Interpolate z values along the edges
• Interpolate z values along scan line
• Special cases vertices horizontal edge

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Back-Face Culled Z-Buffered Wire-Frame
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See the Difference
From HW of Andrew Mroczkowski
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Depth Cueing

• Objects that are closer are brighter
• Objects farther away are darker
• Color BaseColor?(z - far)/(near - far)

www.siggraph.org
www.dm.unibo.it/casciola
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Important Reminder

• Recall near and far planes are transformed by
  Nparand Nper
• Parallel projection
• Near z 0
• Far z -1
• Perspective projection
• Near z
• Far z -1

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Ray Casting (Ray Tracing )

• Determines visible surfaces by tracing rays of
  light from the viewers eye to the objects in the
  world.

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Ray Casting

• Determines visible surfaces by tracing rays of
  light from the viewers eye to the objects
• View plane is divided on the pixel grid
• The eye ray is fired from the center of
  projection through each pixel

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Ray Casting

• Select the center of projection and viewplane
  window
• for (each scan line)
• for (each pixel on the scan line)
• Find ray from the center of projections through
  the pixel
• for (each object in the scene)
• if (object is intersected and
  intersection is closest considered
  so far)
• Record the intersection and the object name
• Set pixel color to that of the closest
  intersected object

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Computing Intersections

• Using parametric equation of the line
• Simplifying for speed
• Resulting ray

(x1,y1,z1)
(x0,y0,z0)
1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Computing Intersections with Sphere

• Sphere with radius r and center (a,b,c) is
• Intersection is found by substituting values for
  (x,y,z)
• With some straightforward algebraic
  transformations
• Equation is quadratic in terms of t
• No real roots, No intersection
• One real root, Ray grazes the sphere
• Two roots, There are two points of intersection

1994 Foley/VanDam/Finer/Huges/Phillips ICG
33
Computing Intersections with Polygon

• First intersect ray with plane
• The substitution results in
• If denominator is 0,
  ray is parallel to the
  plane
• Project polygon and point orthographically on the
  coordinate plane
• Polygon containment test can
  be performed in 2D

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Polygon Containment Test

• Jordan Curve Theorem
• Point is inside if, for any ray, there is an odd
  number of crossings
• Otherwise it is outside
• Be careful with all the special cases
• Wide variety of other techniques exist

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Why Trace Rays?

• More elegant
• Testbed for techniques
• modeling (reflectance, transport)
• rendering (e.g. Monte Carlo)
• texturing (e.g. hypertexture)
• Easiest photorealistic renderer to implement

Boat reflected in wavy water rendered in
openGLusing an environment map
Compiled from Lecture notes of Dr. John C. Hart
_at_ University of Illinois
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Ray Tracing

• Extension of ray casting
• Idea Continue to bounce the ray in the scene
• Shoot rays to light sources
• Simple and powerful
• Reflections, shadows, transparency and multiple
  light sources
• Can be used to produce highly realistic images

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Ray Traced Image
Blessed State, by Ken Musgrave
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To Ray Trace, We Need Refraction

• Snells Law ?i sin ?i ?t sin ?t
• Let ? ?i /?t sin ?t / sin ?i
• Let m (cos ?i n - i) / sin ?i
• Then
• t sin ?t m - cos ?t n
• (sin ?t / sin ?i) (cos ?i n - i) - cos ?t n
• (? cos ?i - cos ?t )n - ? i

Can be negative for grazing angles when ? gt1, say
when going from glass to air, resulting in total
internal reflection (no refraction).
Compiled from Lecture notes of Dr. John C. Hart
_at_ University of Illinois
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Refraction in Action
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More Refractions
Photon Mapping!
Justin Hansen University of Utah
Henrik Wann Jensen UC, San Diego
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Efficiency Considerations

• Partition the bounding box on a regular grid with
  equal sized extents
• Associate each partition with the list of objects
  contained in it

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Efficiency Considerations

• Ray only intersected with the objects contained
  in the partitions it passes through
• If partitions are examined in the order ray
  travels, we can stop after first intersection is
  found
• Need to check if intersection itself is in the
  partition

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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Computer Graphics II
Image produced by Horace Ip