2D Transformations What is transformations? The geometrical changes of an object from a current state to modified state. Why the transformations is needed? To manipulate the initially created object and to display the modified object without having to redraw it.

2 ways Object Transformation Alter the coordinates descriptions an object Translation, rotation, scaling etc. Coordinate system unchanged Coordinate transformation Produce a different coordinate system 2D Transformations

Matrix Math Why do we use matrix? More convenient organization of data. More efficient processing Enable the combination of various concatenations Matrix addition and subtraction a b c d  a  c b  d =

Matrix Math How about it? Matrix Multiplication Dot product a b c d e f  abcdabcd efghefgh. = a.e + b.g a.f + b.h c.e + d.g c.f + d.h

Matrix Math What about this? Type of matrix = abab a b Row-vectorColumn-vector

Matrix Math Is there a difference between possible representations?          dfce bfae              f e dc ba  dfbecfae    dc ba fe         dfcebfae    db ca fe       

Matrix Math We’ll use the column-vector representation for a point. Which implies that we use pre-multiplication of the transformation – it appears before the point to be transformed in the equation.

Translation A translation moves all points in an object along the same straight-line path to new positions. The path is represented by a vector, called the translation or shift vector. We can write the components: p' x = p x + t x p' y = p y + t y or in matrix form: P' = P + T tx tx ty ty x’ y’ xyxy txtytxty = + (2, 2) = 6 =4 ?

Rotation A rotation repositions all points in an object along a circular path in the plane centered at the pivot point. First, we’ll assume the pivot is at the origin.  P P’

Rotation Review Trigonometry => cos  = x/r, sin  = y/r x = r. cos , y = r.sin    P(x,y) x y r x’ y’  P’(x’, y’) r  => cos (  +  ) = x’/r x’ = r. cos (  +  ) x’ = r.cos  cos  -r.sin  sin  x’ = x.cos  – y.sin   =>sin (  +  ) = y’/r  y’ = r. sin (  +  ) y’ = r.cos  sin  + r.sin  cos  y’ = x.sin  + y.cos  Identity of Trigonometry

Rotation We can write the components:We can write the components: p ' x = p x cos  – p y sin  p ' x = p x cos  – p y sin  p ' y = p x sin  + p y cos  p ' y = p x sin  + p y cos  or in matrix form:or in matrix form: P' = R P can be clockwise (-ve) or counterclockwise (+ve as our example).  can be clockwise (-ve) or counterclockwise (+ve as our example). Rotation matrixRotation matrix  P(x,y)  x y r x’ y’  P’(x’, y’)

Rotation Example Find the transformed point, P’, caused by rotating P= (5, 1) about the origin through an angle of 90 .

Scaling Scaling changes the size of an object and involves two scale factors, S x and S y for the x- and y- coordinates respectively. Scales are about the origin. We can write the components: p' x = s x p x p' y = s y p y or in matrix form: P' = S P Scale matrix as: P P’

Scaling If the scale factors are in between 0 and 1  the points will be moved closer to the origin  the object will be smaller. P(2, 5) P’ Example : P(2, 5), Sx = 0.5, Sy = 0.5 Find P’ ?

Scaling If the scale factors are in between 0 and 1  the points will be moved closer to the origin  the object will be smaller. P(2, 5) P’ Example : P(2, 5), Sx = 0.5, Sy = 0.5 Find P’ ? If the scale factors are larger than 1  the points will be moved away from the origin  the object will be larger. P’ Example : P(2, 5), Sx = 2, Sy = 2 Find P’ ?

Scaling If the scale factors are the same, S x = S y  uniform scaling Only change in size (as previous example) P(1, 2) P’ If S x  S y  differential scaling. Change in size and shape Example : square  rectangle P(1, 3), S x = 2, S y = 5, P’ ? What does scaling by 1 do? What is that matrix called? What does scaling by a negative value do?

