Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736:...

Computing & Information SciencesKansas State University

CG Basics 4 of 8: ViewingCIS 636/736: (Introduction to) Computer Graphics

CIS 636Introduction to Computer Graphics

William H. Hsu

Department of Computing and Information Sciences, KSU

KSOL course pages: http://snipurl.com/1y5gc

Course web site: http://www.kddresearch.org/Courses/CIS636

Instructor home page: http://www.cis.ksu.edu/~bhsu

Readings:

Slides based on Viewing 1-3, http://www.cs.brown.edu/courses/cs123/lectures.htm Sections 2.6 – 2.7, Eberly 2e – see http://snurl.com/1ye72

OpenGL API Documentation: http://www.opengl.org/documentation/

Lab 2: glFrustum(), gluPerspective() (http://snurl.com/1zt6z)

CG Basics 4 of 8:Detailed Introduction to Viewing



Online Recorded Lectures for CIS 636

Introduction to Computer Graphics Project Topics for CIS 636

Computer Graphics Basics (8) 1. Mathematical Foundations – Week 2

2. Rasterizing and 2-D Clipping – Week 3

3. OpenGL Primer 1 of 3 – Week 3

4. Detailed Introduction to 3-D Viewing – Week 4


6. Polygon Rendering – Week 6


8. Visible Surface Determination – Week 9

Recommended Background Reading for CIS 636

Shared Lectures with CIS 736 (Computer Graphics) Regular in-class lectures (35) and labs (7)

Guidelines for paper reviews – Week 7

Preparing term project presentations, demos for graphics – Week 11



Lecture Outline

History of Projections

Normalizing and Viewing Transformations

Perspective (to Parallel) Transformation

Kinds of Projections Parallel

Orthographic: top, front, side, other

Axonometric: isometric, dimetric, trimetric

Oblique: cavalier, cabinet

Other

Perspective: 1-point, 2-point, 3-point

Viewing in OpenGL



History

Geometrical Constructions

Types of Projection

Projection in Computer Graphics

From 3-D to 2-D: Orthographic and Perspective Projection – Part 1

Adapted from slides for Brown University CS 123, Computer Graphics

© 2000 – 2007, A. van Dam. Used with permission.



Parallel projections used for engineering and architecture because they can be used for measurements

Perspective imitates our eyes or camera, looks more natural

Logical Relationship amongTypes of Projections





Same method as multiview orthographic projections, except projection plane not parallel to any of the coordinate planes; parallel lines are equally foreshortened

Isometric: Angles between all three principal axes are equal (120º). The same scale ratio applies along each axis

Dimetric: Angles between two of the principal axes are equal; need two scale ratios

Trimetric: Angles different between the three principal axes; need three scale ratios

Note: different names for different views, but all part of a continuum of parallel projections of the cube; these differ in where the projection plane is relative to its cube

dimetric

dimetric

isometric

dimetric

orthographic

Carlbom Fig. 3-8

Axonometric Projections





Used for: catalogue illustrations patent office records furniture design structural design

Pros: don’t need multiple views illustrates 3D nature of object measurements can be made to scale along principal axes

Cons: lack of foreshortening creates distorted appearance more useful for rectangular than curved shapes

Construction of isometric projection:projection plane cuts each principal axis by 45°

Example

Carlbom Fig.2.2

Isometric Projection:Special Case of Axonometric





Projectors are at an oblique angle to the projection plane; view cameras have accordion housing, used for skyscrapers

Pros: can present the exact shape of one face of an object (can take accurate

measurements): better for elliptical shapes than axonometric projections, better for “mechanical” viewing

lack of perspective foreshortening makes comparison of sizes easier

displays some of object’s 3D appearance

Cons: objects can look distorted if careful choice not made about position of

projection plane (e.g., circles become ellipses)

lack of foreshortening (not realistic looking)

obliqueperspective

Oblique Projections





Source: http://www.usinternet.com/users/rniederman/star01.htm

View Camera





Construction of an obliqueparallel projection

Plan oblique projection of a city(Carlbom Fig. 2-6)

Front oblique projection of a radio

(Carlbom Fig. 2-4)

Examples of Oblique Projections





Rules for placing projection plane for oblique views

Projection plane should be set parallel to one of: most irregular of principal faces, or one which contains circular or

curved surfaces

longest principal face of object

face of interest

Projection plane parallel to circular face Projection plane not parallel to circular face

