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1GR2-00
GR2Advanced Computer
GraphicsAGR
GR2Advanced Computer
GraphicsAGR
Lecture 2Basic Modelling
2GR2-00
Polygonal RepresentationPolygonal Representation
Any 3D object can be represented as a set of plane, polygonal surfaces
V1
V2V3
V4
V5V8
V7 V6
Note: each vertex part of severalpolygons
3GR2-00
Polygonal RepresentationPolygonal Representation
Objects with curved surfaces can be approximated by polygons - improved approximation by more polygons
4GR2-00
Scene OrganisationScene Organisation
Scene = list of objects Object = list of surfaces Surface = list of polygons Polygon = list of vertices
scene
objectsurfaces polygons
vertices
5GR2-00
Polygon Data StructurePolygon Data Structure
V1
V2V3
V4
V5V8
V7 V6
P1
P2
Object Table
Obj1P1, P2, P3,P4, P5, P6
Object Obj1
Vertex Table
V1X1, Y1, Z1
V2X2, Y2, Z2
. ...
Polygon Table
P1 V1, V2, V3, V4
P2 V1, V5, V6, V2
. ...
6GR2-00
Typical PrimitivesTypical Primitives
Graphics systems such as OpenGL typically support:– triangles, triangle strips and fans– quads, quad strips– polygons
Which way is front?– convention is that normal points
towards you if vertices are specified counter-clockwise
7GR2-00
Modelling Regular ObjectsModelling Regular Objects
Sweeping
Spinning
2D Profilesweep axis
spinning axis
R1 R2
8GR2-00
Sweeping a Circle to Generate a Cylinder as
Polygons
Sweeping a Circle to Generate a Cylinder as
Polygons
vertices at z=0
vertices at z=depthV1
V2
V3V4
V5
V6 V8
V7
V10
V9
V11
V12V13
V14
V15V16
V17
V18
V1[x] = R; V1[y] = 0; V1[z] = 0V2[x] = R cos ; V2[y] = R sin ; V2[z] = 0 (=/4)Vk[x] = R cos k; Vk[y] = R sin k; Vk[z] = 0wherek = 2 (k - 1 )/8, k=1,2,..8
9GR2-00
Exercise and Further Reading
Exercise and Further Reading
Spinning:– Work out formulae to spin an
outline (in the xy plane) about the y-axis
READING:– Hearn and Baker, Chapter 10
10GR2-00
Complex PrimitivesComplex Primitives
Some systems such as VRML have cylinders, cones, etc as primitives– polygonal representation calculated
automatically OpenGL has a utility library (GLU)
which contains various high-level primitives– again converted to polygons
For conventional graphics hardware:– POLYGONS RULE!
11GR2-00
Automatic Generation of Polygonal Objects
Automatic Generation of Polygonal Objects
3D scanners - or laser rangers - are able to generate computer representations of objects– object sits on rotating table– contour outline generated for a
given height– scanner moves up a level and next
contour created– successive contours stitched
together to give polygonal representation
12GR2-00
A PuzzleA Puzzle
13GR2-00
Modelling Objects and Creating Worlds
Modelling Objects and Creating Worlds
We have seen how boundary boundary representations representations of simple objects can be created
Typically each object is created in its own co-ordinate systemco-ordinate system
To create a world, we need to understand how to transform objects so as to place them in the right place - translationtranslation, at the right size - scalingscaling, in the right orientation- rotationrotation
14GR2-00
TransformationsTransformations
The basic linear transformations are:– translation: P = P + T, where T is
translation vector– scaling: P’ = S P, where S is a scaling
matrix– rotation: P’ = R P, where R is a rotation
matrix As in 2D graphics, we use
homogeneoushomogeneous co-ordinates in order to express all transformations as matrices and allow them to be combined easily
15GR2-00
Homogeneous Co-ordinates
Homogeneous Co-ordinates
In homogeneous coordinates, a 3D point P = (x,y,z)T
is represented as:P = (x,y,z,1)T
That is, a point in 4D space, with its ‘extra’ co-ordinate equal to 1
NoteNote: in homogeneous co-ordinates, multiplication by a constant leaves point unchanged– ie (x, y, z, 1)T = (wx, wy, wz, w)T
16GR2-00
TranslationTranslation
Suppose we want to translate P (x,y,z)T by a distance (Tx, Ty, Tz)T
We express P as (x, y, z, 1)T and form a translation matrix T as below
The translated point is P’
T P
x’y’z’1
P’ =
1 0 0 Tx0 1 0 Ty0 0 1 Tz0 0 0 1
xyz1
= x + Txy + Tyz + Tz1
=
17GR2-00
ScalingScaling
Scaling by Sx, Sy, Sz relative to relative to the originthe origin:
x’y’z’1
Sx 0 0 00 Sy 0 00 0 Sz 00 0 0 1
xyz1
P’ = S P
= = Sx . xSy . ySz . z1
18GR2-00
RotationRotation
Rotation is specified with with respect to an axis respect to an axis - easiest to start with co-ordinate axes
To rotate about the x-axis:
a positive angle corresponds to counterclockwise direction lookingat origin from positive position on axis
EXERCISE: write down the matrices for rotation about y and z axes
x’y’z’1
= 1 0 0 00 cos -sin 00 sin cos 00 0 0 1
xyz1
P’ = Rz () P
19GR2-00
Composite Transformations
Composite Transformations
The attraction of homogeneous co-ordinates is that a sequence of transformations may be encapsulated as a single matrix
For example, scaling with respect to a with respect to a fixed position (a,b,c) fixed position (a,b,c) can be achieved by:– translate fixed point to origin- say, T(-a,-b,-c)– scale- S– translate fixed point back to its starting
position- T(a,b,c) Thus: P’ = T(a,b,c) S T(-a,-b,-c) P = M P
20GR2-00
Rotation about a Specified Axis
Rotation about a Specified Axis
It is useful to be able to rotate about any axis in 3D space
This is achieved by composing 7 elementary transformations
21GR2-00
Rotation through about Specified Axis
Rotation through about Specified Axis
x
y
z
x
y
zrotate throughrequ’d angle,
x
y
z
x
y
z
P2
P1x
y
z
P2
P1x
y
z
initial positiontranslate P1to origin
rotate so that P2 lies on z-axis(2 rotations)
rotate axisto orig orientation
translate back
22GR2-00
Inverse TransformationsInverse Transformations
As in this example, it is often useful to calculate the inverse of a transformation– ie the transformation that returns to
original state Translation: T-1 (a, b, c) = T (-a, -
b, -c) Scaling: S-1 ( Sx, Sy, Sz ) =
S ............ Rotation: R-1
z () = Rz (-)
23GR2-00
Rotation about Specified Axis
Rotation about Specified Axis
Thus the sequence is:
T-1 R-1x() R-1
y() Rz() Ry() Rx() T EXERCISE: How are and
calculated? READING:
– Hearn and Baker, chapter 11.
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