3D Graphics : Computer Graphics Fundamentals

3D Transformations
 Geometric transformations and object modeling in 3D are
extended from 2D methods by including considerations for the
z coordinate.
Translation
 A point is translated from position P=(x,y, z) to position
P’=(x’,y’, z’) with matrix operation
1 0 0
0 1 0
0 0 1
1 0 0 0 1 1
x
y
z
x t x
y t y
z t z
     
     
      
     
     
     
x
y
z
x x t
y y t
z z t
  
  
  
 , ,x y z
 , ,x y z  
x
y
z

Rotation
 Positive rotation angles produce anticlockwise
rotations about a coordinate axis

X- ais Rotation
xx
coszsinz
sinzcos
'
'
'





y
yy

Scaling
 The matrix expression for the scaling transformation
of a position P = (x, y, z) relative to coordinate origin
can be written as






































11000
000
000
000
1
'
'
'
z
y
x
s
s
s
z
y
x
z
y
x
x
y
z
x
y
z
z
'
'
'
.zz
.
.
s
syy
sxx
y
x




 The matrix representation for an arbitrary fixed-point
(xf, yf, zf) can be expressed as:
















1000
)1(00
)1(00
)1(00
),,(),,(),,(
fzz
fyy
fxx
fffzyxfff
zss
yss
xss
zyxTsssSzyxT

Reflections
 The matrix expression for the reflection
transformation of a position P = (x, y, z) relative to xy
plane can be written as:
 similarly, as reflections relative to yz plane and xz
plane, respectively.

Shear
 The matrix expression for the shearing transformation
of a position P = (x, y, z)
Transformation in z axis













1000
0100
010
001
y
x
z
sh
sh
SH

Transformation in y axis
Transformation in x axis













1000
010
0010
001
z
x
y
sh
sh
SH













1000
010
001
0001
z
y
x
sh
sh
SH

3D Display Methods
 3D graphics deals with generating and displaying
three dimensional objects in a two-dimensional
space(eg: display screen).
 In addition to color and brightness, a 3-D pixels adds
a depth property that indicates where the point lies on
the imaginary z-axis.
 To generate realistic picture we have to first setup a
coordinate reference for camera. This co-ordinate
reference defines the position and orientation for the
plane of the camera.

 This plane used to
display a view of the
object
 Object description
has to transfer to the
camera reference co-
ordinates and
projected onto the
selected display
plane.

Parallel Projection
Project points on the object surface along parallel
lines onto the display plane.
Parallel lines are still parallel after projection.
Used in engineering and architectural drawings.
Views maintain relative proportions of the object.
Top View Side View
Front View

Perspective Projection
• Project points to the display plane along converging
paths.
• This is the way that our eyes and a camera lens form
images and so the displays are more realistic.
• Parallel lines appear to converge to a distant point in
the background.
• Distant objects appear smaller than objects closer to
the viewing position.

Depth Cueing
To easily identify the front and back of display objects.
Depth information can be included using various
methods.
A simple method to vary the intensity of objects
according to their distance from the viewing position.
Eg: lines closest to the viewing position are displayed
with the higher intensities and lines farther away are
displayed with lower intensities.

 Application :modeling the effect of the atmosphere
on the pixel intensity of objects. More distant objects
appear dimmer to us than nearer objects due to light
scattering by dust particles, smoke etc.

Visible line and surface identification
• Highlight the visible lines or display them in different
color
• Display nonvisible lines as dashed lines
• Remove the nonvisible lines

Surface rendering
• Set the surface intensity of objects according to
Lighting conditions in the scene
Assigned surface characteristics
 Lighting specifications include the intensity and
positions of light sources and the general background
illumination required for a scene.
 Surface properties include
degree of transparency
and how rough or smooth
of the surfaces

Exploded and Cutaway Views
 To maintain a hierarchical structures to include
internal details.
 These views show the internal structure and
relationships of the object parts

Stereoscopic Views
 To display objects using stereoscopic views
Stereoscopic devices present 2 views of scene
 One for left eye.
 Other for right eye.
 These two views displayed on alternate refresh cycle
of a raster monitor
 Then viewed through glasses that alternately darken
first one lens then the other in synchronized with the
monitor refresh cycle.

