CS 354Transformation, Clipping, and CullingMark KilgardUniversity of TexasJanuary 31, 2012

CS 354 2

Today’s material

Homework #2 assigned, due Thursday start of class! Math problems http://www.cs.utexas.edu/~mjk/teaching/cs354_s12/hw2.pdf

In-class quiz Lecture topic: graphics math & transformations

How do vertex positions transformed to NDC space? Generalized clipping and culling

Assignment Reading

Chapter 6, 310-322 Next project (Project #1) to be assigned Thursday, February 2

Building a 3D object model loader Due Thursday, February 16

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Course Information Reminders Piazza

Working well now https://piazza.com/utexas/cs354

Public CS course web site http://www.cs.utexas.edu/~mjk/teaching/cs354_s12/

Lecture slides in PDF form Now has class lecture schedule

Slideshare.net http://www.slideshare.net/Mark_Kilgard

Lecture slides for web browser viewing

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My Office Hours

Tuesday, before class Painter (PAI) 5.35 8:45 a.m. to 9:15

Thursday, after class ACE 6.302 11:00 a.m. to 12:00

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Last time, this time

Last lecture, we discussed Setup for interpolation Basic hidden surface removal via depth

testing Simplistic pixel updates

This lecture Getting coordinates from object space to

NDC space Generalized clipping and culling

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Daily Quiz

1. Scenario: Depth buffering is enabled, the depth buffer is cleared to 1.0, the depth function is “less than”; and color writes are enabled (blending is disabled)…

A stream of shaded fragments are rasterized to the pixel location (24.5, 11.5) in the following order with their depth and color indicated:

1: [ depth = 0.7, red ]2: [ depth = 0.9, green ]3: [ depth = 0.3, pink ]4: [ depth = 0.4, green ]

What is the final depth and color of the pixel at (24.5, 11.5)?

2. Setting up the plane equation for an interpolated attribute for a triangle to rasterize involves which of the following?

A) Solving a system of equations with 3 unknownsB) Watertight perimeter computationC) Evaluating the quadratic equationD) Inverting a 5x5 matrix

3. For barycentric triangle interpolation, the three barycentric weights for each vertex sum to what value?

On a sheet of paper• Write your EID, name, and date• Write #1, #2, #3, followed by its answer

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Programmer’s View:OpenGL API Example

Let’s draw a triangleglShadeModel(GL_SMOOTH); // smooth color interpolationglEnable(GL_DEPTH_TEST); // enable hidden surface removal

glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT);glBegin(GL_TRIANGLES); { // every 3 vertexes makes a triangle glColor4ub(255, 0, 0, 255); // RGBA=(1,0,0,100%) glVertex3f(-0.8, 0.8, 0.3); // XYZ=(-8/10,8/10,3/10)

glColor4ub(0, 255, 0, 255); // RGBA=(0,1,0,100%) glVertex3f( 0.8, 0.8, -0.2); // XYZ=(8/10,8/10,-2/10)

glColor4ub(0, 0, 255, 255); // RGBA=(0,0,1,100%) glVertex3f( 0.0, -0.8, -0.2); // XYZ=(0,-8/10,-2/10)} glEnd();

Pro Tip: use curly braces to “bracket” nested OpenGLusage; no semantic meaning, just highlights grouping

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Programmer’s View:GLUT API Example

Windowing code#include <GL/glut.h> // includes necessary OpenGL headers

void display() { // << insert code on prior slide here >> glutSwapBuffers();}

void main(int argc, char **argv) { // request double-buffered color window with depth buffer glutInitDisplayMode(GLUT_RGBA | GLUT_DOUBLE | GLUT_DEPTH); glutInit(&argc, argv); glutCreateWindow(“simple triangle”); glutDisplayFunc(display); // function to render window glutMainLoop();}

FYI: GLUT = OpenGL Utility Toolkit

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A Simplified Graphics PipelineApplication

Vertex batching & assembly

Triangle assembly

Triangle clipping

Triangle rasterization

Fragment shading

Depth testing

Color update

Application-

OpenGL API boundary

Framebuffer

NDC to window space

Depth buffer

severaloperations

left outfor simplicityin explaining

the simple_triangleexample

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A few more steps expandedApplication

