3d transformation - vis center projection • isometric: •angles between the projections of all...

3D Transformation

2

Camera Analogy

• 3D is just like taking a photograph

camera

tripod model

viewingvolume

3

Camera Analogy and

Transformations• Projection transformations

• adjust the lens of the camera

• Viewing transformations• tripod–define position and orientation of the

viewing volume in the world

• Modeling transformations• moving the model

• Viewport transformations• enlarge or reduce the physical photograph

4

Coordinate Systems and

Transformations

• Steps in Forming an Image• specify geometry (world coordinates)

• specify camera (camera coordinates)

• project (window coordinates)

• map to viewport (screen coordinates)

• Each step uses transformations

• Every transformation is equivalent to a change in coordinate systems (frames)

Tuesday, October 20, 2009 © Minglun Gong 6

3D Coordinates• Left-handed • Right-handed

X

Y

O

Z

X

Z

O

Y


Homogeneous Coordinate

• Represent 3D point in 4D space.• (x,y,z,w)

• Normalize: (x/w,y/w,z/w,1)

• Represent 3D transformation using 4×4 matrices.

glTranslate(x,y,z)


Translation

1000

100

010

001

,,z

y

x

zyxt

t

t

tttT

X

Y

O

Z

w

wtz

wty

wtx

w

z

y

x

z

y

x


Scaling

1000

000

000

000

,,z

y

x

zyxs

s

s

sssS

X

Y

O

Z

w

zs

ys

xs

w

z

y

x

z

y

x

glScalef(sx, sy, sz)


Rotation about X Axis

1000

0cossin0

0sincos0

0001

xR

X

Y

O

ZglRotate(1, 0, 0, Θ)


Rotation about Y Axis

1000

0cos0sin

0010

0sin0cos

yR

X

Y

O



Rotation about Z Axis

1000

0100

00cossin

00sincos

zR

X

Y

O



3D Rotation is Not

Commutative

1000

0coscoscossinsin

0sincos0

0sincossinsincos

xy RR

1000

0coscossinsincos

0cossincossinsin

0sin0cos

yx RR


Affine Transformations• The product of any

translation, rotation, & scale matrices.

• The 3×3 upper-left submatrix R gives the aggregated rotation and scaling.

• The vector T gives the subsequent aggregate translation.

1000

333231

232221

131211

z

y

x

trrr

trrr

trrr

M

TRPMPP


Composition of 3D

Transformations

X

Y

O

Z

P

Q

• How to rotate θ° about line PQ.• Translate P to the

origin.• Rotate about the X &

Y axis so that PQ align with Z axis.

• Rotate θ° about the Z axis.

• Rotate about the X & Y axis so that PQ return to original direction.

• Translate the origin back to point P.

3D Projection


Outline

• Classification of different types of projections

• Parallel projection

• Perspective projection


Projection• Geometric projection:

• Projectors are straight lines.

• Planar projection:

• Projection surface is a plane.

Projection

Geometric Non-geometric

Planar

Non-planar


Planar Geometric Projection• Parallel:

• Different projectors parallel to each other.

• Preserve parallelism.

• Perspective:

• Different projectors converge to the center of projection.

• Does not preserve parallelism.

Planar Geometric Projection

Parallel Perspective

Orthographic

Oblique

One-point

Two-point

Three-point


Parallel Projection• Orthographic:

• Projectors are perpendicular to the projection plane.

• Oblique:

• Projectors are NOT perpendicular to the projection plane. Projection

surface

Projectors


Orthographic Projection• Elevation:

• Projectors are parallel to a principle axis.

• Axonometric:

• Projectors are NOT parallel to any principle axis.

Orthographic

Elevation Axonometric

Top

Front

Side

Isometric

Dimetric

Trimetric


Elevations• Used for:

• Engineering drawings.• Architecture drawings.

• Advantage:• Preserve distance and

angle.• All views are at same

scale.• Disadvantage:

• Hard to understand the 3D shape even all three elevations are given.

X

Y

O

Z

Top

Front


Matrix Representations

1000

0000

0010

0001

frontP

1000

0000

0010

0100

sideP

1000

0000

0100

0001

topP


Axonometric Projection• Isometric:

• Angles between the projections of all three principal axes are equal (120º).

• Dimetric:

• Angles between the projections of two of the principal axes are equal.

• Trimetric:

• All three angles are different.

Isometric

Dimetric


Isometric Projection• Used for:

• Patent office records;

• Furniture design;

• Video games (SimCity).

