10 3d modeling bw

8/3/2019 10 3D Modeling BW

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Instructor: Oscar Au


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3-D Objects

Objects are virtual entities in a continuous environment

Include points, lines, shapes in 2-D and volume in 3-D 3-D modeling involves representation, position,manipulation, lighting, and rendering

Co-ordinate systems

The object is located in theWorld Coordinate System

(WCS) which is normally a right-handed Cartesian system

Camera defines its Camera

Coordinate System (CCS)

x

y

z

x

y

z

left-handedsystem

right-handed

system

Cameraview-point


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Operation on OCS

Each object has an anchor point

(origin) and all its vertices are definedin the object coordinate system

Matrix operation can be performed on

the OCS

x

y

z

(x2,y2,z2)(x1,y1,z1)

Rotation by roll about the z-axis

=

z

yx

rollrollrollroll

z

yx

100

0)cos()sin(0)sin()cos(

'

''

rollx

y

z

(x2,y2,z2)(x1,y1,z1)


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x

y

z

(x1,y1,z1)

=

z

yx

yawyaw

yawyaw

z

yx

)cos(0)sin(

010)sin(0)cos(

'

''

yaw

(x2,y

2,z

2)

=

z

yx

pitchpitch

pitchpitch

z

yx

)cos()sin(0

)sin()cos(0001

'

''

pitch

x

y

z

(x1,y1,z1)

(x2,y2,z2)

Rotation by yaw about the y-axis

Rotation by pitch about the x-axis


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+

=

z

y

x

t

t

t

z

y

x

z

y

x

'

'

'

x

y

z

(x1,y1,z1)

(x2

,y2

,z2

)

(tX,tY,tZ)

Note: scaling should not be performed on rigid body

x

y

z

=

z

y

x

S

S

S

z

y

x

Z

Y

X

00

00

00

'

'

'(x

1

,y1

,z1

)(x2,y2,z2)

Scaling by (Sx, Sy, Sz)

Note: translation cannot be done by matrix multiplication

Translation by (tx, ty, tz)


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Rotation, scaling and translation all by matrix multiplication

=

110001

'

'

'

z

y

x

TAAA

TAAA

TAAA

z

y

x

ZZZZYZX

YYZYYYX

XXZXYXX

General matrix operation

Deformation

Cannot be done by linear matrix operation General principle: ( )zyxFx X ,,'=

( )zyxFy Y ,,'=

( )zyxFz Z ,,'=


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Tapering ( )zxFx 1'=

( )zyFy 2'=

zz='

Axis Twisting

Bending

( )[ ] ( )[ ]

( )[ ] ( )[ ]zz

zFyzFxy

zFyzFxx

=+=

=

'cossin'

sincos'

( )( ) 0

0

/1cos'

/1sin'

'

ykzz

ykzy

xx

+=+=

=

1/k: curvature of the bending

y0: center of bending = k(y-y0)


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Object representation in WCS

The place where objects are assembled

Objects usually represented by a pointer to their origins

A WCS can become another OCS in a new WCS

xWCS

yWCS

zWCS

Rotation and scaling in WCS (1) is obtained by simple rotation

To obtain (2), either convert all pointsin the OCS to WCS, or calculate

(2)

(1)

the necessary rotation in OCS

Scaling can also be done in WCS or OCS (preferred)

Hierarchy and object-oriented approach preferred


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Simple approach

Assume the camera is located at the origin of the WCS

facing the z-direction Assume also a projection screen is located at adistance dbehind the camera on the x-y plane

x

y

z

d

x

y

(x1,y1,z1)

(x1,y1)

cameraprojection

screen

coordinateof projectedimage

Note thatphysically, theprojected image is

in the left-handedcoordinate system


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Calculating new coordinates on the projection

Using similar triangles

x1 = x1d/z

x

y

z

(x1,y1,z1)

d

y1

z

(x1,y1,z1)

x1

d

z1

z1

y1

x1

y1 = y1d/z


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Perspective information

The distance dof the screen serves as a scaling factor

The new coordinates are divided by the zvalue of the oldsystem. Thus the further away the object, the smaller it ison the projection screen.