Vertex batching & assembly

Lighting

View frustum clipping

Triangle rasterization

Fragment shading

Depth testing

Color update

Application-

OpenGL API boundary

Framebuffer

NDC to window space

Depth buffer

Vertex transformation

User defined clipping

Back face culling

Perspective divide

Triangle assemblyTexture coordinate generation

was just “triangle clipping” before

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Conceptual Vertex Transformation

glVertex*API

commands

Modelviewmatrix

User-definedclip planes

View-frustumclip planes

to primitiverasterization

object-space coordinates

(xo,yo,zo,wo) eye-space coordinates

(xe,ye,ze,we)

clipped eye-space coordinates

clipped clip-space coordinates Perspective

divisionProjection

matrix

Viewport + Depth Rangetransformation

(xc,yc,zc,wc)

window-spacecoordinates

(xw,yw,zw,1/wc)

normalized device coordinates (NDC)

(xn,yn,zn,1/wc)

clip-spacecoordinates

(xc,yc,zc,wc)

(xe,ye,ze,we)

(xe,ye,ze,we)

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User Clip Planes in Practice

[ParaView]

[IVoR]

Primarily used scientific visualizationand Computer Aided Design (CAD) applications

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Remember Back toclipspace Example

Six user-defined clip planes enabled toclip teapot to left scene’s clip space view

Without user clip planes

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Four-component positions!

Conventional geometry represents 3D points at (x,y,z) positions Affine 3D positions, Cartesian coordinates

Projective position concept Use fourth coordinate: W

So (x,y,z,w) (x/w, y/w, z/w) is the corresponding affine 3D position Known as “homogeneous coordinates”

Advantages Represents perspective cleanly Allows rasterization of external triangles Puts off (expensive) division

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Example, All Identical Positions Affine 3D

(x,y,z)

Projective 3D (x,y,z,w) → (x/w,y/w,z/w)

(2,-5,10) (2,-5,10,1)

(4,-10,20,2)

(1,-2.5,5,0.5)

(-2,5,-10,-1)

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Affine View FrustumClip Equations

The idea of a [-1,+1]3 view frustum cube Regions outside this cube get clipped Regions inside the cube get rasterized

Equations -1 ≤ xc ≤ +1

-1 ≤ yc ≤ +1

-1 ≤ zc ≤ +1

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Projective View FrustumClip Equations

Generalizes clip cube as a projective space Uses (xc,yc,zc,wc) clip-space coordinates

Equations -wc ≤ xc ≤ +wc

-wc ≤ yc ≤ +wc

-wc ≤ zc ≤ +wc

Notice Impossible for wc < 0 to survive clipping

Interpretation: wc is distance in front of the eye So negative wc values are “behind your head”

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glVertex3f Generalized

glVertex3f(x,y,z) Provides (x,y,z) affine position Implicit w of 1 supplied

Alternatively glVertex4f(x,y,z,w) Provides (x,y,z,w) projective position Value of w is explicit Rarely used in practice but does work

… Also glVertex2f(x,y) Implicit z of 0 and w of 1 supplied

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Vertex Transformation

So far, we’ve specified vertex positions directly in NDC coordinates Simple approach Everything in [-1,+1]3 cube gets drawn

You can now handle the truth OpenGL actually provides a projective transformation

from the glVertex3f parameters to NDC coordinates Expressed as a combination of two 4x4 matrix

transforms on the (x,y,z,w) object-space position Initial value of both matrices are identity

transforms Hence OpenGL initially takes glVertex* positions

directly to NDC space The simple_trianlge example relies on this

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Vertex Transformation

Object-space vertex position transformed by a general linear projective transformation Expressed as a 4x4 matrix

o

o

o

o

c

c

c

c

w

z

y

x

mmmm

mmmm

mmmm

mmmm

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Matrix Multiplication

v’ = M v Where M is a matrix and v is a vector Use: coordinate system changes

v = M-1 v’ Inverse matrix multiplication Uses: back projection, plane equation setup

