2D and 3D Transformation

April 22, 2023204481 Foundation of Computer Graphics 1

Pradondet NilaguptaDept. of Computer Engineering

Kasetsart University

Transformations and Matrices Transformations are functions Matrices are functions representations Matrices represent linear transformation {2x2 Matrices} {2D Linear Transformation}

Transformations (1/3) What are they?

changing something to something else via rules mathematics: mapping between values in a range set

and domain set (function/relation) geometric: translate, rotate, scale, shear,…

Why are they important to graphics? moving objects on screen / in space mapping from model space to screen space specifying parent/child relationships …

Transformation (2/3) Translation

Moving an object Scale

Changing the size of an object

wold wnew

xnew = xold + tx; ynew = yold + ty

sx=wnew/wold sy=hnew/hold

xnew = sxxold ynew = syyold

Transformation (3/3)

To rotate a line or polygon, we must rotate each of its vertices

Original Data y Shear x Shear

What is a 2D Linear Transform?

.y and x vectorsand ascalar for ,)y(T)x(aT)yxa(T:Definition

)y,x2()y,x2(yy),xx(2:say 2,by x,in Scale

11001010

Example

), 00( yx ), 002( yx

), 11( yx ), 112( yx

), 002( yx

), 112( yx yyxx 1010 ,22

), 00( yx

), 11( yx

yyxx 1010 ),(2

yyxx 1010 ),(

yyxx 1010 ),(2

yyxx 1010 ),(

yyxx 1010 ,22 Scale in x by 2

yyxx 1010 ),(2 yyxx 1010 ),(2

Transformations: Translation (1/2)

A translation is a straight line movement of an object from one position to another.

A point (x,y) is transformed to the point (x’,y’) by adding the translation distances Tx and Ty:

x’ = x + Tx

y’ = y + Ty

Transformations: Translation(2/2)

moving a point by a given tx and ty amount

e.g. point P is translated to point P’

moving a line by a given tx and ty amount

translate each of the 2 endpoints

)10,5(P

)10,15(P

)20,5(1P

)10,5(2P )10,5(1P

)0,5(2P

Transformations: Rotation (1/4)

Objects rotated according to angle of rotation theta ()

Suppose a point P(x,y) is transformed to the point P'(x',y') by an anti-clockwise rotation about the origin by an angle of degrees, then:

Given x = r cos , y = r sin x’ = x cos – y sin y’ = y sin + y cos

Rotation P by anticlockwise relative to origin (0,0)

),( yxP

Rotation about an arbitary pivot point (xR,yR)Step 1: translation of the object by (-xR,-yR)

x1 = x - xR

y1 = y - yRStep 2: rotation about the origin

x2 = x1 cos() - y1sin ()y2 = y1cos() - x1sin ()

Step 3: translation of the rotated object by (xR,yR)x’ = xr + x2

y’ = yr + y2

object can be rotated around an arbitrary point (xr,yr) known as rotation or pivot point by: x' = xr + (x - xr) cos() - (y - yr) sin ()

y' = yr + (x - xr) sin ()+(y - yr) cos()

Transformations: Scaling (1/5)

Scaling changes the size of an object Achieved by applying scaling factors

sx and sy Scaling factors are applied to the X

and Y co-ordinates of points defining an object’s

uniform scaling is produced when sx and sy have same value i.e. sx = sy

non-uniform scaling is produced when sx and sx are not equal - e.g. an ellipse from a circle. i.e. sx sy

x2 = sxx1 y2 = syy1

Simple scaling - relative to (0,0)

General form:

y*syx*sx

),(1 yxP),(1 yxP

)3,2(1P

)1,3(2P

)3,4(1P

)1,6(2P

Ex: sx = 2 and sy=1

If the point (xf,yf) is to be the fixed point, the transformation is:

x' = xf + (x - xf) Sx y' = yf + (y - yf) Sy

This can be rearranged to give:

x' = x Sx + (1 - Sx) xf y' = y Sy + (1 - Sy) yf

which is a combination of a scaling about the origin and a translation.

Transformation as Matrices

Scale:x’ = sxxy’ = syy

Rotation:x’ = xcos - ysin y’ = xsin + ycos

Translation:x’ = x + tx

y’ = y + ty

cossinsincos

Transformations: Shear (1/2)

Shx 101

Shear in x:

)1,( a)1,0(

Transformations: Shear (2/2)

Shear in y:

),1( b

)0,1()0,0(

bShy 1

Shear in x then in y

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

),(1 bab

),( 1a

),( 1 aba

),( 11 baba

),( 11 a

)1,1( b

),1( b

Shear in y then in x

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

),(1 b

),( 1a

),( 1 aba

),( 11 abba

),( 11 a

)1,1( b

),1( b

Homogeneous coordinate

As translations do not have a 2 x 2 matrix representation, we introduce homogeneous coordinates to allow a 3 x 3 matrix representation.

