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2D Transformations : 

2D Transformations x y x y x y

2D Transformation : 

2D Transformation Given a 2D object, transformation is to change the object’s Position (translation) Size (scaling) Orientation (rotation) Shapes (shear) Apply a sequence of matrix multiplication to the object vertices

Point representation : 

Point representation We can use a column vector (a 2x1 matrix) to represent a 2D point x y A general form of linear transformation can be written as: x’ = ax + by + c OR y’ = dx + ey + f

Translation : 

Translation Re-position a point along a straight line Given a point (x,y), and the translation distance (tx,ty) The new point: (x’, y’) x’ = x + tx y’ = y + ty OR P’ = P + T where P’ = x’ p = x T = tx y’ y ty tx ty

3x3 2D Translation Matrix : 

3x3 2D Translation Matrix Use 3 x 1 vector Note that now it becomes a matrix-vector multiplication

Translation : 

Translation How to translate an object with multiple vertices?

2D Rotation : 

2D Rotation Default rotation center: Origin (0,0) q > 0 : Rotate counter clockwise < 0 : Rotate clockwise

Rotation : 

Rotation (x,y) -> Rotate about the origin by q How to compute (x’, y’) ? x = r cos (f) y = r sin (f) r x’ = r cos (f + q) y = r sin (f + q)

Rotation : 

Rotation r x = r cos (f) y = r sin (f) x’ = r cos (f + q) y = r sin (f + q) x’ = r cos (f + q) = r cos(f) cos(q) – r sin(f) sin(q) = x cos(q) – y sin(q) y’ = r sin (f + q) = r sin(f) cos(q) + r cos(f)sin(q) = y cos(q) + x sin(q)

Rotation : 

Rotation r x’ = x cos(q) – y sin(q) y’ = y cos(q) + x sin(q) Matrix form? 3 x 3?

3x3 2D Rotation Matrix : 

3x3 2D Rotation Matrix

Slide 12: 

Rotation How to rotate an object with multiple vertices?

2D Scaling : 

2D Scaling Scale: Alter the size of an object by a scaling factor (Sx, Sy), i.e.

2D Scaling : 

2D Scaling Not only the object size is changed, it also moved!! Usually this is an undesirable effect We will discuss later (soon) how to fix it

3x3 2D Scaling Matrix : 

3x3 2D Scaling Matrix

Put it all together : 

Put it all together Translation: x’ x tx y’ y ty Rotation: x’ cos(q) -sin(q) x y’ sin(q) cos(q) y Scaling: x’ Sx 0 x y’ 0 Sy y = + = * = *

Or, 3x3 Matrix representations : 

Or, 3x3 Matrix representations Translation: Rotation: Scaling: Why use 3x3 matrices? x’ cos(q) -sin(q) 0 x y’ sin(q) cos(q) 0 * y 1 0 0 1 1 = x’ Sx 0 0 x y’ = 0 Sy 0 * y 1 0 0 1 1

Why use 3x3 matrices? : 

Why use 3x3 matrices? So that we can perform all transformations using matrix/vector multiplications This allows us to pre-multiply all the matrices together The point (x,y) needs to be represented as (x,y,1) -> this is called Homogeneous coordinates!

Shearing : 

Shearing Y coordinates are unaffected, but x cordinates are translated linearly with y That is: y’ = y x’ = x + y * h

Slide 20: 

Shearing in y

Slide 21: 

Rotation Revisit The standard rotation matrix is used to rotate about the origin (0,0) cos(q) -sin(q) 0 sin(q) cos(q) 0 0 0 1

Arbitrary Rotation Center : 

Arbitrary Rotation Center To rotate about an arbitrary point P (px,py) by q: Translate the object so that P will coincide with the origin: T(-px, -py) Rotate the object: R(q) Translate the object back: T(px,py)

Slide 23: 

Arbitrary Rotation Center Translate the object so that P will coincide with the origin: T(-px, -py) Rotate the object: R(q) Translate the object back: T(px,py) Put in matrix form: T(px,py) R(q) T(-px, -py) * P

Slide 24: 

Scaling Revisit The standard scaling matrix will only anchor at (0,0) Sx 0 0 0 Sy 0 0 0 1

Slide 25: 

Arbitrary Scaling Pivot To scale about an arbitrary pivot point P (px,py): Translate the object so that P will coincide with the origin: T(-px, -py) Rotate the object: S(sx, sy) Translate the object back: T(px,py) (px,py)

Affine Transformation : 

Affine Transformation Translation, Scaling, Rotation, Shearing are all affine transformation Affine transformation – transformed point P’ (x’,y’) is a linear combination of the original point P (x,y), i.e. x’ m11 m12 m13 x y’ = m21 m22 m23 y 1 0 0 1 1 Any 2D affine transformation can be decomposed into a rotation, followed by a scaling, followed by a shearing, and followed by a translation. Affine matrix = translation x shearing x scaling x rotation

Composing Transformation : 

Composing Transformation Composing Transformation – the process of applying several transformation in succession to form one overall transformation If we apply transform a point P using M1 matrix first, and then transform using M2, and then M3, then we have: (M3 x (M2 x (M1 x P ))) = M3 x M2 x M1 x P

Composing Transformation : 

Composing Transformation Matrix multiplication is associative M3 x M2 x M1 = (M3 x M2) x M1 = M3 x (M2 x M1) Transformation products may not be commutative A x B != B x A Some cases where A x B = B x A A B translation translation scaling scaling rotation rotation uniform scaling rotation (sx = sy)

Transformation order matters! : 

Transformation order matters! Example: rotation and translation are not commutative Translate (5,0) and then Rotate 60 degree OR Rotate 60 degree and then translate (5,0)?? Rotate and then translate !!

How OpenGL does it? : 

How OpenGL does it? OpenGL’s transformation functions are meant to be used in 3D No problem for 2D though – just ignore the z dimension Translation: glTranslatef(d)(tx, ty, tz) -> glTranslatef(d)tx,ty,0) for 2D

Slide 31: 

How OpenGL does it? Rotation: glRotatef(d)(angle, vx, vy, vz) -> glRotatef(d)(angle, 0,0,1) for 2D (vx, vy, vz) – rotation axis x y You can imagine z is pointing out of the slide

OpenGL Transformation Composition : 

OpenGL Transformation Composition A global modeling transformation matrix (GL_MODELVIEW, called it M here) glMatrixMode(GL_MODELVIEW) The user is responsible to reset it if necessary glLoadIdentity() -> M = 1 0 0 0 1 0 0 0 1

Slide 33: 

OpenGL Transformation Composition Matrices for performing user-specified transformations are multiplied to the model view global matrix For example, 1 0 1 glTranslated(1,1 0); M = M x 0 1 1 0 0 1 All the vertices P defined within glBegin() will first go through the transformation (modeling transformation) P’ = M x P

Transformation Pipeline : 

Transformation Pipeline Modeling transformation …

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