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Two Transforms in Sequence

OpenGL thinks of the projective transform as really two matrix transforms

o

o

o

o

e

e

e

e

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MVMVMVMV

MVMVMVMV

MVMVMVMV

MVMVMVMV

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e

e

e

e

c

c

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PPPP

PPPP

PPPP

PPPP

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FIRSTobject-space

toeye-space

SECONDeye-space

toclip-space

16 Multiply-Addoperations

Another16 Multiply-Addoperations

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Modelview-Projection Transform

Matrixes can associate (combine) Combination of the modelview and projection

matrix = modelview-projection matrix or often simply the “MVP” matrix

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MVMVMVMV

MVMVMVMV

MVMVMVMV

MVMVMVMV

PPPP

PPPP

PPPP

PPPP

MVPMVPMVPMVP

MVPMVPMVPMVP

MVPMVPMVPMVP

MVPMVPMVPMVP

concatenation is64 Multiply-Addoperations, done by OpenGL driver

Matrix multiplicationis associative (but not commutative)

A(BC) = (AB)C, but ABC≠CBA

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Setting the Modelview and Projection Matrices

Easy way glLoadMatrixf(const GLfloat *matrix)

matrix is 4x4 array of floats Warning: stored in column-major order

So like FORTRAN, not C

Replaces the current matrix with the new matrix

Which matrix is replaced? Could be modelview or projection matrix? Depends on last call to glMatrixMode

glMatrixMode(GL_MODELVIEW) glMatrixMode(GL_PROJECTION)

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Careful: Beware of Selectors

Danger: glMatrixMode is error-prone If you forget the state of glMatrixMode, you can easily

update the wrong matrix Leads to very frustrating bugs Problem is glMatrixMode sets a selector which

changes which state “other” matrix commands update

The better API would have the matrix mode as a parameter to matrix commands, instead of a selector

New EXT_direct_state_access extension has a “direct” command for setting matrix state glMatrixLoadfEXT(GL_MODELVIEW, matrix) glMatrixLoadfEXT(GL_PROJECTION, matrix)

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Many Matrix Manipulation Commands

glLoadIdentity() Resets matrix to identity matrix Restores initial setting

glMultMatrixf(const GLfloat *m) Multiplies current matrix by

specified 4x4 matrix glTranslatef(x,y,z)

Applies 3D translation to the current matrix

glScalef(xs,ys,zs) Applies 3D (non-uniform) scaling

to the current matrix glRotatef(angle,x,y,z)

Applies rotation around a 3D axis to the current matrix

glPushMatrix() Pushes current matrix to the matrix

state Matrix now can be modified and

later popped back glPopMatrix()

The way to pop back to the prior matrix

Push and pop is good for hierarchical modeling

glOrtho(left, right, bottom, top, near, far) Multiplies current matrix by a 3D

orthographic “box” Maps a box to the NDC

[-1,+1]3 cube glFrustum(left, right, bottom, top,

near, far) Multiplies current matrix by a 3D

perspective frustum Maps a camera view to the NDC

[-1,+1]3 cube

OpenGL API conventionf-suffixed means takes single-precision float parametersd-suffixed means takes double-precision parametersExamples: glTranslated, glScaled, glRotated

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EXT_direct_state_accessVersions of Matrix Commands

Conventional Command

Selector-freeDSA Command

glLoadMatrixf glMatrixLoadfEXT

glLoadTransposeMatrixf glMatrixLoadTransposefEXT

glMultMatrixf glMatrixMultfEXT

glMultTransposeMatrixf glMatrixMultTransposefEXT

glLoadIdentity glMatrixLoadIdentityEXT

glRotatef glMatrixRotatefEXT

glScalef glMatrixScalefEXT

glTranslatef glMatrixTranslatefEXT

glOrthof glMatrixOrthoEXT

glFrustumf glMatrixFrustumEXT

glPushMatrix glMatrixPushEXT

glPopMatrix glMatrixPopEXT

ConvetionsAll f-suffixed entrypoints have d-suffixedversions for doubleprecision too

