SI23 Introduction to Computer Graphics

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SI23Introduction to Computer

Graphics

SI23Introduction to Computer

Graphics

Lecture 11 – 3D Graphics Transformation Pipeline: Modelling and ViewingGetting Started with OpenGL

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3D Transformation Pipeline

3D Transformation Pipeline

3D graphics objects pass through a series of transformations before they are displayed– Objects created in modelling co-

ordinates

mod’gco-ords

worldco-ords

viewingco-ords

ModellingTransform’n

ViewingTransform’n

ProjectionTransform’n

Position object in world

Position worldwith respectto camera

Project viewfrom cameraonto plane

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Modelling Objects and Creating Worlds

Modelling Objects and Creating Worlds

We have seen how boundary representations of simple objects can be created

Typically each object is created in its own co-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- rotationrotationThis process is known as MODELLING

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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

homogeneous co-ordinates in order to express all transformations as matrices and allow them to be combined easily

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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

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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

=

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ScalingScaling

Scaling by Sx, Sy, Sz relative to the 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

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RotationRotation

Rotation is specified with 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’ = Rx () P

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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

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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

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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

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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 (…….)

Exercise: Check T-1 T = I (identity matrix)

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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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Interlude: QuestionInterlude: Question

Why does a mirror reflect left-right and not up-down?

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Getting Started with OpenGL

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What is OpenGL?What is OpenGL?

OpenGL provides a set of routines (API) for advanced 3D graphics– derived from Silicon Graphics GL– acknowledged industry standard, even on PCs

(OpenGL graphics cards available)– integrates 3D drawing into X (and other

window systems such as MS Windows)– draws simple primitives (points, lines,

polygons) but NOT complex primitives such as spheres

– provides control over transformations, lighting, etc

– Mesa is publically available equivalent

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Geometric PrimitivesGeometric Primitives

Defined by a group of vertices - for example to draw a triangle:

glBegin (GL_POLYGON);

glVertex3i (0, 0, 0);

glVertex3i (0, 1, 0);

glVertex3i (1, 0, 1);

glEnd();

See OpenGL supplementpp3-6 for output primitives

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Modelling, Viewing and Projection

Modelling, Viewing and Projection

OpenGL maintains two matrix transformation modes– MODELVIEW

to specify modelling transformations, and transformations to align camera

– PROJECTION

to specify the type of projection (parallel or perspective) and clipping planes

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ModellingModelling

For modelling… set the matrix mode, and create the transformation...

Thus to set a scaling on each axis...

glMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glScalef(sx,sy,sz); This creates a 4x4 modelview matrix Other transformation functions are:

glRotatef(angle, ux, uy, uz);

glTranslatef(tx, ty, tz);See p10 of OpenGLsupplement

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OpenGL Utility Library (GLU)

OpenGL Utility Toolkit (GLUT)

OpenGL Utility Library (GLU)

OpenGL Utility Toolkit (GLUT)

GLU:– useful set of utility routines written in terms of

OpenGL …

GLUT:– Set of routines to provide an interface to the

underlying windowing system - plus many useful high-level primitives (even a teapot - glutSolidTeapot()!)

– Allows you to write ‘event driven’ applications– you specify call back functions which are

executed when an event (eg window resize) occurs See pp1-3 of OpenGL

supplement

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How to Get StartedHow to Get Started

Look at the SI23 resources page:– http://www.comp.leeds.ac.uk/kwb/

si23/ resources.html Points you to:

– example programs– information about GLUT– information about OpenGL– information about Mesa 3D– a simple exercise

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Viewing Transformation

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Camera PositionCamera Position

In OpenGL (and many other graphics systems) the camera is placed at a fixed position

– At origin– Looking down

negative z-axis– Upright direction in

positive y-axis

x

y

z

VIEWING transforms the world so that it is in the requiredposition with respect to this camera

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Specifying the Viewing Transformation

Specifying the Viewing Transformation

OpenGL will build this transformation for us from:

– Where camera is to be

– Point we are looking at

– Upright direction

Becomes part of an overall MODELVIEW matrix

Eye position

Look atposition

Uprightposition

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Specifying the Viewing Transformation in OpenGL

Specifying the Viewing Transformation in OpenGL

For viewing, use gluLookAt()to create a view transformation matrixgluLookAt(eyex,eyey,eyez, lookx,looky,lookz, upx,upy,upz)

ThusglMatrixMode(GL_MODELVIEW);

glLoadIdentity();

glScalef(sx,sy,sz);

gluLookAt(eyex,eyey,eyez, lookx,looky,lookz, upx,upy,upz);

creates a model-view matrixSee pp11-12 ofOpenGL Supp.

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Viewing Pipeline So FarViewing Pipeline So Far

We now should understand the viewing pipeline

mod’gco-ords

worldco-ords

viewingco-ords

ModellingTransform’n

ViewingTransform’n

The next stage is the projection transformation….Next lecture!

ProjectionTransform’n

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Perspective and Parallel Projection

Perspective and Parallel Projection

perspective parallel