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2. 02/21/02 (c) 2001 University of Wisconsin, CS559 Today Viewing Orthographic viewing Homework 3
3. 02/21/02 (c) 2001 University of Wisconsin, CS559 Modeling 101 For the moment assume that all geometry
4. 02/21/02 (c) 2001 University of Wisconsin, CS559 Modeling and OpenGL In OpenGL, all geometry is specified
5. 02/21/02 (c) 2001 University of Wisconsin, CS559 Rendering Generate an image showing the contents of some
6. 02/21/02 (c) 2001 University of Wisconsin, CS559 Graphics Pipeline Graphics hardware employs a sequence of coordinate
7. 02/21/02 (c) 2001 University of Wisconsin, CS559 Local Coordinate Space It is easiest to define individual
8. 02/21/02 (c) 2001 University of Wisconsin, CS559 Global Coordinate System Everything in the world is transformed
9. 02/21/02 (c) 2001 University of Wisconsin, CS559 View Space Associate a set of axes with the
10. 02/21/02 (c) 2001 University of Wisconsin, CS559 3D Screen Space Transform view space into a cube:
11. 02/21/02 (c) 2001 University of Wisconsin, CS559 Window Space Also called screen space (confusing) Convert the
12. 02/21/02 (c) 2001 University of Wisconsin, CS559 3D Screen to Window Transform Typically, windows are specified
13. 02/21/02 (c) 2001 University of Wisconsin, CS559 3D Screen to Window Transform How much do we
14. 02/21/02 (c) 2001 University of Wisconsin, CS559 3D Screen to Window Transform (-1,-1) (1,1) (xmin,ymin) (xmax,ymax)
15. 02/21/02 (c) 2001 University of Wisconsin, CS559 Orthographic Projection Orthographic projection projects all the points in
16. 02/21/02 (c) 2001 University of Wisconsin, CS559 Simple Orthographic Projection Specify the region of space that
17. 02/21/02 (c) 2001 University of Wisconsin, CS559 Rendering the Volume To project, map the view volume
18. 02/21/02 (c) 2001 University of Wisconsin, CS559 General Orthographic Projection We could look at the world
19. 02/21/02 (c) 2001 University of Wisconsin, CS559 Specifying a View The location of the image plane
20. 02/21/02 (c) 2001 University of Wisconsin, CS559 Getting there… We wish to end up in the
21. 02/21/02 (c) 2001 University of Wisconsin, CS559 View Space Given our camera definition: Which point is
22. 02/21/02 (c) 2001 University of Wisconsin, CS559 View Space The origin is at the center of
23. 02/21/02 (c) 2001 University of Wisconsin, CS559 World to View We must translate the world so
24. 02/21/02 (c) 2001 University of Wisconsin, CS559 All Together We apply a translation and then a
25. 02/21/02 (c) 2001 University of Wisconsin, CS559 OpenGL and Transformations OpenGL internally stores several matrices that
26. 02/21/02 (c) 2001 University of Wisconsin, CS559 OpenGL Camera The default OpenGL image plane has u
28. Скачать презентацию

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02/21/02
(c) 2001 University of Wisconsin, CS559
Today
Viewing
Orthographic viewing
Homework 3

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02/21/02
(c) 2001 University of Wisconsin, CS559
Modeling 101
For the moment assume that

all geometry consists of points, lines and faces
Line: A segment between two endpoints
Face: A planar area bounded by line segments
Any face can be triangulated (broken into triangles)

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02/21/02
(c) 2001 University of Wisconsin, CS559
Modeling and OpenGL
In OpenGL, all geometry

is specified by stating which type of object and then giving the vertices that define it
glBegin(…) …glEnd()
glVertex[34][fdv]
Three or four components (regular or homogeneous)
Float, double or vector (eg float[3])
Chapter 2 of the red book

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02/21/02
(c) 2001 University of Wisconsin, CS559
Rendering
Generate an image showing the contents

of some region of space
The region is called the view volume, and it is defined by the user
Determine where each object should go in the image
Viewing, Projection
Determine which object is in front at each pixel
Hidden surface elimination, Hidden surface removal, Visibility
Determine what color it is
Lighting, Shading

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02/21/02
(c) 2001 University of Wisconsin, CS559
Graphics Pipeline
Graphics hardware employs a sequence

of coordinate systems
The location of the geometry is expressed in each coordinate system in turn, and modified along the way
The movement of geometry through these spaces is considered a pipeline

Local Coordinate Space

World Coordinate Space

View Space

3D Screen Space

Display Space

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02/21/02
(c) 2001 University of Wisconsin, CS559
Local Coordinate Space
It is easiest to

define individual objects in a local coordinate system
For instance, a cube is easiest to define with faces parallel to the coordinate axis
Key idea: Object instantiation
Define an object in a local coordinate system
Use it multiple times by copying it and transforming it into the global system
This is the only effective way to have libraries of 3D objects, and such libraries do exist