All selector-freeroutines take aGLenum matrix modeparameter, sameas glMatrixMode

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Scale, Rotate, andTranslate Example

Add matrix codeglShadeModel(GL_SMOOTH); // smooth color interpolationglEnable(GL_DEPTH_TEST); // enable hidden surface removal

glMatrixMode(GL_MODELVIEW);glLoadIdentity(); // reset to null transformglScalef(0.7, 0.4, 1.0); // scale by 70% in X and 40% in Y (and 100% in Z)glRotatef(30, 0, 0, 1); // rotate in XY plane (around Z axis) 30 degreesglTranslatef(-0.1, 0.3, 0.1);// shift -0.1 to the left and 0.3 up (and 0.1 back in Z)

glClear(GL_COLOR_BUFFER_BIT|GL_DEPTH_BUFFER_BIT);glBegin(GL_TRIANGLES); { // every 3 vertexes makes a triangle glColor4ub(255, 0, 0, 255); // RGBA=(1,0,0,100%) glVertex3f(-0.8, 0.8, 0.3); // XYZ=(-8/10,8/10,3/10)

glColor4ub(0, 255, 0, 255); // RGBA=(0,1,0,100%) glVertex3f( 0.8, 0.8, -0.2); // XYZ=(8/10,8/10,-2/10)

glColor4ub(0, 0, 255, 255); // RGBA=(0,0,1,100%) glVertex3f( 0.0, -0.8, -0.2); // XYZ=(0,-8/10,-2/10)} glEnd();

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Load Identity Transform

Prototype glLoadIdentity()

Replace current matrix with this matrix

1000

0100

0010

0001

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Non-uniform Scale Transform

Prototype glScalef(GLfloat xs, GLfloat ys, GLfloat zs)

Post-concatenates this matrix

1000

000

000

000

zs

ys

xs

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Translate Transform

Prototype glTranslatef(GLfloat x, GLfloat y, GLfloat z)

Post-concatenates this matrix

1000

100

010

001

z

y

x

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Rotation Transform

Prototype glRotatef(GLfloat a, // angle in degrees

GLfloat x, GLfloat y, GLfloat z) Post-concatenates a rotation matrix

1000

0cossin0

0sincos0

0001

aa

aa

glRotatef(a,1,0,0)

1000

0cos0sin

0010

0sin0cos

aa

aa

glRotatef(a,0,1,0)

1000

0100

00cossin

00sincos

aa

aa

glRotatef(a,0,0,1)

Axis could be arbitrary un-normalized direction vector

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Orthographic Transform

Prototype glOrthof(GLfloat left, GLfloat right,

GLfloat bottom, GLfloat top, GLfloat near, GLfloat far)

Post-concatenates an orthographic matrix

1000

200

02

0

002

nf

nf

nf

bt

bt

bt

lr

lr

lr

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Frustum Transform

Prototype glFrustumf(GLfloat left, GLfloat right,

GLfloat bottom, GLfloat top, GLfloat near, GLfloat far)

Post-concatenates a frustum matrix

0100

2)(00

02

0

002

nf

fn

nf

nfbt

bt

bt

nlr

lr

lr

n

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Two Transforms Concatenated Multiplying the position by

First by the modelview Then second by the projection …is equivalent to multiplying by their

concatenation so

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MVPMVPMVPMVP

MVPMVPMVPMVP

MVPMVPMVPMVP

MVPMVPMVPMVP

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Just 16 Multiply-Addoperations… even though 2 (combined) transformation matrices!