• Advantage:

• Illustrates 3D nature of object.

• Preserve distance along principal axes.

• Disadvantage:

• Do not preserve angle.

• Lack of foreshortening.


Perspective Projection

Projectionsurface

Projectors

• Foreshortening effect:

• Objects further from the center of projection appear smaller.

• Vanishing point:

• Projections of parallel lines that are not parallel to the projection plane converge to a point.

Center ofProjection


One-point & Two-point

Projections• One-point projection:

• The projection plane is parallel to two of the principle axes.

• Two-point projection:

• The projection plane is parallel to only one of the principle axes.


Renaissance Art• Brunelleschi invented

method of determining perspective projections in early 1400‟s.

• Dutch artist, Johannes Vermeer, may used camera obscura to create perspective effect.

• „Lady Standing at the Virginals‟


Matrix Representation I• Configuration:

• The center of projection is placed at (0,0)

• The function for projection plane is z=f:

• For 3D point (x,y,z,1):

• w = z / f;

• u = x / w = x * f / z

• v = y / w = y * f / z

0100

0100

0010

0001

f

P

f

Projectionplane

Z

X

u(x,z)

Center ofProjection


Matrix Representation II• Configuration:

• The center of projection is placed at (0,-f);

• The function for projection plane is z=0:

• For 3D point (x,y,z,1):

• w = z / f + 1;

• u = x / w

• = x * f / (z + f)

• v = y / w

• = y * f / (z + f)

1100

0000

0010

0001

f

P

f

Projectionplane

Z

X

u(x,z)

Center ofProjection

3D Viewing


Outline

• View specification.• Steps involved in 3D viewing.

• Coordinate systems.

• Camera parameters.

• View volume.• Clipping planes.


View Specification

• A popular way is to specify a virtual camera in the 3D scene.

• How to specify a virtual camera?• External parameters:

• Specify the position and orientation of the camera.

• Internal parameters:

• Specify how the scene is projected onto an image.


Steps Involved

View Volume Clipping

3D Trans-formation

World coordinates

3D Projection

Camera coordinates

Screen coordinates

External parameters

Internal parameters

2D Trans-formation

Raster coordinates


Camera Coordinate System• Also called:

• View-reference coordinate system;

• Eye coordinate system.

• Settings:

• Left hand system.

• The center of projection is at origin;

• The camera looks toward +Z;

• X is to the right and Y is up.

Z´

Y´

X

Y

Z

X´


Screen Coordinate System• Also called:

• Normalized-projection coordinate system;

• Settings:

• All visible objects are projected into a normalized window;

• All the coordinate values are within the interval: [-1,1].

X

Y

1-1

-1

1


Raster Coordinate System• Also called:

• Device coordinate system.

• Settings:

• The range of the coordinate values depends on the resolution.

X

Y

width

heigh

t


External Parameters• Position:

• Center of projection: C

• Orientation:

• View direction: V

• Up direction: U

• The camera coordinate system:

• Z´ = V

• X´ = V × U

• Y´ = X´ × V

V(Z´)

C

Y´

X

Y

Z

X´

U


3D Transformation Defined

• Transformation from world coordinates to camera coordinates is the composition of the following transformations:• Translate by –C, bring the center of projection

to origin;

• Rotate about X & Y axes so that V align with Z axis;

• Rotate about Z axis so that U lies on Y-Z plane.

• Transformation from camera coordinates to world coordinates is the reverse matrix.


Internal Parameters

w

• Focal length:

• Distance between the center of projection and the projection plane: f.

• 50mm for “standard” lens.

• Image (film) size:

• Width & height: 2w×2h.

• 36×24mm for 35mm film.

• Field of view:

• Angle: θh, θv.

• tan(θh/2) = w/f.


3D Projection Defined

11

f

wzyf

wzxf

fz

z

wy

wx

z

y

x

0100

0100

0010

0001

f

w

w

P

1

1

1

f

wh

f

h

w


Clipping

• Points on the z=0 plane cannot be projected.• The weight is zero after projection.

• Mapped to infinity.

• Have to eliminate the potion of scene that is behind the viewpoint before projection.

• To cut computational cost, we can also eliminate all objects that are outside the viewing volume.


View Volume• Defines:

• The portion of the world that projects onto the projection plane.


• Truncated pyramid.

• Parallel projection

• Parallelepiped.

X

Y

Z


Clipping Planes

X

Y

Z

Front clipping plane

Back clipping plane

• Front clipping plane:

• Also called near clipping plane or hither plane.

