cpsc/enel P 1

by

Brian Wyvill

University of Calgary

by

Brian Wyvill

University of Calgary

ComputerGraphicsComputerGraphics

3D transformationsand Viewing 3D Scenes

Zs=1

Zs=0

∆Zs1

∆Zs2

Zs= 1 − D/Ze

1 − D/F

Ze=FZe=D ∆Ze ∆Ze

xeye

ze

yw

zw

eye (Vfrom)Centre of Projection

xw

View to point

What we see

roll angle

Viewing or Near Plane

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

Viewing Frustrum

Yon or far Plane

Sy

d

f

(y, z)

cpsc/enel P 2

3D Coordinate Systems

x

y

zx

y

z

Right Handed System

Left Handed System

Right Handed System,

Axis (+)ve rotationx y to zy z to xz x to y

As if looking down the axisof rotation towards the origin, anti−clockwise is positive.

F&VD.1 p217F&VD.2 p180

cpsc/enel P 3

3D Transformations

Represented by 4x4 matrices. A point (x,y,z) is represented in homogeneous coordinates as (x,y,z,W) where W != 0

1 0 0 0

0 cosθ −sinθ 0

0 sinθ cosθ 0

0 0 0 1

Translation Using Column Vectors Scaling

Rotation About x−axis Rotation about y−axis Rotation about z−axis

cosθ −sinθ 0 0

sinθ cosθ 0 0

0 0 1 0

0 0 0 1

cosθ 0 sinθ 0

0 1 0 0

−sinθ 0 cosθ 0

0 0 0 1

1 0 0 Tx

0 1 0 Ty

0 0 1 Tz

0 0 0 1

Sx 0 0 0

0 Sy 0 0

0 0 Sz 0

0 0 0 1

cpsc/enel P 4

3D TransformationsContinued

For example a point 1 unit along the z−axis rotated 90o about y:

A point 1 unit along the x−axisz rotates positively to x about y * =

0 0 1 0

0 1 0 0

−1 0 0 0

0 0 0 1

0

0

1

1

1

0

0

1

x

z

y

cpsc/enel P 5

Special Orthogonal Matrices1. For the upper 2x2 matrix, the row vectors are unit

2. (cosθ −sinθ) and (sinθ cosθ) are perpendicular to eachother: (cosθ −sinθ).(sinθ cosθ) = cosθsinθ−sinθcosθ=0

3. take the determinant: cosθcosθ+sinθsinθ=1

Properties 1 and 2 are also true of the vectors formed by the columns: (cosθ sinθ) and (−sinθ cosθ) This defines a special orthogonal matrix

A transformation matrix which has such an upper 2x2 iscalled orthogonal i.e. transformations preserve angles andlengths. Matrices comprised of rotation and translationare orthogonal e.g. square when rotated stays a square

Similarly in 3D the upper left 3x3 matrix comprise mutuallyperpendicular unit vectors and the submatrix has adeterminant of unity.

cosθ −sinθ 0

sinθ cosθ 0

0 0 1

1 0 0

0 cosθ −sinθ

0 sinθ cosθ

cosθ 0 sinθ

0 1 0

−sinθ 0 cosθ

cosθ −sinθ 0

sinθ cosθ 0

0 0 1

cosθ −sinθ tx

sinθ cosθ ty

0 0 1

cpsc/enel P 6

Affine TransformationsA transformation matrix composed of scale, rotate and translate is affine, parallel lines are preserved but not lengths and angles.

square rotated and then scaled by 2 1. Lengths and angles of the square are not preserved but the sides remain parallel.

Shear transforms are also affine:

Shear in x: Shear in y:1 shx 0

0 1 0

0 0 1

1 0 0

shy 1 0

0 0 1

cpsc/enel P 7

The inverse of these transformations is:

Translation use −Tx −Ty −TzScaling use 1/Sx 1/Sy 1/SzRotation negate the angle of rotation

For an orthogonal matrix (rotation, translation) the transpose is the inverse.

To compute this simply exchange row and column indices.

Inverse of Transformations

Inverse

exchange

row and columns

for orthogonal matrix

1 0 0 0

0 cosθ −sinθ 0

0 sinθ cosθ 0

0 0 0 1

1 0 0 0

0 cosθ sinθ 0

0 −sinθ cosθ 0

0 0 0 1

cpsc/enel P 8

Composing 3D Transformations

Start position End position Translate to the origin

Rotate around y−axis Rotate around x−axis Rotate around z−axis

Rz.Rx.Ry.T

cpsc/enel P 9

Rotation About Arbitrary Axis

Start position

T to origin R y−axis R x−axis Rotate around z−axisby desired amount

Transform to z−axis Transform back to end position

TT.RTy.RT

x.Rz.Rx.Ry.T

cpsc/enel P 1 0

More Orthogonal Matrix PropertiesMultiply one of the row vectors (transposed) of the upper left 3x3 rotation matrixby the rotation matrix rotates the vector onto one of the principal axes.