3D Object Representation
 Graphics scenes contain many different kinds of objects
and material surfaces
 Trees, flowers, clouds, rocks, water, bricks, wood paneling,
rubber, paper, steel, glass, plastic and cloth
 Polygon and Quadric surfaces: For simple Euclidean
objects eg: polyhedron and ellipsoid
 Spline surfaces and construction: For curved surfaces
eg: aircraft wings , gears
 Procedural methods – Fractals: For natural objects eg:
cloud, grass
 Octree Encoding: For internal features of objects eg:CT
image

Representation schemes categories into 2
Boundary representation(B –reps)
 A set of surfaces that separate the object interior from the
environment
 Eg) Polygon facets, spline patches
Space-partitioning representation
 Used to describe interior properties.
 Partitioning the spatial region into a set of small, non
overlapping, contiguous solids (usually cubes)
 Eg) octree representation

Polygon Surfaces
 Most commonly used boundary representation.
Polygon table
 Specify a polygon surfaces using vertex coordinates and attribute parameter.
Polygon data table organized into 2 group.
1. Geometric data table: vertex coordinate and parameter to identify the spatial
orientation.
3 lists
Vertex table –coordinate values of each vertex.
Edge table - pointer back to vertex table to identify the vertices for polygon edge.
Polygon table- pointer back to edge table to identify the edges for each polygon

2. Attribute table: Degree of transparency and surface
reflectivity etc.
Some consistency checks of the geometric data table:
 Every vertex is listed as an endpoint for at least 2
edges.
 Every edge is part of at least one polygon.
 Every polygon is closed.
 Each polygon has at least one shared edge.

Plane Equation
 The equation for a plane surface expressed at the
form
Ax+By+Cz+D=0
 We can obtain the values of A,B,C,D by solving 3
plane equations using the coordinate values for 3
noncollinear points in the plane(x1,y1, z1), (x2,y2, z2),
(x3,y3, z3).
3,2,1,1z)/()/()/( k  kDCyDBxDA kk

If we substitute any arbitrary point (x,y, z) into this
equation, then,
Ax + By + Cz + D ≠0 implies that the point (x,y,z) not on a
surface
Ax + By + Cz + D < 0 implies that the point (x,y,z) is inside the
surface.
Ax + By + Cz + D >0 implies that the point (x,y,z) is outside the
surface.

Polygon Meshes
 Object surfaces are tiled to specify the surface facets
with mesh function.
Triangle strip - Produce n-2 connected triangles ,for n
vertices
Quadrilateral mesh - Produce (n-1)×(m-1) quadrilateral for
n×m array of vertices.

Quadric Surfaces
 Described with second degree equations
Quadric surfaces include:
 Spheres
 Ellipsoids
 Torus

Sphere
 A spherical surface with radius r
centred on the origin is defined as the
set of points (x, y, z) that satisfy the
equation
 This can also be done in parametric
form using latitude and longitude
angles
2222
rzyx 



sin
sincos
coscos
rz
ry
rx


 22
 
 
y axis
z axis
x axis
P ( x, y, z )
θ
φ
r

Ellipsoid
 An extension of a spherical surface
Where the radii in three mutually perpendicular
directions ,have different values.
 parametric form using latitude and longitude angles
1
222





















zyx r
z
r
y
r
x



sin
sincos
coscos
z
y
x
rz
ry
rx



22
 
 

Torus
 Doughnut –shaped object.
 parametric form using latitude and longitude angles
1
2
2
22
































zyx r
z
r
y
r
x
r



sin
sin)cos(
cos)cos(
z
y
x
rz
rry
rrx



 
 

SuperQuadrics
 A generalization of quadric surfaces, formed by
including additional parameters into quadric
equations
 Increased flexibility for adjusting object shapes.
Superellipse
1
/2/2















S
y
S
x r
y
r
x

 When s=1 ,get an ordinary ellipse
 Parametric representation.


s
y
s
x
ry
rx
sin
cos


 

Superellipsoid
 For s1=s2=1 ,get an ordinary ellipsoid
 Parametric representation.
1
1
12
21 /2
//2/2





























S
z
SSS
y
S
x r
z
r
y
r
x



1
21
21
sin
sincos
coscos
s
z
ss
y
ss
x
rz
ry
rx


 22
 
 

Spline Representation
 Spline is a flexible strip used to produce a smooth curve through a designed set of
points.
 Spline mathematically describe with a piecewise cubic polynomial function whose
first and second derivative are continuous across the various curve section.

