Download - CS 431/636 Advanced Rendering Techniques

CS 431/636 Advanced Rendering Techniques

Dr. David Breen

University Crossings 149

Tuesday 6PM 8:50PM

Presentation 2

4/7/09

Start Up

Any questions from last time? Go over sampling image plane? Or intersection algorithms?

Slide Credits

Leonard McMillan, Seth Teller, Fredo Durand, Barb Cutler - MIT

G. Drew Kessler, Larry Hodges - Georgia Institute of Technology

John Hart - University of Illinois Rick Parent - Ohio State University

More Geometry & Intersections

Ray/Plane Intersection

Ray is defined by R(t) = Ro + Rd*t where t 0

Ro = Origin of ray at (xo, yo, zo)

Rd = Direction of ray [xd, yd, zd] (unit vector)

Plane is defined by [A, B, C, D]

Ax + By + Cz + D = 0 for a point in the plane

Normal Vector, N = [A, B, C] (unit vector)

A2 + B2 + C2 = 1

D = - N • P0 (P0 - point in plane)

Ray/Plane (cont.)

Substitute the ray equation into the plane equation:

A(xo + xdt) + B(yo + ydt) + C(zo +zdt) + D = 0

Solve for t:t = -(Axo + Byo + Czo + D) / (Axd + Byd + Czd)

t = -(N • Ro - N • P0 ) / (N • Rd)

What Can Happen?

N • Rd = 0N • Rd > 0

t < 0

Ro

t > 0

Ro

Ray/Plane Summary

Intersection point:(xi, yi, zi) = (xo + xdti, yo + ydti, zo + zdti)

n Calculate N • Rd and compare it to zero.

n Calculate ti and compare it to zero.

n Compute intersection point.n Flip normal if N • Rd is positive

Ray-Parallelepiped Intersection Axis-aligned From (X1, Y1, Z1) to (X2, Y2, Z2) Ray P(t)=Ro+Rdt

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

Naïve ray-box Intersection Use 6 plane equations Compute all 6 intersection Check that points are inside box

Ax+By+Cz+D ≤ 0

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

Factoring out computation Pairs of planes have the same

normal Normals have only one non-0

component Do computations one dimension at a

time Maintain tnear and tfar (closest and

farthest so far)

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

Test if parallel

If Rdx = 0, then ray is parallel If Rox < X1 or Rox > x2 return false

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

If not parallel

Calculate intersection distance t1 and t2 t1 = (X1-Rox)/Rdx

t2 = (X2-Rox)/Rdx

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

t1

t2

Test 1 Maintain tnear and tfar

If t1 > t2, swap if t1 > tnear, tnear = t1 if t2 < tfar, tfar = t2

If tnear > tfar, box is missed

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

t1x t2xtnear

t1y

t2ytfar

tfar

Test 2

If tfar < 0, box is behind

y=Y2

y=Y1

x=X1 x=X2

Ro

Rd

t1xt2xtfar

Algorithm recap Do for all 3 axes

Calculate intersection distance t1 and t2 Maintain tnear and tfar If tnear > tfar, box is missed If tfar < 0, box is behind

If box survived tests, return intersection at tnear

If tnear is negative, return tfar

y=Y2

y=Y1

x=X1 x=X2Ro

Rd

tnear

t1y

tfar

Ray/Ellipsoid Intersection Ray/Cylinder Intersection

Ellipsoid's surface is defined by the set of points {(xs, ys, zs)} satisfying the equation:

(xs/ a)2 + (ys/ b)2 + (zs/ c)2 - 1 = 0

Cylinder's surface is defined by the set of points {(xs, ys, zs)} satisfying the equation:

(xs)2 + (ys)2 - r2 = 0 -z0 ≤ zs ≤ z0

Centers at origin

Substitute ray equation into surface equations

This is a quadratic equation in t: At2 + Bt + C = 0

Analyze as before Solve for t with quadratic formula Plug t back into ray equation - Done Well,… not exactly

Ray/Ellipsoid Intersection Ray/Cylinder Intersection

Ray/Cylinder Intersection

Is intersection point Pi between -Z0 and Z0? If not, Pi is not valid Also need to do intersection test with

z = -Z0 , Z0 plane If (Pix)2 + (Piy)2 ≤ r2, you’ve intersected a

“cap” Which valid intersection is closer?

Superquadrics

€

x

a1

⎛

⎝ ⎜

⎞

⎠ ⎟

2ε 2

+y

a2

⎛

⎝ ⎜

⎞

⎠ ⎟

2ε 2

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

ε 2ε1

+z

a3

⎛

⎝ ⎜

⎞

⎠ ⎟

2ε1

=1

Bezier Patch Patch of order (n, m) can be defined in terms of a set

of (n + 1)(m + 1) control points Pi+1,j+1 for integer indices i = 0 to n, j = 0 to m.