• Remove objects that are behind or too close to the center of projection.

• Back clipping plane:

• Also called far clipping plane or yon plane.

• Sometime at infinite.

• Remove objects that are too far away and the projection are too small.


Clipping Planes (Cont’d)• Surrounding planes:

• Defined by the four edges of the image.

• It is difficult to do clipping in the camera coordinates.

• Clipping can be done after projection.

X

Y

Z

52

Transformations in OpenGL

• Modeling

• Viewing• orient camera

• projection

• Animation

• Map to screen

55

3D Transformations

• A vertex is transformed by 4 x 4 matrices• all affine operations are matrix

multiplications• all matrices are stored column-major in

OpenGL• matrices are always post-multiplied• product of matrix and vector is

v

M

151173

141062

13951

12840

mmmm

mmmm

mmmm

mmmm

M

56

Specifying Transformations• Programmer has two styles of specifying

transformations• specify matrices (glLoadMatrix, glMultMatrix)

• specify operation (glRotate, glOrtho)

• Programmer does not have to remember the exact matrices• check appendix of Red Book (Programming

Guide)

57

Programming

Transformations

• Prior to rendering, view, locate, and orient:• eye/camera position

• 3D geometry

• Manage the matrices• including matrix stack

• Combine (composite) transformations

58

vertex

Modelview

Matrix

Projection

Matrix

Perspective

Division

Viewport

Transform

Modelview

Modelview

Projection

l

l

l

object eye clip normalized

devicewindow

• other calculations here• material color• shade model (flat)• polygon rendering mode• polygon culling• clipping

Transformation

Pipeline CPU DL

Poly.Per

Vertex

Raster Frag FB

Pixel

Texture

59

Matrix Operations

• Specify Current Matrix Stack

• Other Matrix or Stack Operations

• Viewport• usually same as window size• viewport aspect ratio should be same as

projection transformation or resulting image may be distorted

60

Projection Transformation

• Shape of viewing frustum


• Orthographic parallel projection

• calls with z values near zero

61

Applying Projection

Transformations

• Typical use (orthographic projection)glMatrixMode( GL_PROJECTION );

glLoadIdentity();

glOrtho( left, right, bottom, top, zNear, zFar );

62

Viewing Transformations

• Position the camera/eye in the scene• place the tripod down; aim camera

• To “fly through” a scene• change viewing transformation and

redraw scene

•

• up vector determines unique orientation• careful of degenerate positions

tripod

63

Projection Tutorial

64

Modeling Transformations

• Move object

• Rotate object around arbitrary axis

• angle is in degrees

• Dilate (stretch or shrink) or mirror object

zyx

65

Transformation Tutorial

66

Connection: Viewing and

Modeling

• Moving camera is equivalent to moving every object in the world towards a stationary camera

• Viewing transformations are equivalent to several modeling transformationsgluLookAt() has its own command

can make your own polar view or pilot view

67

Projection is left handed

• Projection transformations (gluPerspective, glOrtho) are left handed• think of zNear and zFar as distance from view

point

• Everything else is right handed, including the vertexes to be rendered

xx

yy

z+

z+

left handed right handed

68

Common Transformation

Usage

• Examples of resize() routine

• restate projection & viewing transformations

• Usually called when window resized

• Registered as callback

69

resize(): Perspective &

LookAtvoid resize( int w, int h ){

glViewport( 0, 0, (GLsizei) w, (GLsizei) h );glMatrixMode( GL_PROJECTION );glLoadIdentity();

gluPerspective( 65.0, (GLfloat) w / h,

1.0, 100.0 );glMatrixMode( GL_MODELVIEW );glLoadIdentity();

gluLookAt( 0.0, 0.0, 5.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0 );

}

71

resize(): Ortho (part 1)

void resize( int width, int height )

{

GLdouble aspect = (GLdouble) width / height;

GLdouble left = -2.5, right = 2.5;

GLdouble bottom = -2.5, top = 2.5;

glViewport( 0, 0, (GLsizei) w, (GLsizei) h );

glMatrixMode( GL_PROJECTION );

glLoadIdentity();

… continued …

72

if ( aspect < 1.0 ) {

left /= aspect;

right /= aspect;

} else {

bottom *= aspect;

top *= aspect;

}

glOrtho( left, right, bottom, top, near, far );

glMatrixMode( GL_MODELVIEW );

glLoadIdentity();

}

resize(): Ortho (part 2)

3d transformation - vis center projection • isometric: •angles between the projections of all...

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