Example and Principles:Oblique View





Cavalier: Angle between projectors and projection plane is 45º. Perpendicular faces are projected at full scale

Cabinet: Angle between projectors and projection plane is arctan(2) = 63.4º. Perpendicular faces are projected at 50% scale

cavalier projection of unit cube

cabinet projection of unit cube

Main Types of Oblique Projections





multiview orthographic

cavalier cabinet

Carlbom Fig. 3-2

Examples of Orthographic andOblique Projections





Assume object face of interest lies in principal plane, i.e., parallel to xy, yz, or

zx planes. (DOP = Direction of Projection, VPN = View Plane Normal)

1) Multiview Orthographic

– VPN || a principal coordinate axis

– DOP || VPN

– shows single face, exact measurements

2) Axonometric


– DOP || VPN

– adjacent faces, none exact, uniformly foreshortened (as a function of angle between face normal and DOP)

3) Oblique


– DOP || VPN

– adjacent faces, one exact, others uniformly foreshortened

Parallel Projections: Summary





Used for: advertising presentation drawings for architecture, industrial design, engineering fine art

Pros: gives a realistic view and feeling for 3D form of object

Cons: does not preserve shape of object or scale (except where object intersects

projection plane)

Different from a parallel projection because parallel lines not parallel to the projection plane converge size of object is diminished with distance foreshortening is not uniform

Perspective Projections





One Point Perspective (z-axis vanishing point)

Three Point Perspective(z, x, and y-axis vanishing points)

Two Point Perspective (z, and x-axis vanishing points)

z

Vanishing Points [1]

For right-angled forms whose face normals are perpendicular to the x, y, z coordinate axes, the number of vanishing points = number of principal coordinate axes intersected by projection plane





What happens if same form is turned so that its face normals are not perpendicular to x, y, z coordinate axes?

• Although projection plane only intersects one axis (z), three vanishing points were created

• Note: can achieve final results which are identical to previous situation in which projection plane intersected all three axes

• New viewing situation: cube rotated, face normals no longer perpendicular to any principal axes

• Note: unprojected cube depicted here with parallel projection

Perspective drawing of rotated cube

Vanishing Points [2]





We’ve seen two pyramid geometries for understanding perspective projection:

Combining these 2 views:

1. perspective image is intersection of a plane with light rays from object to eye (COP)

2. perspective image is result of foreshortening due to convergence of some parallel lines toward vanishing points

Vanishing Points and View Point [1]





Project parallel lines AB, CD on xy plane Projectors from the eye to AB and CD define two planes, which meet

in a line which contains the view point, or eye This line does not intersect the projection plane (XY), because parallel

to it. Therefore there is no vanishing point






Lines AB and CD (this time with A and C behind projection plane) projected on xy plane: A’B and C’D

Note: A’B not parallel to C’D

Projectors from eye to A’B and C’D define two planes which meet in a line containing view point

This line intersects projection plane

Point of intersection is vanishing point

C

A






Analogous to size of film used in a camera Determines proportion of width to height of image displayed on screen Square viewing window has aspect ratio of 1:1 Movie theater “letterbox” format has aspect ratio of 2:1 NTSC television has aspect ratio of 4:3, HDTV is 16:9

Ko

dak H

DT

V 1

6:9

NT

SC

4: 3

Aspect Ratio





Determines amount of perspective distortion in picture parallel projection: none

wide-angle lens: lots

In a frustum, two viewing angles Width and height angles

We specify Height angle

Get Width angle from

Aspect ratio * Height angle

Choosing View angle Analogous to photographer choosing a specific type of lens

e.g., wide-angle or telephoto (narrow angle) lens

View Angle [1]





Lenses made for distance shots often have a nearly parallel viewing angle

and cause little perspective distortion, though they foreshorten depth

Wide-angle lenses cause a lot of perspective distortion

Resulting pictures

View Angle [2]





Real cameras have a roll of film that captures pictures

Synthetic camera “film” is a rectangle on an infinite film plane that contains image of scene

Why haven’t we talked about the “film” in our synthetic camera, other than mentioning its aspect ratio?

How is the film plane positioned relative to the other parts of the camera? Does it lie between the near and far clipping planes? Behind them?