Also: the middle row rotates into the y axisand the bottom row rotates into the z axis

−sinθ

0

cosθ

cosθ 0 sinθ

0 1 0−sinθ 0 cosθ

=

−sinθcosθ+cosθsinθ

0

cosθcosθ+sinθsinθ

= 0

0

1

Y rotation

0

cosθ

−sinθ

1 0 0

0 cosθ −sinθ

0 sinθ cosθ=

0

cosθcosθ+sinθsinθ

sinθcosθ−cosθsinθ

= 0

1

0

X rotation

cosθ

−sinθ

0

cosθ −sinθ 0

sinθ cosθ 0

0 0 1

=

cosθcosθ+sinθsinθ

sinθcosθ−cosθsinθ

0

= 1

0

0

Z rotation

similarly

cpsc/enel P 1 1

Using Orthogonal Matrix Properties

The example required Rz.Rx.Ry.T let R =r1x r2x r3x 0

r1y r2y r3y 0

r1z r2z r3z 00 0 0 1

The upper left 3x3 matrix handles the rotationthe bottom row rotates a vector into the z−axis.Consider the magenta vector in the example,call it V

The matrix will rotate (r1z r2z r3z)T into the z−axis but it also rotated V into the z−axis so

V is also (r1z r2z r3z)T ||V||

Similarly (r1x r2x r3x)T will be rotated into the x−axis. Given

a vector Q in the plane of the dinosaur that is also rotated into the zy plane with V then

(r1x r2x r3x)T = VxQ finally (r1y r2y r3y)T=VxVxQ

||VxQ|| ||VxVxQ||

V

Q

cpsc/enel P 1 2

Change of Coordinate System

0

1

2

3

45

6

A point system j can be transformed to system i by applying a matrix Mi j

Pi = Mi j * Pj

E.g. the point at (2,4)j is also at (3,5)i

For (2,4)j Scale by 0.5,0.5 = 1,2Translate by 2,3 = (3,5)i

For (3,5)i Translate by −2,−3 (1,2) Scale by (2,2) = (2,4)j

Similarly to transform from k to j with Mj k

Pi = Mi j * Mj k * Pk

Also Mj i = Mi j−1

System i

System j

cpsc/enel P 1 3

The Eye System

xeye

ze

yw

zw

eye (Vfrom)Centre of Projection

xw

View to point

What we see

roll angle

Viewing or Near Plane

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Viewing Frustrum

Yon or far Plane

F&VD.1 p229F&VD.2 p193

cpsc/enel P 1 4

Transformations Into the Eye SystemIn this example the Vfrom is 5,6,4 The Vto is 2,1,0 the un−normalisedview vector is Vfrom−Vto = 3,5,4 the roll angle is 200

Rotate around the y Rotate around the x Translate to Vfrom

Eye system alignedwith world

Make left handedscale (1 1 −1)

Rotate aroundthe z (roll)

cpsc/enel P 1 5

Sy

Sx

Zeye

Yeye

Xeye

The Eye Coordinate System

cpsc/enel P 1 6

Perspective

For a point in the eye system

y/z = Sy/d

x/z = Sx/d

Clipping boundariesfor viewing frustrum

y/z = Sy/d above and below

x/z = Sx/d left and right

z = d Near Planez = f Yon Plane (far plane)

For a point to be visible:

−z <= x*d/Sx <= z

−z <= y*d/Sy <= z

d<=z<=f

Sy

d

f

(z, y)

Sy

Eye System

+−+−

Zeye

cpsc/enel P 1 7

Preserving Z depthThe Perspective Transformation

The Perspective Transformation Matrix Uses the lens equation to transform from viewing frustrum to viewbox. Clip to viewbox. Preserves relative depth, stright lines and planes. Z coordinates are scaled between 0 and 1 d(1−d/f)

1

(1−d/f)−1

1

d

1/Sx 0 0 0

0 1/Sy 0 0

0 0

0 0 0

Sy

f

(y, z)

d

f

d

P1

P2

P3

Sy

f

(y, z)

d

f

d

P1 obscures P2 not P3 must perform divisonto determine if points are on the same projector.

Parallel projection merelycompare x and y.

cpsc/enel P 1 8

The Perspective Transformation

Sy

f

(y, z)

d

f

d

Why Perform the Perspective Transformation

1. It is inefficient to make depth comparisons along projectors thatare not parallel to the axes.

2. It is inefficient to clip to non−rectangular boundaries.

The transformation must preserve straight lines and planes.e.g. line or plane in eye space: Axe + Bye + Cze + D = 0 transforms toa line in the screen space. Planes will end up as flat since the screeWe want xs = (xe / ze) * (d / S) ys = (ye / ze) * (d / S)

The zs transform only has to preserve the depth relationship in Z.

The Z−relationship is of the form zs= α + β/ze

cpsc/enel P 1 9

zs is of the form α + β/ze we want the following conditions:

zs = 0 when ze = d

zs = ! when ze = F

0=α + β/d α= −β/d1=α + β/F α= 1−β/F

−β/d=1−β/F

β = −d/(1−d/F)α = 1/(1−d/F)

The Perspective Transformationcontinued

but zs = α + β/ze

substituting

zs = 1/(1−d/F) −d/(1−d/F)/ze

zs =(1−d/ze) (1−d/F)

cpsc/enel P 2 0

Viewbox and Perspective

x = xe/Sx

y = ye/Sy

z =

W = Ze/d

d(1−d/f)

Ze

(1−d/f)−1

Changing from Homogeneous coordinates to cartesian screen system

xs = =

ys =

zs =

Ze/d

(xe/Sx) xed

ZeSx

Ze/d

(ye/Sy) yed

ZeSy

(1−d/Ze)

(1−d/f)d(1−d/f)

Ze − d d/Ze =

d(1−d/f)1

(1−d/f)−1

1

d

1/Sx 0 0 0

0 1/Sy 0 0

0 0

0 0 0

x y z w = xe ye ze 1*

cpsc/enel P 2 1

Determining Z−Depth

What is the problem with linear interpolation method?Beware of large polygonsBeware of coincident polygons

Perspective transformation (viewbox tranformation)to normalize z to range [0:1]

Zs=1

Zs=0

∆Zs1

∆Zs2

Zs= 1 − D/Ze

1 − D/F

Ze=FZe=D ∆Ze ∆Ze