David Luebke1/10/2016 CS 551 / 645: Introductory Computer Graphics David Luebke [email protected] cs551

David Luebke 04/21/23

CS 551 / 645: Introductory Computer Graphics

David Luebke

[email protected]

http://www.cs.virginia.edu/~cs551


Side Note

Shifts versus additions


Where We’ve Been

Till now we’ve talked about a 2-D world We’ve covered:

– Display technology– Color– Framebuffers– Drawing 2-D lines (and drawing them fast)– Drawing 2-D polygons (especially triangles)– Clipping polygons


Where We’re Going

3-D graphics (finally!) Next lectures:

– The graphics pipeline: the big picture– Rigid-body transforms

Math

– Homogeneous coordinates Math

– The viewing transform Math

– The projection transform Math


3-D Graphics: A Whirlwind Tour

Transform

Illuminate

Transform

Clip

Project

Rasterize

Model & CameraModel & CameraParametersParameters Rendering PipelineRendering Pipeline FramebufferFramebuffer DisplayDisplay


The Display You Know

Transform

Illuminate

Transform

Clip

Project

Rasterize



The Framebuffer You Know

Transform

Illuminate

Transform

Clip

Project

Rasterize



The Rendering Pipeline

Transform

Illuminate

Transform

Clip

Project

Rasterize


Why do we call it a pipeline?


2-D Rendering You Know

Transform

Illuminate

Transform

Clip

Project

Rasterize



Edge Equations You Know

V2

V1

V0 Find edge

equations



V2

V1

V0

- +

Disable processors where plugging (Xproc,Yproc) into edge equation of {V0,V1} evaluates to negative



V2

V1

V0 Find edge

equations



V2

V1

V0

+-




V2

V1

V0

+

-




V2

V1

V0 All enabled

processors evaluate linear expressions for triangle color, depth, etc.


Edge Walking (You Know?)

Traditional method:– Find edge equations

{V0V1} {V0V2} {V2V1}– Find and fill spans

where scanlines cross the edge

– Just a bilinearinterpolation!

V0

V2

V1


The Rendering Pipeline: 3-D

Transform

Illuminate

Transform

Clip

Project

Rasterize




ModelingTransforms

Scene graphObject geometry

LightingCalculations

ViewingTransform

Clipping

ProjectionTransform

Result:Result:

• All vertices of scene in shared 3-D “world” coordinate All vertices of scene in shared 3-D “world” coordinate systemsystem

• Vertices shaded according to lighting modelVertices shaded according to lighting model

• Scene vertices in 3-D “view” or “camera” coordinate Scene vertices in 3-D “view” or “camera” coordinate systemsystem

• Exactly those vertices & portions of polygons in view Exactly those vertices & portions of polygons in view frustumfrustum

• 2-D screen coordinates of clipped vertices2-D screen coordinates of clipped vertices





Clipping

Result:Result:






ModelingTransforms

ViewingTransform

ProjectionTransform


Rendering: Transformations

So far, discussion has been in screen space But model is stored in model space

(a.k.a. object space or world space) Three sets of geometric transformations:

– Modeling transforms– Viewing transforms– Projection transforms



Modeling transforms– Size, place, scale, and rotate objects parts of the

model w.r.t. each other– Object coordinates world coordinates

Z

X

Y

X

Z

Y



Viewing transform– Rotate & translate the world to lie directly in front of

the camera Typically place camera at origin Typically looking down -Z axis

– World coordinates view coordinates



Projection transform– Apply perspective foreshortening

Distant = small: the pinhole camera model

– View coordinates screen coordinates



All these transformations involve shifting coordinate systems (i.e., basis sets)

That’s what matrices do… Represent coordinates as vectors, transforms

as matrices

Multiply matrices = concatenate transforms!