€

(u,v)∈ [0,1]2

p(u,v) =i= 0

n

∑ Bin (u)B j

m (v)j= 0

m

∑ Pi+1, j +1

Bin (u) =

n

i

⎛

⎝ ⎜

⎞

⎠ ⎟u

i(1− u)n−i

Tesselate Patches and Superquadrics

Utah Teapot

Modeled by 32 Bézier Patches Control points available at

http://www.holmes3d.net/graphics/teapot

SMF Triangle Meshesv -1 -1 -1v 1 -1 -1 v -1 1 -1 v 1 1 -1 v -1 -1 1 v 1 -1 1 v -1 1 1 v 1 1 1 f 1 3 4 f 1 4 2 f 5 6 8 f 5 8 7 f 1 2 6 f 1 6 5 f 3 7 8 f 3 8 4 f 1 5 7 f 1 7 3 f 2 4 8 f 2 8 6

vertices

triangles

Draw data structure

Triangle Meshes (.iv)

Transformations & Hierarchical Models

Add an extra dimension• in 2D, we use 3 x 3 matrices• In 3D, we use 4 x 4 matrices

Each point has an extra value, w

Homogeneous Coordinates

x'

y'

z'

w'

=

x

y

z

w

a

e

i

m

b

f

j

n

c

g

k

o

d

h

l

p

p' = M p

Most of the time w = 1, and we can ignore it

Homogeneous Coordinates

x'

y'

z'

1

=

x

y

z

1

a

e

i

0

b

f

j

0

c

g

k

0

d

h

l

1

Translate (tx, ty, tz)

Why bother with the extra dimension?Because now translations can be encoded in the matrix!

Translate(c,0,0)

x

y

p p'

c

x'

y'

z'

1

=

x

y

z

1

1

0

0

0

0

1

0

0

0

0

1

0

tx

ty

tz

1

x'

y'

z'

Scale (sx, sy, sz) Isotropic (uniform)

scaling: sx = sy = sz

You only have to implement uniform scaling

x'

y'

z'

1

=

x

y

z

1

sx

0

0

0

0

sy

0

0

0

0

sz

0

0

0

0

1

Scale(s,s,s)

x

p

p'

qq'

y

Rotation

About z axis

x'

y'

z'

1

=

x

y

z

1

cos

sin

0

0

-sin

cos

0

0

0

0

1

0

0

0

0

1

ZRotate()

x

y

z

p

p'

θ

Rotation

About x axis:

About y axis:

x'

y'

z'

1

=

x

y

z

1

0

cos sin

0

0

-sin cos

0

1

0

0

0

0

0

0

1

x'

y'

z'

1

=

x

y

z

1

cos 0

-sin 0

sin 0

cos 0

0

1

0

0

0

0

0

1

Rotation About (kx, ky, kz), an

arbitrary unit vector

(Rodrigues Formula)

x'

y'

z'

1

=

x

y

z

1

kxkx(1-c)+c

kykx(1-c)+kzs

kzkx(1-c)-kys

0

0

0

0

1

kykx(1-c)-kzs

kyky(1-c)+c

kzky(1-c)+kxs

0

kxkz(1-c)+kys

kykz(1-c)-kxs

kzkz(1-c)+c

0

where c = cos & s = sin

Rotate(k, )

x

y

z

θ

k

How are transforms combined?

(0,0)

(1,1)(2,2)

(0,0)

(5,3)

(3,1)Scale(2,2) Translate(3,1)

TS =2

0

0

2

0

0

1

0

0

1

3

1

2

0

0

2

3

1=

Scale then Translate

Use matrix multiplication: p' = T ( S p ) = ((TS) p)

Caution: matrix multiplication is NOT commutative!

0 0 1 0 0 1 0 0 1

Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p

Translate then Scale: p' = S ( T p ) = ST p

(0,0)

(1,1)(4,2)

(3,1)

(8,4)

(6,2)

(0,0)

(1,1)(2,2)

(0,0)

(5,3)

(3,1)Scale(2,2) Translate(3,1)

Translate(3,1) Scale(2,2)

TS =2

0

0

0

2

0

0

0

1

1

0

0

0

1

0

3

1

1

ST =2

0

0

2

0

0

1

0

0

1

3

1

Non-commutative CompositionScale then Translate: p' = T ( S p ) = TS p

2

0

0

0

2

0

3

1

1

2

0

0

2

6

2

=

=

Translate then Scale: p' = S ( T p ) = ST p

0 0 1 0 0 1 0 0 1

Transformations in Ray Tracing

Transformations in Modeling

Position objects in a scene Change the shape of objects Create multiple copies of objects Projection for virtual cameras Animations

Scene Description

Scene

LightsCamera ObjectsMaterialsBackground

Simple Scene Description File

Camera { center 0 0 10 direction 0 0 -1 up 0 1 0 }

Lights { numLights 1 DirectionalLight { direction -0.5 -0.5 -1 color 1 1 1 } }

Background { color 0.2 0 0.6 }

Materials { numMaterials <n>

<MATERIALS> }

Group { numObjects <n> <OBJECTS> }

Hierarchical Models

Logical organization of scene

Group { numObjects 3 Group { numObjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numObjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