The new coordinate system needs another variable z = zfor each point to determine the front and backrelationship.

If 2 points at different distance results in the same (x,y),only the one with smaller zis displayed.


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Field of viewing

The distance dalso defines the viewing angle

y

d

ltan() = l/d

z

Viewableregion

Camera position

Camera can be considered as an object and be placedat any location in the WCS by matrix transformation

Question: for a point (x, y, z) in the WCS, what is its

new position in CCS after camera motion?


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Converting points in WCS to CCS

Assume camera is at the origin of WCS at beginning

Camera is then moved to a new location by a series ofmatrix transformation in the form of

( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )

1000,,,,,,

,,,,,,

,,,,,,

ZZZZYZX

YYZYYYX

XXZXYXX

TAAA

TAAA

TAAA

This can be viewed as WCS moved w.r.t. CCS. We can

perform the following transforms in same sequence( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

1000

,,,,,,

,,,,,,

,,,,,,

ZZZZYZX

YYZYYYX

XXZXYXX

TAAA

TAAA

TAAA


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Zooming is equivalent to moving the screen away from

the camera lens (increasing magnification factor d).

Moving camera closer to the object also results in theobject enlarged.

But they result in different distortion

original

screen

zViewableregion

zoomedscreen

closeup

screen

camera position

Zooming and camera position


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Example:

reference image placing cameracloser to object

zoom in object

Depth of focus

In real situations, objects far fromthe focal planes should be blurred

simulate this by applying blurring

function that depends on z value


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Edge represented object

Objects formed by lines defined by vertices

Used for quick visualization and transparent view

(1,5)(1,2)(0,4)(0,3)(0,1)Line

(0,1,0)(0,0,1)(1,0,1)(1,0,0)(0,0,0)Point43210Index

x

y

z

0

11

1

03

21

4 7

65

Difficult to remove lines masked by other surfaces

Not very common for modeling


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Planar polygons

Formed by a chain of straight lines

(0,3,6,7,0)

(5,6,8,5)

(4,5,8,9,4)

(2,3,6,5,2)

(1,2,5,4,1)

Polygons

(0,1,2)(3,0,0)(4,0,2)(0,0,2)(0,0,0)Vertices

53210Index

x

z

13

5

4

9

2

68

y

Difficult in representing curved surface

Points are not free to move along different axes

1

5

2

?

Example: reducing the z-coordinate ofvertex (2) destroy the planar property onpolygon (0)

4


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Planar triangles

Try to resolve the point motion in planar polygons by

breaking it down to triangles

Easier to edit object

Still a problem: how to representdisjoint boundaries such as holes?

Breakdown to more triangles

Subtraction space may be needed

x

z1

3

5

9

2

6

8y

4

(2,3,5)(5,8,9)(4,5,9)(2,4,5)(1,2,4)triangles

(0,1,2)(3,0,0)(4,0,2)(0,0,2)(0,0,0)Vertices

53210Index

Boundary 1

Boundary 2


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Sectional polygons

Slice a 3-D object into cross-sections, and store each

cross-section in polygon form Object surface mesh is reconstructed by connectingcontours with those above and below

Require a central axis to align planes

Object

Central axis

Sectionalrepresentation

2 planes are necessary to

represent a simple object like this


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Number of planes can be varied depending on thecomplexity of an object

Polar coordinates with the central axis as origin providean easy description of planner polygons

Still difficult to represent holes and back-curving lines

X-Section in polar coordinate X-Section with holes andback curving edges

holes back-curvingedges


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Extrusion

One of the short-cut to overcome the tedious cross-

section in building up complex models A 2-D cross-section is extended to create a cylinder-like3-D object

x

y

z

x

y

z

X-Section

extrusiondirection

result

Can re-define size and rotation inextrusion

z


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Object of revolution

Begin with a cross-section of an object and then rotate

the cross-section around a central axis Can represent a complicated object with very few datapoints

The cross-section can be off-axis to form torus-likeobjects y

x

z


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Constructive Solid Geometry (CSG)