The Homogeneous coordinate corresponding to the point (x,y) is the triple (xh, yh, w) where:

xh = wx yh = wy

For the two dimensional transformations we can set w = 1.

Matrix representation

1),( y

1000000

1000cossin0sincos

1001001

Basic Transformation (1/3)

Translation

Rotation

P x y x yt tt t PS t( , , ) ( , , )

cos sinsin cos ( )1 1

Scaling

Composite Transformation

Suppose we wished to perform multiple transformations on a point:

P2 T3,1P1P3 S2, 2P2

P4 R30P3

M R30S2,2T3,1

P4 MP1

Example of Composite Transformation(1/3)

A scaling transformation at an arbitrary angle is a combination of two rotations and a scaling:

R(-t) S(Sx,Sy) R(t)

A rotation about an arbitrary point (xf,yf) by and angle t anti-clockwise has matrix:

T(-xf,-yf) R(t) T(xf,yf)

Reflection about the y-axis Reflection about the x-axis

100010001

Reflection about the origin Reflection about the line y=x

100010001

100001010

3D Transformation

Basic 3D Transformations

Translation Scale Rotation Shear As in 2D, we use homogeneous coordinates

(x,y,z,w), so that transformations may be composited together via matrix multiplication.

3D Translation and Scaling

TP = (x + tx, y + ty, z + tz)

SP = (sxx, syy, szz)

1000100010001

1000000000000

3D Rotation (1/4)

Positive Rotations are defined as follows:

Axis of rotation is Direction of positive rotation isx y to zy z to xz x to y

3D Rotation (2/4)

Rotation about x-axis Rx(ß)P

10000cossin00sincos00001

)0,1,0(

)1,0,0(

3D Rotation (3/4)

Rotation about y-axis Ry(ß)P

10000cos0sin00100sin0cos

)0,0,1(

)1,0,0(

3D Rotation (4/4)

Rotation about z-axis Rz(ß)P

1000010000cossin00sincos

3D Shear

xy Shear: SHxyP

10000100010001

Rotation About An Arbitary Axis (1/3)

1. Translate one end of the axis to the origin

2. Rotate about the y-axis and angle

3. Rotate about the x-axis through an angle

X4. When U is aligned with the z-axis, apply the original rotation, RR, about the z-axis.5. Apply the inverses of the transformations in reverse order.

T-1 Ry(ß) Rx(-µ) R Rx(µ) Ry(-ß) T P

2D and 3D Transformation

Documents

Computer Graphics: 2-Viewing · • View transformation. Pipeline 3D World Coordinates 2D Device Coordinates Transformation into Viewport for display 2D Device (Raster coordinates)

Transformation between 2D and 3D Covalent Organic

Transformations. Content Coordinate systems 2D Transformations 3D Transformations Transformation composition Rotating about a pivot Rotating about an

2D-3D Camera Fusion for Visual Odometry in Outdoor ... · Visual odometry is generally carried out by relying on 2D-2D, 3D-3D, or 2D-3D information. 2D-2D based methods typically

Free from line between 2D & 3D Excess-Hybrid CAD 2D CAD 3D … · 2019-04-11 · CATIA V4/V5 (Option) - 3D 2D - 2D Dimension 3D - 21) Drawing 3D - Parameters CAM Milling/Wire cut

Computer Graphics 2D & 3D Transformation. 2D Transformation transform composition: multiple transform on the same object (same reference point or line!)

2d 3d Conversion

Chapter 2 2d Transformation

2D/3D Shape Manipulation , 3D Printing

View in 2D & 3D - 计算机图形学 Viewing.pdf · Windowing Concepts Computer Graphics 5. 2D Viewing Transformation Computer Graphics 6. ... •Clipping is done in 3D by clipping

Transformation Matrices - United States Naval Academy ... · 3D Kinematics So now we’re pretty good at 2D kinematics, and we can expand this to 3D kinematics. Transformation Matrices

UNIT-1 : 2D AND 3D TRANSFORMATION & VIEWING

2D Transformations 3D Transformations OpenGL Transformationsabbirsaleh.weebly.com/.../1/0/4510327/4_transformations.pdf · 2018. 9. 6. · 3D Transformations OpenGL Transformation

Automated Transformation of 2D Vector-Based Plans to 3D ... · Automated Transformation of 2D Vector-Based Plans to 3D Geovirtual Environments transformation parameters can be assigned

2D/3D Shape Manipulation, 3D Printing

Geometric Transformation-2D

2D Transformation

The 3D Rasterization Pipeline€¦ · Viewport Transformation Scan Conversion Viewing Transformation This is a pipelined sequence of operations to draw 3D primitives into a 2D image

Analyse multifractale 2D et 3D l'aide de la transformation en ondelettes

CAD/CAM Technology in Automoibile Engineering-2D AND 3D transformation