2D Transformations

Outline2D Affine Transformations

2D Transformations

Dr. Rahul Rai

Department of Mechanical and Aerospace EngineeringUniversity at Buffalo - SUNY

February 24, 2014

Most of the figures are adopted from Foley/VanDam. Only few of the

figures have been created by course team!!!

Dr. Rahul Rai 2D Transformations


2D Affine Transformations




I All represented as matrix operations on vectors! Parallel linespreserved, angles/lengths not

I TranslateI RotateI ScaleI ShearI Reflect

Translation Rotation Uniform scaling

Non-uniform scaling

Shearing Reflection




I Example 1: rotation and non uniform scale on unit cube

I Example 2: shear first in x, then in yI Note:

I Preserves parallelsI Does not preserve lengths and angles



2D Transforms: Translation

I Rigid motion of points to new locations

I Defined with column vectors:

Before translation After translation



2D Transforms: Scale

I Stretching of points along axis:

I In matrix form:

I or just:

Before scaling After scaling





I Rotation of points about the origin:x’=x cos θ − y sin θ

y’=x sin θ + y cos θ

I In matrix form:

I or just:

Before rotation After rotation



2D Transforms: Rotation

I Substitute the 1st two equations into the 2nd two to get thegeneral equation:x=r cosφ

y=r sinφx’=r cos(θ + φ) = r cosφ cos θ − r sinφ sin θy’=r sin(θ + φ) = r cosφ sin θ + r sinφ cos θ



Homogenous Coordinates

I Translation is treated differently from scaling and rotationP’=P+TP’=S*PP’=R*P

I Homogenous coordinates: allows all transformations to betreated as matrix multiplications

I Example: A 2D point (x,y) is the line (x,y,w), where w is anyreal number, in 3D homogenous coordinates.To get the point, homogenize by dividing by w (i.e., w=1)



Recall our Affine Transformations

Translation Rotation Uniform scaling

Non-uniform scaling

Shearing Reflection



Matrix representation of 2D Affine Transformations

I Translation:

I Scale:

I Rotation:

I Shear:

I Reflection:



Composition of 2D Transforms

I Rotate about a point P1I Translate P1 to originI RotateI Translate back to P1

I T (x1, y1)R(θ)T (−x1,−y1)




I Scale object around point P1I Translate P1 to originI ScaleI Translate back to P1




I Scale + rotate object around point P1 and move to P2I Translate P1 to originI ScaleI RotateI Translate back to P1




I Be sure to multiply transformations in proper order!


Documents

2D Transformations