Intend to construct complicated objects from a set of

simple primitives such as cube, cone, sphere etc. Mathematical equations used to describe geometry

e.g. sphere is given by x2 + y2 + z2 = r2

To allow object interaction such as addition andsubtraction


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Mathematical construction

A set of inequalities is used to determine if a point is

inside or outside the solid boundary

points outside sphere: x2+ y2+ z2> r2

points inside sphere: x2+ y2+ z2< r2

More complex geometry can be constructedby logical operations of the inequalities

x2+ y2+ z2> r12

e.g. points of hollow sphere with inner radius r1 and outer radiusr2have to satisfy both

x2+ y2+ z2< r22and


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Spatial subdivision

Concept similar to a 3-D bitmap,

where objects are divided into smallcubes

Larger cubes are identified to savestorage space

More advanced models (will not be covered)

Spine objects (you may regard it as extrusion withcurved central axis)

Fractals

Soft objects interacting with neighboring objects


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Definition of rendering

Viewing algorithm of how objects are drawn and

visualized by a viewer Usually objects are rendered in order

Different algorithms to handle different representations

Wire frame or line rendering

Simplest for rendering for quick visualization

Only required to figure out the vertices and connectrelated pairs

Overlapping/depth information need not be considered

Sometimes difficult to understand the object


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Opaque surface rendering

Define the color of a polygon/surface and color the

entire surface with the single color Relatively simple, except that the visual order of thesurface has to be calculated

Project each object onto the cameras viewing screenand determine the paint color by comparing order withexisting points

Complex surface rendering Need to consider texture, perspective, lighting and otherfactors

Interaction between objects also needs to be considered


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Ray tracing

Traces light backward from eye to scene to light

sourceTake into account perfect specular interaction

Ray-traced examples:

http://www.cse.buffalo.edu/pub/WWW/povray/


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Ray tracing

Use analogy from thermal heat diffusion to model the

energy emitted from each surface patchHandle diffused interaction

http://freespace.virgin.net/hugo.elias/radiosity/radiosity.htm


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Difficulty in simulating realistic illumination

Illumination of an object consists of many sources from

multiple reflection of surrounding surfaces (similar toreverberation in audio)

It is very difficult to trace all the sources to calculate theright color and luminosity or the resulting surface

Need to develop models to achieve trade-off betweenrealism and efficient rendering

www.siggraph.org/education/materials/

HyperGraph/radiosity/overview_1.htm


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The simplest source with parameters of position, colorand intensity

The illumination intensity at a certain point iscalculated by inverse square law and angle of incident

(px,py,pz)

(x,y,z)

normal vectorn= [0, 1, 0]

s= [px-x, py-y, pzz]

-s n = |s||n| cos()

cos() = -s n / |s|

Recall vector dot product

( )3

z)y,(x,atonilluminatis

nsIr

rr

= whereI is the intensity at the source

Point light source


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Spot light

Similar to point source except that intensity falls off

rapidly at location outside the cone of illumination(px,py,pz)

s

u

(x,y,z) (x0,y0,z0)

( )ususrr

rr

=)cos(

A point given by uiswithin the illumination

cone if < /2

- Position: (px,py,pz)- Direction: s

- Cone angle:

illumination can be calculated as afunction of and distance from source

example:( )

=

4

cosonIlluminati3

gsource

u

nuIr

rr

n

gcontrol the fading

sharpness of the sport


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Multiple light sources

Superposition principle can be used to calculate final

equalization necessary to prevent saturation

Shadows

Need a model to handle the interaction between lightsources and objects

Simple method: treat opaque objects as negative lightsources to subtract out illumination

The magnitude of the negative light source defines thetransparency of the object surface to a third object


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Ambient light reflection It refers to constant illumination naturally exists due tomultiple reflections that enable us to see objects

No light source is associated with it and independentof orientation and distance

Simply calculated by multiplying the light source colorcomponent and the reflection coefficient

