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Geometrical Transformation

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Outline• General Transform

• 3D Objects

• Quaternion & 3D Track Ball

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Modeling Transform• Specify transformation for objects

– Allow definitions of objects in own coordinate systems

– Allow use of object definition multiple times in a scene

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Overview• 2D transformations

– Basic 2-D transformations– Matrix representation– Matrix composition

• 3D transformations– Basic 3-D transformation– Same as 2-D

• Transformation Hierarchies– Scene graphs

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2-D Transformations

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2-D Transformations

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2-D Transformations

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2-D Transformations

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2-D Transformations

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2-D Transformations

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Basic 2D Transformations

xtxx '

ytyy '

xSxx '

ySyy '

xShyy y 'yShxx x '

sincos' yxx cossin' yyy

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Basic 2D Transformations

xtxx '

ytyy '

xSxx '

ySyy '

yShxx x 'xShyy y '

sincos' yxx cossin' yyy

y

x

Syy

Sxx

'

'

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Basic 2D Transformations

xtxx 'ytyy '

xSxx '

ySyy '

sincos' yxx cossin' yyy

yShxx x 'xShyy y '

cos)(sin)('

sin)(cos)('

yx

yx

SySxy

SySxx

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Rotation around the origin (2-D)

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Rotation around the origin (2-D)

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Rotation around the origin (2-D)

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Rotation (3-D)

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Rotation (3-D)

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Basic 2D Transformations

xtxx 'ytyy '

xSxx '

ySyy '

yShxx x 'xShyy y '

sincos' yxx cossin' yyy

cos)(sin)('

sin)(cos)('

yx

yx

SySxy

SySxx

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Basic 2D Transformations

xtxx '

ytyy '

xSxx '

ySyy '

yShxx x 'xShyy y '

sincos' yxx cossin' yyy

yyx

xyx

tSySxy

tSySxx

cos)(sin)('

sin)(cos)('

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Basic 2D Transformations

xtxx '

ytyy '

xSxx '

ySyy '

yShxx x 'xShyy y '

sincos' yxx cossin' yyy

yyx

xyx

tSySxy

tSySxx

cos)(sin)('

sin)(cos)('

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Matrix Representation

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Matrix Representation

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2x2 Matrix

xsx x 'ysy y '

xsys

y

x

s

s

y

x

y

x

0

0

'

'

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Scaling

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Scaling Around A Point

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2x2 Matrix

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Shear (2-D)

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Shear (3-D)

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2x2 Matrix

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2x2 Matrix

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2D Translation

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Basic 2D Transformations

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Homogeneous Coordinates

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Linear Transformations

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Affine Transformations

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Projective Transformations

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Matrix Composition

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Matrix Composition

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Matrix Composition

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Matrix Composition

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3D Transformations

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Basic 3D Transformations

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Basic 3D Transformations

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GENERAL ROTATION ABOUT ANAXIS

An axis in space is specified by a point P and a vector direction .Suppose that we wish to rotate an object about this arbitrary axis.

t

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Developing the General Rotation Matrix

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Developing the General Rotation Matrix

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Developing the General Rotation Matrix

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Developing the General Rotation Matrix

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Developing the General Rotation Matrix

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Developing the General Rotation Matrix

• Be careful …………

Z

X

(+,+)

(-,-)In both cases, tan(y/x) are positive.So, we need to carefully chooseit by checking the signs of x and y

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Developing the General Rotation Matrix

Another problem is: rotation interpolation is not easy and not goodreported in many papers.

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Angular displacement glRotate( , Ax,Ay,Az)

• (,n) defines an angular displacement of about an axis u or n for rotating a vector v

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sin)()cos1)((cos

sin)(cos))(()(

sin)(cos][][][][ ||||||

vnvnnv

vnnvnvnvn

vnvvvRvRvvRvR

The above formula is a matrix form, so we can use Matrix to compute rotation

In above equation, v=(x,y,z)T and n=(ax,ay,az)T

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Inverse Transformation

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Inverse Transformation

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Transform points, lines, planes etc.

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Transforming Normals

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Transformation Hierarchies

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OpenGL transformation Matrices

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OpenGL transformation Matrices

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OpenGL transformation Matrices

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OpenGL transformation Matrices

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Transformation Example 1

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Transformation Example 2

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Transformation Example 2

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Hierarchical Scene Graph

This topics will be taught in future or the next semester!!

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Applications

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Applications

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Applications

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