2/24/04© University of Wisconsin, CS559 Spring 2004 Last Time 3D Transformations –Most are...

2/24/04 © University of Wisconsin, CS559 Spring 2004

Last Time

• 3D Transformations– Most are natural extensions of 2D transforms

– Rotations can be represented in many ways: Orthonormal 3x3 sub-matrix, Quaternions, Axis-Angle, Exponential Map, Euler Angles

• Coordinate Systems– Local/Object, Global/World, Eye/View/Camera, Canonical View,

Window/Screen

• Canonical to Window mappings

• Orthogonal projection when the camera axes are aligned with the world axes


Today

• General Orthogonal viewing

• Perspective viewing


Simple Orthographic Projection

z

y

x

• The view volume is a rectilinear box for orthographic projection

• Assume that the viewer is looking in the –z direction, with x to the right and y up– Can move but not rotate the camera

– Assuming a right-handed coordinate system

• The view volume is defined by:– 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, (r>l)

– a top plane at y=t

– and a bottom plane at y=b, (b<t)(r,b,n)

(l,t,f)


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 similar to the one for canonical → window:

viewcanonicalviewcanonical

view

view

view

canonical

canonical

canonical

z

y

x

fnfnfn

btbtbt

lrlrlr

fn

bt

lr

fn

bt

lr

z

y

x

xMx

11000

200

020

002

1000

2100

2010

2001

1000

0200

0020

0002

1


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?


Specifying a View

• The location of the eye in space– A point in space for the center of projection, (ex,ey,ez)

• The direction in which we are looking: gaze direction– Specified as a vector: (gx,gy,gz)

– This vector will be normal to the image plane

• A direction that we want to appear up in the image– (upx,upy,upz), 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 eye and image plane, not the world


General Orthographic

(0,0) x

y

e

image plane g

b,n

b,f

t,n

t,f

Subtle point: it doesn’t precisely matter where we put the image plane


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 eye

• We must:– Define such a coordinate system, view space

– Transform points from the world space into view space

– Apply our simple projection from before


View Space

• Given our camera definition:– Which point is the origin of view space?

– Which direction is the normal to the view plane, w?

– 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?


View Space

• The origin is at the eye: (ex,ey,ez)

• The normal vector is the normalized viewing direction:– It is actually more general to think of it as the vector perpendicular

to the image plane

• We know which way up should be, and we know we have a right handed system, so u=up×w, normalized:

• We have two vectors in a right handed system, so to get the third: v=w×u

gw ˆ

u


World to View

• We must translate so the origin is at (ex,ey,ez)

• 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 w should be (0,0,1)

• The matrix that does that is:

1000

0

0

0

zyx

zyx

zyx

www

vvv

uuu


All Together

• We apply a translation and then a rotation, so the result is:

• And to go all the way from world to screen:

10001000

100

010

001

1000

0

0

0

ew

ew

ew

Mzyx

zyx

zyx

z

y

x

zyx

zyx

zyx

viewworld www

vvv

uuu

e

e

e

www

vvv

uuu

worldcanonicalworldcanonical

viewworldcanonicalviewcanonicalworld

xMx

MMM


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


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->canonical 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


Left vs Right Handed View Space

• You can define u as right, v as up, and n as toward the viewer: a right handed system uv=w– Advantage: Standard mathematical way of doing things

• You can also define u as right, v as up and n as into the scene: a left handed system vu=w– Advantage: Bigger n values mean points are further away

• OpenGL is right handed

• Many older systems, notably the Renderman standard developed by Pixar, are left handed


Perspective Projection

• Abstract camera model - box with a small hole in it

• Pinhole cameras work in practice - camera obscura, etc


Distant Objects Are Smaller


Parallel lines meet

common to draw film planein front of the focal point


Vanishing points

• Each set of parallel lines (=direction) meets at a different point: The vanishing point for this direction– Classic artistic perspective is 3-

point perspective

• Sets of parallel lines on the same plane lead to collinear vanishing points: the horizon for that plane

• Good way to spot faked images


Basic Perspective Projection

• Assume you have transformed to view space, with x to the right, y up, and z back toward the viewer

• Assume the origin of view space is at the center of projection (the eye)

• Define a focal distance, d, and put the image plane there (note d is negative)


Basic Perspective Projection

• If you know P(xv,yv,zv) and d, what is P(xs,ys)?– Where does a point in view space end up on the screen?

xv

yv

-zvd

P(xv,yv,zv)P(xs,ys)


Basic Case

• Similar triangles gives:

v

vs

z

x

d

x

v

vs

z

y

d

y

yv

-zv

P(xv,yv,zv)P(xs,ys)

View Plane

d


Simple Perspective Transformation

• Using homogeneous coordinates we can write:

dzz

y

x

d

y

x

v

v

v

v

s

s

vs

d

PP

0100

0100

0010

0001


Perspective View Volume

• Recall the orthographic view volume, defined by a near, far, left, right, top and bottom plane

• The perspective view volume is also defined by near, far, left, right, top and bottom planes – the clip planes– Near and far planes are parallel to the image plane: zv=n, zv=f

– Other planes all pass through the center of projection (the origin of view space)

– The left and right planes intersect the image plane in vertical lines

– The top and bottom planes intersect in horizontal lines


Clipping Planes

xv

-zv

Near Clip Plane

Far Clip PlaneView

Volume

Left ClipPlane

Right ClipPlane

fn l

r


Where is the Image Plane?