 A spline curve is specified using a set of coordinate
position called control points , which indicates the
general shape of the curve.
 There are two ways to fit a curve to
these points:
 Interpolation - the curve passes
through all of the control points.
 Approximation - the
curve does not pass
through all of the control
points, that are fitted to the
general control-point path

 The spline curve is defined, modified and manipulated
with operation on the control points.
 The boundary formed by the set of control points for a
spline is known as a convex hull

 A polyline connecting the control points is known as
a control graph.
 Usually displayed to help designers keep track of
their splines.

Parametric Continuity Condition
 For the smooth transition from one section of a
piecewise parametric curve to the next, impose
continuity condition at the connection points.
 Each section of a spline is described with parametric
coordinate functions
)(zz
)(
)(
21
u
uuuuyy
uxx




Zero –order Parametric continuity (C0 )
 Simply means that the curves meet. That is x,y and z evaluated
at u2 for the first curve section are equal to the values of x,y
and z evaluated at u1 for the next curve section.
First –order Parametric continuity (C1 )
 The first parametric derivatives(tangent lines) of the
coordinate functions for two successive curve sections are
equal at their joining point.
Second –order Parametric continuity (C2 )
 Both first and second parametric derivatives of the two curve
sections are the same at the intersection.

Geometric Continuity Condition
 In Geometric Continuity ,only require parametric derivatives
of the two sections to be proportional to each other at their
common boundary
Zero –order Geometric continuity (G0 )
 Same as Zero –order parametric continuity. That is the two
curves sections must have the same coordinate position at the
boundary point.
First –order Geometric continuity (G1 )
 The first parametric derivatives(tangent lines) of the
coordinate functions for two successive curve sections are
proportional at their joining point.
Second –order Geometric continuity (G2 )
 Both first and second parametric derivatives of the two curve
sections are proportional at their boundary.

Spline Specification
 Three methods for specifying a spline representation.
1. We can state the set of boundary conditions that are
imposed on the spline.
2. We can state the matrix that characterizes the spline.
3. We can state the set of blending functions.

 Parametric cubic polynomial representation for the x
coordinate of a spline section
 Boundary condition set on the endpoint coordinates
x(0) and x(1) and on first parametric derivatives at
the endpoints x’(0) and x’(1).
10,)( 23
 uducubuaux xxxx

 From the boundary condition, obtain the matrix that
characterizes the spline curve.
geomspline
geomspline
MMUux
MMC
CU
x
d
x
c
x
b
x
a
uuuux





















)(
123)(

Finally the polynomial representation


3
0
)()(
k
kk uBFgux

Bezier Curve and Surfaces
 This spline approximation method developed by the
French engineer Pierre Bezier for use in the design of
Renault automobile bodies.
 Easy to implement.
 Available in CAD system, graphic package, drawing
and painting packages.
Bezier Curve
 A Bezier curve can be fitted to any number of control
points.
 Given n+1 control points position
pk=(xk, yk, zk) 0≤k≤n

 The coordinate positions are blended to produce the
position vector P(u) which describes the path of the
Bezier polynomial function between p0 and pn
 The Bezier blending functions BEZk,n(u) are the
Bernstein polynomials


n
k
nkk uuBEZpuP
0
, 10),()(
knk
nk uuknCuBEZ 
 )1(),()(,

where parameters C(n,k) are the binomial coefficients
The individual curve coordinates can be given as
follows
)!(!
!
),(
knk
n
knC




n
k
nkk uBEZxux
0
, )()(


n
k
nkk uBEZzuz
0
, )()(


n
k
nkk uBEZyuy
0
, )()(

Properties Of Bezier Curves
 Bezier Curve is a polynomial of degree one less than
the number of control points