R = IR-sourcex RR-object

G = IG-sourcex RG-objectB = IB-sourcex RB-object

Observed color

Note that the source color and surface color is in the

subtraction color space


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Diffused reflection

It simulates reflection by rough surfaces that incident

lights are reflected in all directions Similar to ambient light reflection, perspective info dueto distance and orientation w.r.t. light sources areincluded

Easier to be represented using the HSV model suchthat the positional data can be represented by theintensity components without affecting the color

Example: surfaces are rendered by samecolor with different brightness


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Reflection generated by polished surfaces creates a

highlight of the illuminating light source andsurrounding environment

Surface brightness is the sum of its ambient reflection

together with specular highlight

Perfectly reflected light follows the law of reflectionwith incident angle equal to the reflected angle

Depending on different material, some small amount ofdiffusion due to micro-surface roughness may occurresulting in slightly blurred surface

Specular reflection


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(px,py,pz)

s r

(x,y,z)

(x,y,z)v

( )gssourceKIR cos=

- specular reflectioncomponent is modeled by

Example: g=10 gives a rough plastic effect whileg=150 gives metallic-like surface with small highlight

Have to find in term of vectors in the Cartesiansystem calculated from relative coordinates, we have

Isource is the source intensityKS is the specular coefficientgis the gross surface factor

( )( ) ( )[ ]gssource svvnsnKIRrrrrrr

= 2


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Examples insurface rendering


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Render order

In 2-D, the bottom-most plane is colored first and then

color is applied upward towards the top In 3-D, there are different orders to color the scene

Object order:

for each primitive Pdofor each pixel q within Pdo

update frame buffer based on color and visibility of P at q

Image order:for each pixel qdo

for each primitive P covering qdocompute Ps contribution to q, outputting q when finished


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Remove all planes that face away from the camera

x

z

y

Can be tested by the dot productor the normal vector and the viewvector

The plane is facing away if anglebetween the vectors are > 90degree

forwardfacing

backfacing

Back-face removal

Assume solid objects with no 2-D plane exist

Only one side of a surface will be displayed, indicatedby the direction of its normal vector


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Bitmap surface representation

Some surface texture can only be described by bitmap

but not object

Assume the texture is represented by a bitmap with u-vcoordinate

Can be mapped to any surface in WCS by complicated3-D mathematical transforms

The texture map defines the background ambient color

and additional effects are added by superpositionprinciple


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Scaling problem

A distance from the camera which the texture map and

the object plane align must be defined When the plane is very close to the camera, the entireplane may be mapped only to 1-2 pixels in the texture

Similarly when the plane is very far, the entire texturemay will be scaled to a few pixels

Example: cylindrical mapping

Assume a texture map with U-Vcoordinate is mapped toa cylinder with radius rand height h

A point Pi(xi,yi,xi) will be mapped into T(u,v) in the texture

space


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xz

y

h

r

u

vU

V

1

1

T(u,v)

0

P(xi,yi,zi)

The portion of the arc is:

v = yi/h

= cos-1(zi/r)

Vertical coordinate:

Horizontal coordinate:

u =/2

= cos-1

(zi/r)/2 Note: reflection of nearby objectscan be considered as texturemapping from camera screen


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Further info Alan Watt, 3D Computer Graphics, 3rd Ed., Addison-Wesley 2000

Olin Lathrop, The Way Computer Graphics Works, John Wiley &

Sons, 1997 http://www.embedinc.com/book/index.htm Ray-tracing examples:

http://www.cse.buffalo.edu/pub/WWW/povray/

RadiosityReference:

http://www.siggraph.org/education/materials/HyperGraph/radiosity/radiosity.htmhttp://freespace.virgin.net/hugo.elias/radiosity/radiosity.htm

Polygonal representation and rendering:http://cis.csuohio.edu/~arndt/graphics.ppt

Bezier Curves: www.moshplant.com/direct-or/bezierwww.math.ucla.edu/~baker/java/hoefer/Bezier.htm

Parametric Patches:http://www.cs.wpi.edu/~matt/courses/cs563/talks/surface/bez_surf.html

10 3d modeling bw

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