Turns out that fine positioning of Film plane doesn’t matter. Here’s why:

for a parallel view volume, as long as the film plane lies in front of the scene, parallel projection onto film plane will look the same no matter how far away film plane is from scene

same is true for perspective view volumes, because the last step of computing the perspective projection is a transformation that stretches the perspective volume into a parallel volume

In general, it is convenient to think of the film plane as lying at the eye point (Position)



“Where’s My Film?”



Some camera models take a Focal length Focal Length is a measure of ideal focusing range; approximates behavior of

real camera lens Objects at distance of Focal length from camera are rendered in focus; objects

closer or farther away than Focal length get blurred Focal length used in conjunction with clipping planes Only objects within view volume are rendered, whether blurred or not. Objects

outside of view volume still get discarded

Focal Length





Non-oblique view volume:

Oblique view volume:

Look vector at an angleto film plane

Limitations of this Camera Model

Look vector perpendicular to film plane

Can create the following view volumes: perspective: positive view angle parallel: zero view angle

Cannot create oblique view volume Non-oblique vs. oblique view volumes:

For example, view cameras with bellows are used to take pictures of (tall) buildings.

Film plane is parallel to the façade, while camera points up This is oblique view volume, with façade undistorted





Carlbom, Ingrid and Paciorek, Joseph, “Planar Geometric Projections

and Viewing Transformations,” Computing Surveys, Vol. 10, No. 4

December 1978

Kemp, Martin, The Science of Art, Yale University Press, 1992

Mitchell, William J., The Reconfigured Eye, MIT Press, 1992

Foley, van Dam, et. al., Computer Graphics: Principles and Practice,

Addison-Wesley, 1995

Wernecke, Josie, The Inventor Mentor, Addison-Wesley, 1994

References





Projection in Computer Graphics





More concrete way to say the same thing as orientation soon you’ll learn how to express orientation in terms of Look and Up vectors

Look Vector the direction the camera is pointing three degrees of freedom; can be any vector in 3-space

Up Vector determines how the camera is rotated around the Look vector for example, whether you’re holding the camera horizontally or vertically (or

in between) projection of Up vector must be in the plane perpendicular to the look vector

(this allows Up vector to be specified at an arbitrary angle to its Look vector)

Up vectorLook vector

Position

projection of Up vector

Look and Up Vectors





D = P1 – P0 = (x1 – x0, y1 – y0)

Leave PEi as an arbitrary point on the clip edge; it’s a free

variable and drops out

Clip Edgei Normal Ni PEiP0-PEi

left: x = xmin (-1,0) (xmin, y) (x0- xmin,y0-y)

right: x = xmax (1,0) (xmax,y) (x0- xmax,y0-y)

bottom: y = ymin (0,-1) (x, ymin) (x0-x,y0- ymin)

top: y = ymax (0,1) (x, ymax) (x0-x,y0- ymax)

DiN

iEPP

iN

t

)0

(

)01

(

)min0

(

xx

xx

)01

(

)max0

(

xx

xx

)01

(

)min0

(

yy

yy

)01

(

)max0

(

yy

yy

Calculations for Parametric Line Clipping Algorithm

Clipping:Parametric Clipping, Review





Volume of space between Front and Back clipping planes defines what camera can see

Position of planes defined by distance along Look vector

Objects appearing outside of view volume don’t get drawn

Objects intersecting view volume get clipped

Front clipping plane

Back clipping plane

Clipping:Front and Back Clipping Planes

[1]





Reasons for Front (near) clipping plane:

Don’t want to draw things too close to the camera would block view of rest of scene

objects would be prone to distortion

Don’t want to draw things behind camera wouldn’t expect to see things behind the camera

in the case of perspective camera, if we decided to draw things behind camera, they would appear upside-down and inside-out because of perspective transformation

Reasons for Back (far) clipping plane:

Don’t want to draw objects too far away from camera distant objects may appear too small to be visually significant, but still take long

time to render

by discarding them we lose a small amount of detail but reclaim a lot of rendering time

alternately, scene may be filled with many significant objects

for visual clarity, we may wish to declutter scene by rendering those nearest camera and discarding rest


[2]





Have you ever played a video game and all of the sudden some object pops up in the background (e.g. a tree in a racing game)? That’s the object coming inside the far clip plane.

