CS 536 Overview Computer Graphics Backface Culling, Z ...david/Classes/CS536/Lectures/L... · 1 2 CS 536 Computer Graphics Backface Culling, Z-buffering, Ray Casting, Ray Tracing

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CS 536Computer Graphics

Backface Culling,Z-buffering,

Ray Casting, Ray TracingWeek 8, Lecture 17

David Breen, William Regli and Maxim PeysakhovDepartment of Computer Science

Drexel University

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Overview

• Rendering topics– Z-buffering– Back-Face Culling– Ray Tracing (Ray Casting)

1994 Foley /VanDam/Finer/Huges/Phi ll ips ICG

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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– Right-hand vertex ordering defines outward normal– Polygons not facing the viewer called Back-Facing

• Back-Face Culling is a technique for eliminating polygons for these kinds of models

• On average eliminates half of the polygons– Could be done for performance reasons

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Back-Face Culling

A

B

C

D

F

E

G

H

X

Z

• After canonical transformation, examine normal Nk (xk , yk, zk)to the face.

• If zk < 0, face is a Back-Face - don’t draw it

© More general test looks at Nk•V

© V - View vector

• The only test necessary for a single convex polyhedron 1994 Foley /VanDam/Finer/Huges/Phi ll ips ICG

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Normal for Triangle

p0p1

p2

nplane n · (p - p0 ) = 0

n = (p1 - p0 ) × (p2 - p0 )

normalize n ¬ n / |n|

p

Note that right-hand rule determines outward face

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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-buffering0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

5 5 5 5 5 5 55 5 5 5 5 55 5 5 5 55 5 5 55 5 55 55

5 5 5 5 5 5 5 05 5 5 5 5 5 0 05 5 5 5 5 0 0 05 5 5 5 0 0 0 05 5 5 0 0 0 0 05 5 0 0 0 0 0 05 0 0 0 0 0 0 00 0 0 0 0 0 0 0

5 5 5 5 5 5 5 05 5 5 5 5 5 0 05 5 5 5 5 0 0 05 5 5 5 0 0 0 05 5 5 0 0 0 0 05 5 0 0 0 0 0 05 0 0 0 0 0 0 00 0 0 0 0 0 0 0

5 5 5 5 5 5 5 05 5 5 5 5 5 0 05 5 5 5 5 0 0 05 5 5 5 0 0 0 06 5 5 3 0 0 0 07 6 5 4 3 0 0 08 7 6 5 4 3 0 00 0 0 0 0 0 0 0

34 35 4 36 5 4 37 6 5 4 38 7 6 5 4 3

+

+

=

=


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Z-bufferingfor (y = 0; y < YMAX; y++)

for (x = 0; x < XMAX; x++) {F[x][y] = BACKGROUND_COLOR;Z[x][y] = ZMIN;

}for (each polygon)

for (each pixel in polygon’s projection) {pz = polygon’s z-value at pixel coordinates (x,y)if (pz > Z[x][y]) { /* New point is closer */

Z[x][y] = pz;F[x][y] = polygon’s color at pixel coordinates

(x,y)}

} 1994 Foley /VanDam/Finer/Huges/Phi ll ips ICG

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Z-buffering w/ front/back clippingfor (y = 0; y < YMAX; y++)

for (x = 0; x < XMAX; x++) {F[x][y] = BACKGROUND_VALUE;Z[x][y] = -1; /* Back value in NPC */

}for (each polygon)

for (each pixel in polygon’s projection) {pz = polygon’s z-value at pixel coordinates (x,y)if (pz < FRONT && pz > Z[x][y]) { /* New point is behind front plane

& closer than previous point */ Z[x][y] = pz;F[x][y] = polygon’s color at pixel coordinates (x,y)