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02/21/02
(c) 2001 University of Wisconsin, CS559
Global Coordinate System
Everything in the world

is transformed into one coordinate system - the global coordinate system
Actually, some things, like dashboards, may be defined in a different space, but we’ll ignore that
Lighting is defined in this space
The locations, brightness’ and types of lights
The camera is defined with respect to this space
Some higher level operations, such as advanced visibility computations, can be done here

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02/21/02
(c) 2001 University of Wisconsin, CS559
View Space
Associate a set of axes

with the image plane
The image plane is the plane in space on which the image should “appear,” like the film plane of a camera
One normal to the image plane
One up in the image plane
One right in the image plane
These three axes define a coordinate system (a rigid body transform of the world system)
Some camera parameters are easiest to define in this space
Focal length, image size
Depth is represented by a single number in this space
The “normal to image plane” coordinate

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02/21/02
(c) 2001 University of Wisconsin, CS559
3D Screen Space
Transform view space into

a cube: [-1,1]×[-1,1]×[-1,1]
The cube is the canonical view volume
Parallel sides make many operations easier
Tasks to do:
Clipping – decide what you can see
Rasterization - decide which pixels are covered
Hidden surface removal - decide what is in front
Shading - decide what color things are

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02/21/02
(c) 2001 University of Wisconsin, CS559
Window Space
Also called screen space (confusing)
Convert

the virtual screen into real screen coordinates
Drop the depth coordinates and translate
The windowing system takes care of this

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02/21/02
(c) 2001 University of Wisconsin, CS559
3D Screen to Window Transform
Typically, windows

are specified by an origin, width and height
Origin is either bottom left or top left corner, expressed as (x,y) on the total visible screen on the monitor or in the framebuffer
This representation can be converted to (xmin,ymin) and (xmax,ymax)
3D Screen Space goes from (-1,-1,-1) to (1,1,1)
Lets say we want to leave z unchanged
What basic transformations will be involved in the total transformation from 3D screen to window coordinates?

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02/21/02
(c) 2001 University of Wisconsin, CS559
3D Screen to Window Transform
How much

do we translate?
How much do we scale?

(-1,-1)

(1,1)

(xmin,ymin)

(xmax,ymax)

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02/21/02
(c) 2001 University of Wisconsin, CS559
3D Screen to Window Transform
(-1,-1)
(1,1)
(xmin,ymin)
(xmax,ymax)

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02/21/02
(c) 2001 University of Wisconsin, CS559
Orthographic Projection
Orthographic projection projects all the

points in the world along parallel lines onto the image plane
Projection lines are perpendicular to the image plane
Like a camera with infinite focal length
The result is that parallel lines in the world project to parallel lines in the image, and ratios of lengths are preserved
This is important in some applications, like medical imaging and some computer aided design tasks

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02/21/02
(c) 2001 University of Wisconsin, CS559
Simple Orthographic Projection
Specify the region of

space that we wish to render as a view volume
Assume that the viewer is looking in the –z direction, with x to the right and y up
Assuming a right-handed coordinate system
The view volume has:
a near plane at z=n
a far plane at z=f , (f < n)
a left plane at x=l
a right plane at x=r
a top plane at y=t
and a bottom plane at y=b

Specify the region of space that we wish to render as a view volume
Assume that the viewer is looking in the –z direction, with x to the right and y up
Assuming a right-handed coordinate system
The view volume has:
a near plane at z=n
a far plane at z=f , (f < n)
a left plane at x=l
a right plane at x=r
a top plane at y=t
and a bottom plane at y=b

(r,b,n)

(l,t,f)

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02/21/02
(c) 2001 University of Wisconsin, CS559
Rendering the Volume
To project, map the

view volume onto the canonical view volume
After that, we know how to map the view volume to the window
The mapping looks just like the one for screen->window:

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02/21/02
(c) 2001 University of Wisconsin, CS559
General Orthographic Projection
We could look at

the world from any direction, not just along –z
The image could rotated in any way about the viewing direction: x need not be right, and y need not be up
How can we specify the view under these circumstances?

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02/21/02
(c) 2001 University of Wisconsin, CS559
Specifying a View
The location of the

image plane in space
A point in space for the center of the image plane, (cx,cy,cz)
The direction in which we are looking
Specified as a vector that points back toward the viewer: (dx,dy,dz)
This vector will be normal to the image plane
A direction that we want to appear up in the image
This vector does not have to be perpendicular to n
We also need the size of the view volume – l,r,t,b,n,f
Specified with respect to the image plane, not the world

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02/21/02
(c) 2001 University of Wisconsin, CS559
Getting there…
We wish to end up

in the “simple” situation, so we need a coordinate system with:
A vector toward the viewer
One pointing right in the image plane
One pointing up in the image plane
The origin at the center of the image
We must:
Define such a coordinate system, view space
Transform points from the world space into view space
Apply our simple projection from before

Слайд 21

02/21/02
(c) 2001 University of Wisconsin, CS559
View Space
Given our camera definition:
Which point

is the origin of view space?
Which direction is the normal to the view plane, n?
How do we find the right vector, u?
How do we find the up vector, v?
Given these points, how do we do the transformation?