The old hack to keep you from noticing the pop-up is to add fog in the distance. A classic example of this is from Turok: Dinosaur Hunter

Now all you notice is fog and how little you can actually see. This practically defeats the purpose of an outdoor environment! And you can still see pop-up from time to time.

Thanks to fast hardware and level of detail algorithms, we can push the far plane back now and fog is much less prevalent

Putting the near clip plane as far away as possible helps Z precision. Sometimes in a game you can position the camera in the right spot so that the front of an object gets clipped letting you see inside of it.


[3]

Then (1997)

Now (2008)





From Position, Look vector, Up vector, Aspect ratio, Height angle, Clipping planes, and (optionally) Focal length together specify a truncated view volume

Truncated view volume is a specification of bounded space that camera can “see”

2D view of 3D scene can be computed from truncated view volume and projected onto film plane

Truncated view volumes come in two flavors: parallel and perspective



View Volume Specification



Limiting view volume useful for eliminating extraneous objects Orthographic parallel projection has width and height view angles of zero

Height

Width

Look vector

Near distance

Position

Far distance

Up vector



Truncated View Volume (Cuboid) for

Orthographic Parallel Projection



Removes objects too far from Position Otherwise would merge into “blobs”

Removes objects too close to Position Would be excessively distorted

Position

Near distance

Far distance

Height angle

Width angle =

Look vector

Up vectorHeight angle • Aspect ratio



Truncated View Volume (Frustum) for

Perspective Projection



Reduce degrees of freedom

Five steps to specifying view volume

1. Position the camera and therefore its view/film plane

2. Point it at what you want to see with camera in desired orientation

3. Define the field of view for a perspective view volume, aspect ratio of film and angle of view:

somewhere between wide angle, normal, and zoom

for a parallel view volume, width and height)

4. Choose perspective or parallel projection

5. Determine focal distance



Stage One:Specifying a View Volume



Placement of view volume (visible part of world) specified by camera’s position and orientation

Position (a point) Look and Up vectors

Shape of view volume specified by horizontal and vertical view angles front and back clipping planes

Perspective projection: projectors intersect at Position Parallel projection: projectors parallel to Look vector, but never intersect (or

intersect at infinity) Coordinate Systems

world coordinates – standard right-handed xyz 3-space viewing reference coordinates – camera-space right handed coordinate system (u, v, n);

origin at Position and axes rotated by orientation; used for transforming arbitrary view into canonical view



Specifying a View Volume



We have now specified an arbitrary view using our viewing parameters Problem: map arbitrary view specification to 2D picture of scene. This is

hard, both for clipping and for projection Solution: reduce to a simpler problem and solve

Note: Look vector along negative, not positive, z-axis is arbitrary but makes math easier

– Need: View specification from which it is easy to take pictures– Canonical view: from the origin, looking down negative z-axis

– parallel projection– sits at origin: Position = (0, 0, 0)– looks along negative z-axis: Look vector = (0, 0, –1)– oriented upright: Up vector = (0, 1, 0)– film plane extending from –1 to 1 in x and y

– think of the scene as lying behind window and we’re looking through the window

up

n

v

u

Arbitrary View Volume too Complex





Our goal is to transform our arbitrary view and the world to the canonical view volume, maintaining the relationship between view volume and world, then take picture for parallel view volume, transformation is affine: made up of translations,

rotations, and scales in the case of a perspective view volume, it also contains a non-affine†

perspective transformation that frustum into a parallel view volume, a cuboid

the composite transformation that will transform the arbitrary view volume to the canonical view volume, named the normalizing transformation, is still a 4x4 homogeneous coordinate matrix that typically has an inverse

easy to clip against this canonical view volume; clipping planes are axis-aligned!

projection using the canonical view volume is even easier: just omit the z-coordinate

for oblique parallel projection, a shearing transform is part of the composite transform, to “de-oblique” the view volume

† Affine transformations preserve parallelism but not lengths and angles. The perspective transformation is a type of non-affine transformation known as a projective transformation, which does not preserve parallelism



Normalizing toCanonical View Volume



Problem of taking a picture has now been reduced to problem of finding correct normalizing transformation

It is a bit tricky to find the rotation component of the normalizing transformation.

However, it is easier to find the inverse of this rotational component (trust us)

So we’ll digress for a moment and focus our attention on the inverse of the normalizing transformation, which is called the viewing transformation.