}}


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• 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: horizontal edge, degenerate

triangle & single vertex

Z Interpolation


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• za = z1 + (|Pa - P1|/|P2 - P1|)(z2 - z1)• zb = z1 + (|Pb - P1|/|P3 - P1|)(z3 - z1)• zp = za + (|Pp - Pa|/|Pb - Pa|)(zb - za)

• P1 = (x1, y1) • P2 = (x2, y2)• P3 = (x3, y3)

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Back-Face Culled & Z-Buffered Wire-Frame

See the Difference

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See the Difference

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See the Difference

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Depth Cueing• Objects that are closer are brighter• Objects farther away are darker• Color = BaseColor*(z - far)/(near - far)

www.dm.unibo.it/~casciola www.siggraph.org

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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


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Select the center of projection and viewplane windowfor (each scan line) {

for (each pixel on the scan line) {Find ray from the center of projections through the pixelfor (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

}}

Ray Casting


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Computing Intersections• Using parametric equation of the line:

• Simplifying for speed:

• Resulting ray:

(x0,y0,z0)

(x1,y1,z1)

)(),(),( 010010010 zztzzyytyyxxtxx −+=−+=−+=

)(),(),( 010101 zzzyyyxxx −=Δ−=Δ−=Δ

ztzzytyyxtxx Δ+=Δ+=Δ+= 000 ,,


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• 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• Solve with quadratic formula

– No real roots (square root is negative). No intersection– One real root. Ray grazes the sphere– Two real roots. There are two points of intersection

Computing Intersections with Sphere

2222 )()()( rczbyax =−+−+−

220

20

20 )()()( rcztzbytyaxtx =−Δ++−Δ++−Δ+

0)()()(

)]()()([2)(22

02

02

0

0002222

=−−+−+−+

+−Δ+−Δ+−Δ+Δ+Δ+Δ

rczbyax

czzbyyaxxttxyx


Relation of t to IntersectionWe want the smallest positive t - call it ti

t0

t1

t1

t0Discriminant = 0

Discriminant < 0

t1

t0

t0 =−B− B2 − 4AC

2A

"

#$$

%

&''

t1 =−B+ B2 − 4AC

2A

"

#$$

%

&''

t0

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Ray-triangle Intersection

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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

0=+++ DCzByAx

)(000

zCyBxADCzByAx

tΔ+Δ+Δ

+++−=


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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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3

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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 re flec ted in wavy water rendered in openGL us ing an env i ronment map

Compi led from: Lec ture notes of Dr. J ohn C. Hart @ Univers i ty o f Il l inois

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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

Bles s ed State , by Ken Mus grave

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• Snell’s Law: hi sin qi = ht sin qt• Let h =hi /ht = sin qt / sin qi• Let m = (cos qi n - i) / sin qi• Then…t = sin qt m - cos qt n

= (sin qt / sin qi) (cos qi n - i) - cos qt n= (h cos qi - cos qt )n - h i

To Ray Trace, We Need Refraction

cos qi n - i

i

-n

qi

qt t = ?

m

i

tt

θη

θθ

22

2

sin1

sin1cos

−=

−=

€

t = η(n ⋅ i) − 1−η2(1− (n ⋅ i)2)( ) n−ηi

Can be negative for grazing angles when h >1, say when going from glass to air, resulting in total internal reflection(no refraction).

Compi led from: Lec ture notes of Dr. J ohn C. Hart @ Univers i ty o f Il l ino is

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Refraction in Action

Turner Whitted, Bell Labs, 1980

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More Refractions

Henrik Wann JensenUC, San Diego

Photon Mapping!

Justin HansenUniversity of Utah 38

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


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• 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


Efficiency Considerations

40Image produced by Horace Ip

Advanced Rendering Techniques

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41Image produced by Dmitriy Bespalov

Advanced Rendering Techniques

Documents

CS 536 Overview Computer Graphics Backface Culling, Z ...david/Classes/CS536/Lectures/L... · 1 2 CS 536 Computer Graphics Backface Culling, Z-buffering, Ray Casting, Ray Tracing