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Perspective Divide

Divide clip-space (x,y,z) by clip-space w To get Normalized Device Coordinate (NDC) space

Means reciprocal operation is done once And done after clipping Minimizes division by zero concern

c

c

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c

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n

n

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wzw

yw

x

z

y

x

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Viewport and Depth Range

Prototypes glViewport(GLint vx, GLint vy, GLsizei vw, GLsizei vh) glDepthRange(GLclampd n, GLclampd f)

Equations Maps NDC space to window space

22

22

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nfznf

vvy

v

vvx

v

z

y

x

n

hyn

h

wxn

w

w

w

w

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Conceptual Vertex Transformation

glVertex*API

commands

Modelviewmatrix

User-definedclip planes

View-frustumclip planes

to primitiverasterization

object-space coordinates

(xo,yo,zo,wo) eye-space coordinates

(xe,ye,ze,we)

clipped eye-space coordinates

clipped clip-space coordinates Perspective

divisionProjection

matrix

Viewport + Depth Rangetransformation

(xc,yc,zc,wc)

window-spacecoordinates

(xw,yw,zw,1/wc)

normalized device coordinates (NDC)

(xn,yn,zn,1/wc)

clip-spacecoordinates

(xc,yc,zc,wc)

(xe,ye,ze,we)

(xe,ye,ze,we)

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View Frustum Clipping Generalizes Cleanly

Recall moving left vertex so it’s X = -1.8 Result is a clipped triangle

(-1.8, 0.8, 0.3, 1)

(-0.8, 0.8, -0.2,1)

(0, -0.8, -0.2, 1)

origin at (0,0,0,1)

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Clipped Triangle Visualized

Clipped and Rasterized Normally Visualization of NDC space

Notice triangle is “poking out” of the cube;this is the reason that should be clipped

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Break Clipped Triangle intoTwo Triangles

But how do we find these “new” vertices?The edge clipping the triangle is the line at X = -1so we know X = -1 at these points—but what about Y?

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Use Ratios to Interpolate Clipped Positions

(-1.8, 0.8, 0.3, 1)

(-0.8, 0.8, -0.2,1)

(0, -0.8, -0.2)

origin at (0,0,0,1)

X = -1Y = (1.8/2.6)×0.8 + (0.8/2.6)×0.8 = 0.8Z = (1.8/2.6)×0.3 + (0.8/2.6)×-0.2 = 0.1461538W = (1.8/2.6)×1 + (0.8/2.6)×1 = 1

-1-(-1.8)=0.8

0.8-(-1)=1.8

0.8-(-1.8)=2.6(-1,0.8,0.146153,1)

Straightforward becauseall the edges are orthogonal

Weights: 1.8/2.6 0.8/2.6, sum to 1

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Use Ratios to Interpolate Clipped Positions

(-1.8, 0.8, 0.3)

(-0.8, 0.8, -0.2)

(0, -0.8, -0.2)

origin at (0,0,0)

0-(-1.8) = 1.8

0-(-1) = 1

X = -1Y = (1/1.8)×0.8 + (0.8/1.8)×-0.8 = 0.08888… Z = (1/1.8)×0.3 + (0.8/1.8)×-0.2 = 0.07777…

(-1,0.0888,0.0777)

-1-(-1.8) = 0.8

Weights: 1/1.8 0.8/1.8, sum to 1

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Generalize to Non-1 W

Affine clipping plane in example -1 ≤ xc

Generalizes to 1 xc + 0 yc + 0zc + 1 wc ≥ 0

Looks like a plane equation A xc + B yc + C zc + D wc ≥ 0 with coefficients A, B, C, and D

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View Frustum Plane Equations

All six view frustum planes can be described by simple projective plane equations

Name A B C D Plane equation

Left 1 0 0 1 1 xc + 1wc ≥ 0

Right -1 0 0 1 -1 xc + 1wc ≥ 0

Bottom 0 1 0 1 1 yc + 1wc ≥ 0

Top 0 -1 0 1 -1 yc + 1wc ≥ 0

Near 0 0 1 1 1 zc + 1wc ≥ 0

Top 0 0 -1 1 -1 zc + 1wc ≥ 0

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Projective Clipping

Each vertex computes its clip distance w.r.t. a plane Plug vertex’s (x,y,z,w) into Ax+By+Cz+Dw≥0 plane equation…

provides a clip distance For two vertexes forming a triangle edge

Both negative? Discard the edge Both positive? Accept the edge (no clipping) One negative, one positive