Слайд 22

02/21/02
(c) 2001 University of Wisconsin, CS559
View Space
The origin is at the

center of the image plane: (cx,cy,cz)
The normal vector is the normalized viewing direction:
We know which way up should be, and we know we have a right handed system, so u=up×n, normalized:
We have two vectors in a right handed system, so to get the third: v=n×u

Слайд 23

02/21/02
(c) 2001 University of Wisconsin, CS559
World to View
We must translate the

world so the origin is at (cx,cy,cz)
To complete the transformation we need to do a rotation
After this rotation:
The direction u in world space should be the direction (1,0,0) in view space
The vector v should be (0,1,0)
The vector n should be (0,0,1)
The matrix that does that is:

Слайд 24

02/21/02
(c) 2001 University of Wisconsin, CS559
All Together
We apply a translation and

then a rotation, so the result is:
And to go all the way from world to screen:

Слайд 25

02/21/02
(c) 2001 University of Wisconsin, CS559
OpenGL and Transformations
OpenGL internally stores several

matrices that control viewing of the scene
The MODELVIEW matrix is intended to capture all the transformations up to the view space
The PROJECTION matrix captures the view to screen conversion
You also specify the mapping from the canonical view volume into window space
Directly through function calls to set up the window
Matrix calls multiply some matrix M onto the current matrix C, resulting in CM
Set view transformation first, then set transformations from local to world space – last one set is first one applied

Слайд 26

02/21/02
(c) 2001 University of Wisconsin, CS559
OpenGL Camera
The default OpenGL image plane

has u aligned with the x axis, v aligned with y, and n aligned with z
Means the default camera looks along the negative z axis
Makes it easy to do 2D drawing (no need for any view transformation)
glOrtho(…) sets the view->screen matrix
Modifies the PROJECTION matrix
gluLookAt(…) sets the world->view matrix
Takes an image center point, a point along the viewing direction and an up vector
Multiplies a world->view matrix onto the current MODELVIEW matrix
You could do this yourself, using glMultMatrix(…) with the matrix from the previous slides

Last Time

Содержание

02/21/02(c) 2001 University of Wisconsin, CS559TodayViewingOrthographic viewingHomework 3

02/21/02(c) 2001 University of Wisconsin, CS559Modeling 101For the moment assume that

02/21/02(c) 2001 University of Wisconsin, CS559Modeling and OpenGLIn OpenGL, all geometry

02/21/02(c) 2001 University of Wisconsin, CS559RenderingGenerate an image showing the contents

02/21/02(c) 2001 University of Wisconsin, CS559Graphics PipelineGraphics hardware employs a sequence

02/21/02(c) 2001 University of Wisconsin, CS559Local Coordinate SpaceIt is easiest to

02/21/02(c) 2001 University of Wisconsin, CS559Global Coordinate SystemEverything in the world

02/21/02(c) 2001 University of Wisconsin, CS559View SpaceAssociate a set of axes

02/21/02(c) 2001 University of Wisconsin, CS5593D Screen SpaceTransform view space into

02/21/02(c) 2001 University of Wisconsin, CS559Window SpaceAlso called screen space (confusing)Convert

02/21/02(c) 2001 University of Wisconsin, CS5593D Screen to Window TransformTypically, windows

02/21/02(c) 2001 University of Wisconsin, CS5593D Screen to Window TransformHow much

02/21/02(c) 2001 University of Wisconsin, CS5593D Screen to Window Transform(-1,-1)(1,1)(xmin,ymin)(xmax,ymax)

02/21/02(c) 2001 University of Wisconsin, CS559Orthographic ProjectionOrthographic projection projects all the

02/21/02(c) 2001 University of Wisconsin, CS559Simple Orthographic ProjectionSpecify the region of

02/21/02(c) 2001 University of Wisconsin, CS559Rendering the VolumeTo project, map the

02/21/02(c) 2001 University of Wisconsin, CS559General Orthographic ProjectionWe could look at

02/21/02(c) 2001 University of Wisconsin, CS559Specifying a ViewThe location of the

02/21/02(c) 2001 University of Wisconsin, CS559Getting there…We wish to end up

02/21/02(c) 2001 University of Wisconsin, CS559View SpaceGiven our camera definition:Which point

02/21/02(c) 2001 University of Wisconsin, CS559View SpaceThe origin is at the

02/21/02(c) 2001 University of Wisconsin, CS559World to ViewWe must translate the

02/21/02(c) 2001 University of Wisconsin, CS559All TogetherWe apply a translation and

02/21/02(c) 2001 University of Wisconsin, CS559OpenGL and TransformationsOpenGL internally stores several

02/21/02(c) 2001 University of Wisconsin, CS559OpenGL CameraThe default OpenGL image plane

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