• Notice that it doesn’t really matter where the image plane is located, once you define the view volume– You can move it forward and backward along the z axis and still get

the same image, only scaled

• But we need to know where it is to define the clipping planes– Assume the left/right/top/bottom planes are defined according to

where they cut the near clip plane

• Or, define the left/right and top/bottom clip planes by the field of view


Clipping Planes

xv

-zv

Near Clip Plane

Far Clip PlaneView

Volume

Left ClipPlane

Right ClipPlane

fFOV


OpenGL

• gluPerspective(…)– Field of view in the y direction (vertical field-of-view)

– Aspect ratio (should match window aspect ratio)

– Near and far clipping planes

– Defines a symmetric view volume

• glFrustum(…)– Give the near and far clip plane, and places where the other clip

planes cross the near plane

– Defines the general case

– Used for stereo viewing, mostly


Perspective Projection Matrices

• We want a matrix that will take points in our perspective view volume and transform them into the orthographic view volume– This matrix will go in our pipeline just before the orthographic

projection matrix

(l,b,n)(r,t,n) (l,b,n)

(r,t,n)


Mapping Lines

• We want to map all the lines through the center of projection to parallel lines– Points on lines through the center of projection map to the same

point on the image– Points on parallel lines map orthographically to the same point on

the image– If we convert the perspective case to the orthographic case, we can

use all our existing methods

• The intersection points of lines with the near clip plane should not change

• The matrix that does this, not surprisingly, looks like the matrix for our simple perspective case


General Perspective

• This matrix leaves points with z=n unchanged

• It is just like the simple projection matrix, but it does some extra things to z to map the depth properly

• We can multiply a homogenous matrix by any number without changing the final point, so the two matrices above have the same effect

0100

00

000

000

0100

00

0010

0001

nffn

n

n

n

fnfnPM


Complete Perspective Projection

• After applying the perspective matrix, we still have to map the orthographic view volume to the canonical view volume:

0100

00

000

000

1000

200

02

0

002

nffn

n

n

fn

fn

fn

bt

bt

bt

lr

lr

lr

POscreenview MMM


OpenGL Perspective Projection

• For OpenGL you give the distance to the near and far clipping planes

• The total perspective projection matrix resulting from a glFrustum call is:

0100

200

02

0

002

fn

nf

fn

fnbt

bt

bt

nlr

lr

lr

n

OpenGLM


Near/Far and Depth Resolution

• It may seem sensible to specify a very near clipping plane and a very far clipping plane– Sure to contain entire scene

• But, a bad idea:– OpenGL only has a finite number of bits to store screen depth– Too large a range reduces resolution in depth - wrong thing may be

considered “in front”– See Shirley for a more complete explanation

• Always place the near plane as far from the viewer as possible, and the far plane as close as possible


Clipping

• Parts of the geometry to be rendered may lie outside the view volume– View volume maps to memory addresses– Out-of-view geometry generates invalid addresses– Geometry outside the view volume also behaves very strangely

under perspective projection• Triangles can be split into two pieces, for instance

• Clipping removes parts of the geometry that are outside the view

• Best done in screen space before perspective divide– Before dividing out the homogeneous coordinate


Clipping

• Points are trivial to clip - just check which side of the clip planes they are on

• Many algorithms for clipping lines exist– Next lecture

• Two main algorithms for clipping polygons exist– Sutherland-Hodgman (today)

– Weiler (next lecture)


Clipping Points

• A point is inside the view volume if it is on the “inside” of all the clipping planes– The normals to the clip planes are considered to point inward, toward the

visible stuff

• Now we see why clipping is done in canonical view space

• For instance, to check against the left plane:– X coordinate in 3D must be > -1

– In homogeneous screen space, same as: xscreen> -wscreen

• In general, a point, p, is “inside” a plane if:– You represent the plane as nxx+nyy+nzz+d=0, with (nx,ny,nz) pointing

inward

– And nxpx+nypy+nzpz+d>0

2/24/04© University of Wisconsin, CS559 Spring 2004 Last Time 3D Transformations –Most are...

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