Combining transformations We have a general transformation of a point: P' = M P + A When we scale or rotate, we set M, and A is the additive identity. When we translate, we set A, and M is the multiplicative identity. To combine multiple transformations, we must explicitly compute each transformed point. It’d be nicer if we could use the same matrix operation all the time. But we’d have to combine multiplication and addition into a single operation.

Homogenous Coordinates Let’s move our problem into 3D. Let point (x, y) in 2D be represented by point (x, y, 1) in the new space. Scaling our new point by any value a puts us somewhere along a particular line: (ax, ay, a). A point in 2D can be represented in many ways in the new space. (2, 4)  (8, 16, 4) or (6, 12, 3) or (2, 4, 1) or etc. yy x x w 

Homogenous Coordinates We can always map back to the original 2D point by dividing by the last coordinate (15, 6, 3) ---  (5, 2). (60, 40, 10) -  ?. Why do we use 1 for the last coordinate? The fact that all the points along each line can be mapped back to the same point in 2D gives this coordinate system its name – homogeneous coordinates.

Matrix Representation Point in column-vector: Our point now has three coordinates. So our matrix is needs to be 3x3. Translation xy1xy1

Matrix Representation Rotation Scaling

Composite Transformation We can represent any sequence of transformations as a single matrix. No special cases when transforming a point – matrix vector. Composite transformations – matrix matrix. Composite transformations: Rotate about an arbitrary point – translate, rotate, translate Scale about an arbitrary point – translate, scale, translate Change coordinate systems – translate, rotate, scale Does the order of operations matter?

Composition Properties Is matrix multiplication associative? (A.B).C = A.(B.C)                          dhlcfldgjcejdhkcfkdgicei bhlaflbgjaejbhkafkbgiaei lk ji dhcfdgce bhafbgae                             lk ji hg fe dc ba                          dhldgjcflcejdhkdgicfkcei bhlbgjaflaejbhkbgiafkaei hlgjhkgi flejfkei dc ba                             lk ji hg fe dc ba ?

Composition Properties Is matrix multiplication commutative? A. B = B. A ?

Order of operations So, it does matter. Let’s look at an example: 1.Translate 2.Rotate 1.Rotate 2.Translate

Composite Transformation Matrix Arrange the transformation matrices in order from right to left. General Pivot- Point Rotation Operation :- 1.Translate (pivot point is moved to origin) 2.Rotate about origin 3.Translate (pivot point is returned to original position ) T(pivot) R(  ) T(–pivot) 1 0 -t x 0 1 -t y cos  -sin  0 sin  cos  t x 0 1 t y cos  -sin  -t x cos  + t y sin  + t x sin  cos  -t x sin  - t y cos  + t y cos  -sin  -t x cos  + t y sin  sin  cos  -t x sin  - t y cos  t x 0 1 t y

Composite Transformation Matrix Example Perform 60  rotation of a point P(2, 5) about a pivot point (1,2). Find P’? cos  -sin  -t x cos  + t y sin  + t x sin  cos  -t x sin  - t y cos  + t y xy1xy = P’ = (-1, 4) Sin 60 = Kos 60 = 1/2

Without using composite homogenus matrix Example Perform 90  rotation of a point P(5, 1) about a pivot point (2, 2). Find P’? 1. Translate pivot point ke asalan ( tx = -2, ty = -2) Titik P(5, 1 )  P’ (3, -1) 2. Rotate P ‘ = 90 degree P’(3, -1) -- > kos 90 -sin 90 3 = = 1 sin 90 kos Translate back ke pivot point (tx = 2, ty = 2) titik (1, 3 )  titik akhir (3, 5)

Composite Transformation Matrix General Fixed-Point Scaling Operation :- 1.Translate (fixed point is moved to origin) 2.Scale with respect to origin 3.Translate (fixed point is returned to original position ) T(fixed) S(scale) T(–fixed) Find the matrix that represents scaling of an object with respect to any fixed point? Given P(6, 8), Sx = 2, Sy = 3 and fixed point (2, 2). Use that matrix to find P’?