Simple Example with Groups

Group { numObjects 3 Group { numObjects 3 Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { numObjects 2 Group { Box { <BOX PARAMS> } Box { <BOX PARAMS> } Box { <BOX PARAMS> } } Group { Box { <BOX PARAMS> } Sphere { <SPHERE PARAMS> } Sphere { <SPHERE PARAMS> } } } Plane { <PLANE PARAMS> } }

Adding Materials

Adding Transformations

Using Transformations

Position the logical groupings of objects within the scene

Transformation in group

Directed Acyclic Graph is more efficient and useful

Processing Model Transformattions

Goal Get everything into world coordinates

Traverse graph/tree in depth-first order Concatenate transformations Can store intermediate transformations Apply/associate final transformation to

primitive at leaf node What about cylinders, superquadrics, etc.?

Transform ray!

Transform the Ray

Map the ray from World Space to Object Space

World Space

r major

r minor

(x,y)

(0,0) Object Space

r = 1

pWS = M pOS

pOS = M-1 pWS

Transform Ray New origin:

New direction:

originOS = M-1 originWS

directionOS = M-1 (originWS + 1 * directionWS) - M-1 originWS

originOS

originWS

directionOS

directionWS

Object SpaceWorld Space

qWS = originWS + tWS * directionWS

qOS = originOS + tOS * directionOS

directionOS = M-1 directionWS

Transforming Points & Directions

Transform point

Transform direction

Map intersection point and normal back to

world coordinates

Homogeneous Coordinates: (x,y,z,w)

W = 0 is a point at infinity (direction)

Transform Normals Why? They’re used for shading

object color only Diffuse Shading

Transforming Normals A surface normal is a property, not a

geometric entity

Correct normal transformation matrix:

See http://www.cgafaq.info/wiki/Transforming_Normals

Given overlapping shapes A and B:

Union Intersection Subtraction

Constructive Solid Geometry (CSG)

How can we implement CSG?

Points on A, Outside of B

Points on B, Outside of A

Points on B, Inside of A

Points on A, Inside of B


Collect all the intersections


Implementing CSG

1. Test "inside" intersections:• Find intersections with A,

test if they are inside/outside B• Find intersections with B,

test if they are inside/outside A

2. Overlapping intervals:• Find the intervals of "inside"

along the ray for A and B• Compute

union/intersection/subtractionof the intervals

Constructive Solid Geometry

Ray Tracing CSG Models

Rick ParentOhio State U.

CSG Form object as booleans of primitive objects

Primitives: sphere, cube, cylinder, cone Boolean operators: union, intersection, difference

Tree structure used to manage operations Leaf nodes are primitive objects Intermediate nodes specify combination operator

CSE 681

Union

Min (tCmin, tB

min )

B

C +

BC

Ray intersects union: at first intersection

Possible ways for 2 spans to overlap

Intersection

++

If ((tCmin< tB

min ) and (tCmax> tB

min ) ): tBmin

Else If ((tBmin< tC

min ) and (tBmax> tC

min ) ): tCmin

Else: none

BC

B

C

First time in B and in C

Difference

-+

BC

If ((tBmin< tC

min ): tBmin

Else if (tCmax< tB

max ): tCmax

Else: none

B

C

First time in B not in C

Difference

+-

BC

If ((tCmin< tB

min): tCmin

Else if (tBmax< tC

max ): tBmax

Else: none

B

C

First time in C not in B

PrimitivesAnything that can be intersected (easily) with a ray

Conics: solve analytically using R(t)Convex polyhedraA plane (a cutting plane is useful)

can be used as a modeling tool (boolean operations) surface model (e.g., polyhedron) computed from CGS

or Can be used as a model representation

keep tree structure and ray trace directly

Controlling the Combinations

-+

+

?

Tree Structure

circlerectangle rectangle

+ +

+ -

T1T2 T3

T4

T5- +

+T1

T2

T3

Tree Structure #1

Tree Structure


+ +

- +

T1T2 T3

T4

T5- +

+T1

T2

T3

Tree Structure #2

Tree Structure

• Intersect ray with leaf nodes (primitive objects)• Combine intersection spans according to intermediate nodes

• union• intersection• difference

• Might create multiple spans

Union of Spans

Intersection of Spans

Difference of Spans

Normals of CSG intersections

Normal of some surface (or its negation)

Union or intersection: positive normal of intersected surface

Difference normals

• Intersection is one of:• tmin of positive object – normal of surface• tmax of negative object – negated normal

Add transformations to tree

http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/model/csg.html

CSE 681

Bounding Volumes


+ +

+ -

T1T2 T3

T4

T5

Construction•Use bounding volumes at leaf nodes• Union bounding volumes at interior nodes

Traversal•Top-down•Test bounding volume at interior

Example

Example (Simon Chorley)

For example:

Wrap Up

Discuss status/problems/issues with this week’s programming assignment

Download - CS 431/636 Advanced Rendering Techniques

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