simple mesh format (smf) 3d clipping

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CS 430/585Computer Graphics I

3D ClippingBackface Culling,

Z-buffering,Ray Casting, Ray Tracing

Week 7, Lecture 14

David Breen, William Regli and Maxim PeysakhovGeometric and Intelligent Computing Laboratory

Department of Computer ScienceDrexel University

http://gicl.cs.drexel.edu 2

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 Garlandhttp://graphics.cs.uiuc.edu/~garland/

• Triangle data

• Vertex indices begin at 1

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3D Clipping

• Cohen-Sutherland and Cyrus-Beck canbe 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 completelyvisible iff both codevalues of endpointsare 0, i.e.

• If line segments arecompletely outsidethe window, then

Pics/Math courtesy of Dave Mount @ 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

bit 1 point ABOVE the view volume y > 1bit 2 point BELOW the view volume y < -1bit 3 point RIGHT OF the view volume x > 1bit 4 point LEFT OF the view volume x < -1bit 5 point BEHIND the view volume z < -1bit 6 point IN FRONT the view volume z > 0

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Cohen-Sutherland for 3Dcomputing intersection points.• Use parametric representation of the line to

compute intersections• So for y=1 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, x=1, x=-1, z=-1, z=0

)()()(

010

010

010

zztzzyytyyxxtxx

-+=

-+=

-+=

)()1(

01

0

yyyt

-

-=

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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

bit 1 point ABOVE the view volume y > -zbit 2 point BELOW the view volume y < zbit 3 point RIGHT OF the view volume x > -zbit 4 point LEFT OF the view volume x < zbit 5 point BEHIND the view volume z < -1bit 6 point IN FRONT the view volume z > zmin

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Cohen-Sutherland for 3Dcomputing intersection points.

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

)()()(

010

010

010

zztzzyytyyxxtxx

-+=

-+=

-+=

)()()(

0101

00

zzyyyzt

---

-=

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More Efficient Alternative?

• Use Cohen-Sutherland to do trivialaccept/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 foreliminating these polygons

• On average eliminates half of the polygons18

Back-Face Culling

A

B

C

D

F

E

G

H

X

Z

• After projection, examinenormal Nk (xk , yk, zk)to the face.

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

©More general testlooks at Nk•V

© V - View vector

• The only test necessaryfor a single convexpolyhedron


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Back-Face Culled Wire-Frame

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Z-buffering

• Z-buffering (depth-buffering) is a visible surfacedetection 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_VALUE;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)

}}


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

Z-buffering


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Back-Face Culled & Z-BufferedWire-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.dm.unibo.it/~casciola www.siggraph.org

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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

†

- vrpz '+Fvrpz '+B

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Ray Casting (Ray Tracing )

• Determinesvisible surfacesby tracing rays oflight from theviewers eye to theobjects in theworld.

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Ray Casting

• Determines visible surfaces by tracing rays oflight 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 -=D-=D-=D

ztzzytyyxtxx D+=D+=D+= 000 ,,


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Computing Intersections withSphere

• 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

2222 )()()( rczbyax =-+-+-

220

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20 )()()( rcztzbytyaxtx =-D++-D++-D+

0)()()(

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

02

02

0

0002222

=--+-+-+

+-D+-D+-D+D+D+D

rczbyaxczzbyyaxxttxyx


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Computing Intersections withPolygon

• 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 thecoordinate plane

• Polygon containment test canbe performed in 2D

0=+++ DCzByAx

)(000

zCyBxADCzByAxt

D+D+D

+++-=


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Polygon Containment Test

• Jordan Curve Theorem:– Point is inside if, for any

ray, there is an oddnumber of crossings

– Otherwise it is outside• Be careful with all the

special cases• Wide variety of other

techniques exist

2

3

6

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Why Trace Rays?• More elegant• Testbed for techniques:

– modeling (reflectance,transport)

– rendering (e.g. MonteCarlo)

– texturing (e.g.hypertexture)

• Easiest photorealisticrenderer to implement Boat reflected in wavy water rendered in openGLusing an

environment map

Compiled from: Lecture notes of Dr. John C. Hart @ University ofIllinois

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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 multiplelight sources

• Can be used to producehighly realistic images

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Ray Traced Image

Blessed State, by Ken Musgrave

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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

qh

qq

22

2

sin1

sin1cos

-=

-=

†

t = h(n ⋅ i) - 1-h2(1- (n ⋅ i)2)( ) n -hi

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

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

Compiled from: Lecture notes of Dr. John C. Hart @ University of Illinois

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Refraction in Action

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More Refractions

Henrik Wann JensenUC, San Diego

Photon Mapping!

Justin HansenUniversity of Utah

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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


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• Ray only intersected with the objects contained in thepartitions it passes through

• If partitions are examined in the order ray travels, wecan stop after first intersection is found

• Need to check if intersection itself is in the partition


Efficiency Considerations

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Computer Graphics II

Image produced by Horace Ip

simple mesh format (smf) 3d clipping

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