Answer 1 0 -t x 0 1 -t y Sx Sy t x 0 1 t y Sx 0 -t x Sx 0 Sy -t y Sy t x 0 1 t y = x =6, y = 8, Sx = 2, Sy = 3, t x =2, t y = 2 Sx 0 -t x Sx + t x 0 Sy -t y Sy + t y ( 2) (3) =

Composite Transformation Matrix General Scaling Direction Operation :- 1. Rotate (scaling direction align with the coordinate axes) 2. Scale with respect to origin 3. Rotate (scaling direction is returned to original position ) R(–  ) S(scale) R(  ) Find the composite transformation matrix by yourself !!

cos  -sin  0 sin  cos  Sx Sy t x 0 1 t y S. T. R cos  -sin  t x sin  cos  t y Sx Sy Sxcos  Sx(-sin  ) Sx t x Sy sin  Sy cos  Sy t y 0 0 1

Other transformations Reflection: x-axisy-axis

Other transformations Reflection: origin line x=y

Other transformations Shear: x-direction y-direction

Coordinate System Transformations We often need to transform points from one coordinate system to another: 1.We might model an object in non-Cartesian space (polar) 2.Objects may be described in their own local system 3.Other reasons: textures, display, etc

Basic 3D transformations: scaling Some of the 3D transformations are just like 2D ones. For example, scaling Scaling

Translation in 3D

Rotation in 3D Rotation now has more possibilities in 3D:

Rotation in 3D What about the inverses of 3D rotations?

Shearing in 3D

Compositing multiple transformations Expressed as matrix multiplications: p’ = M 1 M 2 p means apply transformation M 2 first, followed by M 1 Historically, the vectors in graphics were row vectors, but columns are more popular.

Compositing multiple transformations 3 steps 1) T(-2, -3) 2) R(30) 3) T(2, 3) q’=T(2,3) R(30) T(-2,-3) q

Change of the coordinate frame, Change of basis Let’s first look at a 2D example p=(2, 1) T, what are the coordinates of p with respect to the new basis (coordinate system)? p Specify: an origin, and two vectors

Linear dependence or independence Spanning a space or a subspace, ‘generators’ Basis for a space or a subspace (a set of vectors) Dimension of a subspace (a number) A basis for V is a set of vectors: linearly independent (not too many) and span the space V (not too few) Basis is not unique, coordinates are, when the basis is fixed.

p New basis: O=(4,3) T, e 0 =(-1,0) T, e 1 =(0,-1) T

Let coordinates of p in new basis be (  0,  1 ) T New basis: O=(4,3) T, e 0 =(-1,0) T, e 1 =(0,-1) T Represent as homogeneous coordinates: O=(4,3,1) T, e 0 =(-1,0,0) T, e 1 =(0,-1,0) T For vectors, last component of homogeneous coordinates is 0, for convenience only for the time being.

Basis change: e’_i = B e_i Pt change, x’ = A x x=\sum x_i e_i, x=\sum x_i B^{-1} e’ = \sum B^{-1} x_i e’ x’ = B^{-1} x

Change of basis in 3D one origine, three vectors 3D case Given an orthonormal basis (O, e 0, e 1, e 2 ) e 0, e 1, e 2 are unit length vectors e 0, e 1, e 2 are mutually orthogonal To change basis, need matrix M=(e 0 e 1 e 2 O) It is easy to find the inverse of such a matrix

We can write M as Finding the inverse Transformation between two coordinate systems involves a translation, followed by a general rotation.

Cross product The cross product of two vectors in 3-space is An easy way to remember this is to write it in the form

Cross product: define coordinate system Given two vectors u and v (not parallel) Define a coordinate system such that One of the axis is in the direction of u Another axis is perpendicular to both u and v Solution: Direction of x-axis: e 0 = u Direction of z-axis: e 2 = u x v Direction of y-axis: e 1 = e 2 x e 0 (note:  e 0 x e 2 ) The Gram-Shmidt process