 Bezier Curves always passes through the first and last
control points.
P(0) = p0
P(1) = pn
 Bezier curves are tangent to their first and last edges of
control garph.
 The curve lies within the convex hull as the Bezier
blending functions are all positive and sum to 1
1)(
0
, 
n
k
nk uBEZ

Design Techniques
 Closed Bezier curves are
generated by specifying the
first and last control points at
same position.
 Specifying multiple control
points at a single coordinate
position gives more weight
to that position.

Cubic Bezier Curve
 Cubic Bezier curves are generated with 4 control
points.
 Cubic Bezier curves gives reasonable design flexibility
while avoiding the increased calculations needed with
higher order polynomials.
The blending functions when n = 3
3
3,3
2
3,2
2
3,1
3
3,0
)1(3
)1(3
)1(
uBEZ
uuBEZ
uuBEZ
uBEZ





 At u=0, BEZ0,3=1, and at u=1, BEZ3,3=1. thus, the
curve will always pass through control points P0 and
P3.
 The functions BEZ1,3 and BEZ2,3, influence the shape
of the curve at intermediate values of parameter u.
 The resulting curve tends toward points P1 and P3.

Bezier Surface
 Two sets of orthogonal Bezier curves are used to
design surface.
Pj,k specify the location of the control points.
 

n
k
nkmjkj
m
j
uBEZvBEZpvuP
0
,,,
0
)()(),(

B-Spline Curves and
Surfaces
1. The degree of a B-spline polynomial can be set independently of
the number of control points.
2. B-splines allow local control over the shape of a spline curve
(or surface)

 The point on the curve that corresponds to a knot is
referred to as a knot vector.
 The knot vector divide a B-spline curve into curve
subinterval, each of which is defined on a knot span.

 Given n + 1 control points P0, P1, ..., Pn
 Knot vector U = { u0, u1, ..., un+d }
 The B-spline curve defined by these control points
and knot vector
Pk is kth control point
Blending function Bk,d of degree d-1



n
k
nduuu
dk
B
k
puP
0
12,
maxmin
,)(
,
)(

 Blending functions defined with Cox-deBoox recursive form
)()()(
,0
,1
)(
1,1
1
1,
1
,
1
1,
uB
uu
uu
uB
u
uu
uB
otherwise
uuif
uB
dk
kdk
dk
dk
dk
k
dk
kk
k













 


To change the shape of a B-spline curve,
modify one or more of these control
parameters:
1. The positions of control points
2. The positions of knots
3. The degree of the curve

Uniform B-Spline
 The spacing between knot values is constant.
Non-uniform B-spline
 Unequal spacing between the knot values.
Open uniform B-Spline
 This B-Spline is across between Uniform B-Spline and
non-uniform B-Spline.
 The knot spacing is uniform expect at the ends where
knot values are repeated d times

B-Spline Surfaces
Similar to Bezier surface
 

2
2
221121
1
1 0
,,,
0
)()(),(
n
k
dkdkkk
n
k
vBuBpvuP

Sweep Representations
 Sweep representations are useful for constructing 3
dimensional objects that possess translational,
rotational or other symmetries.
 Objects are specified as a 2 dimensional shape and a
sweep that moves that shape through a region of
space

Octrees
 Octrees are hierarchical tree structures used to
represent solid objects.
 Octrees are particularly useful in applications that
require cross sectional views – for example medical
applications.

Octrees & Quadtrees
 Octrees are based on a two-dimensional
representation scheme called quadtree encoding.
 Quadtree encoding divides a square region of space
into four equal areas until homogeneous regions are
found.
 These regions can then be arranged in a tree

 An octree takes the same approach as quadtrees, but
divides a cube region of 3D space into octants.
 Each region within an octree is referred to as a
volume element or voxel.
 Division is continued until homogeneous regions are
discovered

3D Graphics : Computer Graphics Fundamentals

3D Graphics : Computer Graphics Fundamentals

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