Y

X

Y

X

cossin

sincos



Homogeneous coordinates: represent coordinates in 3 dimensions with a 4-vector– Denoted [x, y, z, w]

Note that typically w = 1 in model coordinates

– To get 3-D coordinates, divide by w:[x’, y’, z’] = [x/w, y/w, z/w]

Transformations are 4x4 matrices Why? To handle translation and projection



OpenGL caveats:– All modeling transforms and the viewing transform

are concatenated into the modelview matrix– A stack of modelview matrices is kept– The projection transform is stored separately in

the projection matrix

See Chapter 3 of the OpenGL book



ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:







Rendering: Lighting

Illuminating a scene: coloring pixels according to some approximation of lighting– Global illumination: solves for lighting of the whole

scene at once– Local illumination: local approximation, typically

lighting each polygon separately

Interactive graphics (e.g., hardware) does only local illumination at run time



ModelingTransforms



ViewingTransform

Clipping

ProjectionTransform

Result:Result:







Rendering: Clipping

Clipping a 3-D primitive returns its intersection with the view frustum:

See Foley & van Dam section 19.1


Rendering: Clipping

Clipping is tricky!

In: 3 verticesIn: 3 verticesOut: 6 verticesOut: 6 vertices

Clip

Clip In: 1 polygonIn: 1 polygonOut: 2 polygonsOut: 2 polygons



Transform

Illuminate

Transform

Clip

Project

Rasterize



Modeling: The Basics

Common interactive 3-D primitives: points, lines, polygons (i.e., triangles)

Organized into objects– Collection of primitives, other objects– Associated matrix for transformations

Instancing: using same geometry for multiple objects – 4 wheels on a car, 2 arms on a robot


Modeling: The Scene Graph

The scene graph captures transformations and object-object relationships in a DAG

Objects in black; blue arrows indicate instancing and each have a matrix

Robot

BodyHead

ArmTrunkLegEyeMouth


Modeling: The Scene Graph

Traverse the scene graph in depth-first order, concatenating transformations

Maintain a matrix stack of transformations

ArmTrunkLegEyeMouth

Head Body

Robot

Foot

MatrixMatrixStackStack

VisitedVisited

UnvisitedUnvisited

ActiveActive


Modeling: The Camera

Finally: need a model of the virtual camera– Can be very sophisticated

Field of view, depth of field, distortion, chromatic aberration…

– Interactive graphics (OpenGL): Pinhole camera model

Field of view Aspect ratio Near & far clipping planes

Camera pose: position & orientation


Modeling: The Camera

These parameters are encapsulated in a projection matrix– Homogeneous coordinates 4x4 matrix!– See OpenGL Appendix F for the matrix

The projection matrix premultiplies the viewing matrix, which premultiplies the modeling matrices– Actually, OpenGL lumps viewing and modeling

transforms into modelview matrix


Rigid-Body Transforms

Goal: object coordinatesworld coordinates Idea: use only transformations that preserve

the shape of the object– Rigid-body or Euclidean transforms – Includes rotation, translation, and scale

To reiterate: we will represent points as column vectors:

z

y

x

zyx ),,(


Vectors and Matrices

Vector algebra operations can be expressed in this matrix form– Dot product:

– Cross product: Note: use

right-handrule!

z

y

x

zyx

b

b

b

aaaba

0cb

0ca

cba

z

y

x

z

y

x

xy

xz

yz

c

c

c

b

b

b

aa

aa

aa

0

0

0


Translations

For convenience we usually describe objects in relation to their own coordinate system– Solar system example

We can translate or move points to a new position by adding offsets to their coordinates:

– Note that this translates all points uniformly

z

y

x

t

t

t

z

y

x

z

y

x

'

'

'


3-D Rotations

Rotations in 2-D are easy:

3-D is more complicated– Need to specify an axis of rotation

– Common pedagogy: express rotation about this axis as the composition of canonical rotations

Canonical rotations: rotation about X-axis, Y-axis, Z-axis

y

x

y

x

cossin

sincos

'

'


3-D Rotations

Basic idea:– Using rotations about X, Y, Z axes, rotate model until

desired axis of rotation coincides with Z-axis– Perform rotation in the X-Y plane (i.e., about Z-axis)– Reverse the initial rotations to get back into the initial

frame of reference

Objections:– Difficult & error prone– Ambiguous: several combinations about the canonical

axis give the same result– For a different approach, see McMillan’s lecture 12

Documents

David Luebke1/10/2016 CS 551 / 645: Introductory Computer Graphics David Luebke [email protected] cs551