Clipping is needed Compute t as s / (s + p) where s and p are clip distances

s is the “inside” distance; p is the “outside” distance Weight all per-vertex attributes based on t Makes new “clipped” vertex on the clip plane Generate 1 triangle if 1 of 3 vertices is inside;

if 2 inside, generate 2 triangles Repeat process (recursively) for all clip planes

Only slightly more complicated than prior clipping algorithm

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Readily Extends toUser-defined Clip Planes

In addition to the six view frustum planes of clip space… OpenGL supports user-defined clip planes Allows slicing into geometry

Operate in eye space instead of clip space Enabled with glEnable(GL_CLIP_PLANE0+num) Plane equation set by glClipPlane

Clip planes are transformed current modelview projection matrix

Plane equation is Axe+Bye+Cze+Wze≥0 Instead of using (xc, yc, zc, wc) as view frustum planes do

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(Clip) Plane Transformation Vertex positions (and direction vectors) are

transformed like column vectors

Plane equations are transformed like row vectors

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MVMVMVMV

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MVMVMVMV

MVMVMVMV

MVMVMVMV

MVMVMVMVwzyx

w

z

y

x clipclipclipclip

e

e

e

e

glClipPlane parameters

glVertex4fparameters

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Conceptual Vertex Transformation

glVertex*API

commands

Modelviewmatrix

User-definedclip planes

View-frustumclip planes

to primitiverasterization

object-space coordinates

(xo,yo,zo,wo) eye-space coordinates

(xe,ye,ze,we)

clipped eye-space coordinates

clipped clip-space coordinates Perspective

divisionProjection

matrix

Viewport + Depth Rangetransformation

(xc,yc,zc,wc)

window-spacecoordinates

(xw,yw,zw,ww)

normalized device coordinates (NDC)

(xn,yn,zn,wn)

clip-spacecoordinates

(xc,yc,zc,wc)

(xe,ye,ze,we)

(xe,ye,ze,we)

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Vertex Shaders in the Pipeline

GeometryProgram

3D Applicationor Game

OpenGL API

GPUFront End

VertexAssembly

VertexShader

Clipping, Setup,and Rasterization

FragmentShader

Texture Fetch

RasterOperations

Framebuffer Access

Memory Interface

CPU – GPU Boundary

OpenGL 3.3

Attribute Fetch

PrimitiveAssembly

Parameter Buffer Readprogrammable

fixed-function

Legend

So far, we’ve discussed “fixed-function” vertex transformationSo far, we’ve discussed “fixed-function” vertex transformation

Modern GPUs make vertex processing Modern GPUs make vertex processing programmableprogrammable

Via vertex shaders!Via vertex shaders!

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Vertex Programmability

Paletted matrixPaletted matrixskinningskinning

Twister vertex programTwister vertex program

Per-vertex Per-vertex cartoon cartoon shadingshading

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Other Vertex Processing Tasks

Per-vertex lighting Given normal vectors at each vertex, view vector, material

parameters, and light parameters, compute the lit appearance of the surface

Texture coordinate generation Texture coordinates help map images (called “textures”) onto

rasterized primitives Often texture coordinates are passed explicitly with each vertex Alternatively, a vertex shader can compute them

Non-linear transformation Animation by blending various matrix transforms Twisting or morphing geometry

Fog computations Arbitrary setup for fragment shaders

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Vertex Shader Example:Key Frame Blending

Frame A Frame BBlended Frame

47%= + 53%

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Key Frame Blending

VertexShader

Frame A

Frame B

Other possible key frames

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Non-linear Vertex Transform + Fragment Shader Setup

2D grid over (s,t)[0,1]Tessellated torus

Fragment program

Vertex program

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Next Lecture

Graphics Math Interpolation, vector math, and number representations for

computer graphics As usual, expect a short quiz on today’s lecture

Know the form of a translate and scale matrix Know the order of operations in transforming a vertex position

Assignments Reading

Chapter 6, 310-322 Next project (Project #1) to be assigned Thursday, February 2

Building a 3D object model loader Due Thursday, February 16 2nd homework is on

class web site,5 pages, 10 problems