The viewing transformation turns the canonical view into the arbitrary view, or (x, y, z) to (u, v, n)

Viewing Transformation Normalizing Transformation





We know the view specification: Position, Look vector, and Up vector We need to derive an affine transformation from these parameters that

will translate and rotate the canonical view into our arbitrary view the scaling of the film (i.e. the cross-section of the view volume) to make a

square cross-section will happen at a later stage, as will clipping

Translation is easy to find: we want to translate the origin to the point Position; therefore, the translation matrix is

Rotation is harder: how do we generate a rotation matrix from the viewing specifications that will turn x, y, z, into u, v, n? a digression on rotation will help answer this

1000

100

010

001

)(z

y

x

Pos

Pos

Pos

PositionT

Building Viewing Transformationfrom View Specification





3 x 3 rotation matrices We learned about 3 x 3 matrices that “rotate” the world (we’re leaving out the

homogeneous coordinate for simplicity) When they do, the three unit vectors that used to point along the x, y, and z

axes are moved to new positions Because it is a rigid-body rotation

the new vectors are still unit vectors the new vectors are still perpendicular to each other the new vectors still satisfy the “right hand rule”

Any matrix transformation that has these three properties is a rotation about some axis by some amount!

Let’s call the three x-axis, y-axis, and z-axis-aligned unit vectors e1, e2, e3

Writing out:

0

0

1

1e

0

1

0

2e

1

0

0

3e



Rotation [1]



Let’s call our rotation matrix M and suppose that it has columns v1, v2, and v3:

When we multiply M by e1, what do we get?

Similarly for e2 and e3:

321 vvvM

23212

0

1

0

vvvvMe

33213

1

0

0

vvvvMe

13211

0

0

1

vvvvMe



Rotation [2]



Thus, for any matrix M, we know that Me1 is the first column of M

If M is a rotation matrix, we know that Me1 (i.e., where e1 got rotated to) must be a unit-length vector (because rotations preserve length)

Since Me1 = v1, the first column of any rotation matrix M must be a unit vector

Also, the vectors e1 and e2 are perpendicular…

So if M is a rotation matrix, the vectors Me1 and Me2 are perpendicular… (if you start with perpendicular vectors and rotate them, they’re still perpendicular)

But these are the first and second columns of M … Ditto for the other two pairs

As we noted in the slide on rotation matrices, for a rotation matrix with columns vi

columns must be unit vectors: ||vi|| = 1

columns are perpendicular: vi • vj = 0 (i j)Adapted from slides for Brown University CS 123, Computer Graphics


Rotation [3]



Therefore (for rotation matrices)

We can write this matrix of vi•vj dot products as

where MT is a matrix whose rows are v1, v2, and v3

Also, for matrices in general, M-1M = I, (actually, M-1 exists only for “well-behaved” matrices)

Therefore, for rotation matrices only we have just shown that M-1 is simply MT

MT is trivial to compute, M-1 takes considerable work: big win!

100

010

001

332313

322212

312111

vvvvvv

vvvvvv

vvvvvv

IMM T



Rotation [4]



Summary If M is a rotation matrix, then its columns are pairwise perpendicular

and have unit length Inversely, if the columns of a matrix are pairwise perpendicular and

have unit length and satisfy the right-hand rule, then the matrix is a rotation

For such a matrix,

100

010

001

MM T



Rotation [5]



Summary

History of Projections

Normalizing and Viewing Transformations


Kinds of Projections Parallel




Other


Viewing in OpenGL



Terminology

World Coordinates

Screen Coordinates

Normalizing Transformation

Viewing Transformation

Duality


Projections Parallel




Other




Next: OpenGL Tutorial 2

Four More Short OpenGL Tutorials from SIGGRAPH 2000

Vicki Shreiner: Animation and Depth Buffering Double buffering

Illumination: light positioning, light models, attenuation

Material properties

Animation basics in OpenGL

Vicki Schreiner: Imaging and Raster Primitives

Ed Angel: Texture Mapping

Dave Shreiner: Advanced Topics Display lists and vertex arrays

Accumulation buffer

Fog

Stencil buffering

Fragment programs (to be concluded in Tutorial 3)

Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736:...

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Transcript of Computing & Information Sciences Kansas State University CG Basics 4 of 8: ViewingCIS 636/736:...