2D Transformation
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This document summarizes the process of 2D transformations and window to viewport transformation in computer graphics. It describes basic 2D transformations including translation, rotation, scaling and their equation representations. It also explains the concept of composite transformations and discusses translation, rotation and scaling as composite transformations. Finally, it provides details about the window to viewport transformation including translating and scaling the window to fit within the viewport boundaries.
![Since, the composite transformations are as follows:
Composite Translation:
The two successive translation vectors (tx1, tyl) and (tx2, ty2)
are applied to a coordinate position P, the final transformed location
P' is calculated as
P’=[T{tx2,ty2}.T{tx1,ty1}].P
In matrix form
100
tyty10
txtx01
100
ty10
tx01
.
100
ty10
tx01
12
12
1
1
2
2
Hence, two successive translations are additive.
Composite Rotation
Two successive rotations applied to point p product the transformed
position
P' = {R(θ2) . R(θ1) }. P
By multiplying the two rotation matrices, we can verify that two
successive rotations are additive
PRP
PRRP
)..('
).().('
12
12
so that the final rotated coordinates with the composite rotation matrix is
100
0)21cos()21sin(
0)21sin(-)21cos(
Composite Scaling
Concatenating transformation matrices for two successive scaling
operations produces the following composite scaling matrix:
100
0Sy.Sy0
00Sx.Sx
100
0Sy0
00Sx
.
100
0Sy0
00Sx
1
1
2
2
OR
).,.().()..( 21211122 sysysxsxSsysxSsysxS
The resulting matrix indicates that successive scaling operations are
multiplicative. To triple the size of an object twice in succession, then the
final size would be nine times of original object.](https://image.slidesharecdn.com/transformation-200611084435/85/2D-Transformation-7-320.jpg)
2D Transformation
- 1. Assignment No.2 Submitted To: Ma’am Aqsa Naeem Subject: Computer Graphics Topics: Transformation 2D transformation Basic Types of transformation Translation Rotation Scaling Composite Transformation Window to viewport Transformation Submitted By Name: Asma Tehseen Roll No 1823110013 (319) Class BSCS-IV Submission Date: 18th may, 2020 Department of Computer Science
- 2. Transformation “In graphics, transformation means to change described object by moving, rotating or scaling that object while following some specific rules.” Or “A transformation is any operation on a point in a space (x,y) that maps the point’s coordinates into a new set of coordinates i. e; (x1,y1).” Simply moves an object from one position to another following this equation: dyyy dxxx oldnew oldnew 2D Transformation “The transformation takes place in 2D plane to re-position graphics, size or orientation of an object on screen is known as 2D Transformation.” Types of Transformation: There are several types of basic transformations: Translation Rotation Scale Translation: “A translation is applied to an object by re-positioning it along a straight line path from one coordinate location to another.” In 2D Translation, a point moves to a specified distance in the x or y direction. It is performed by adding translation distances,tx and ty to the original coordinate position (x,y) to move the point to a new position (x’,y’)
- 3. yty x ' x ' y tx The translation distance pair (tx, ty) is called a translation vector or shift vector. In translation, deformation of image does not occur as every point on the object is translated by the same amount. Translation equation in matrix form is represented as: 2 1 x x P 2 1 ' ' ' x x P y x t t T Row vector form: yxP yx ttT P=(x,y) P1=(2,3) , P2=(1,1) , P3=(3,1) While translation T=(tx,ty) is at (5,3) For P = (2,3) P’1= (2,3) + (5,3) = (7,6) P’2= (1,1) + (5,3) = (6,4) P’3= (3,1) + (5,3) = (8,4) y X (0,0) 1 1 2 2 3 4 5 6 7 8 9 1 3 4 5 6P1(2,3) P2(1,1) P3(3,1) y x 1 2 3 4 5 6 (6,4) (8,4) (7,6) Where P’ = P + T Translated by (5, 3) X’ = x + dx y’ = y + dy P’ = (x’ , y’) x’ = x + tx y’ = y + ty
- 4. Rotation: “A two dimensional rotation is applied on an object by re-positioning it along a circular path in the xy plane.” It causes a point to be moved relative to a central point, without changing the distance of r from the central point. While, Rotation angle θ and Rotation point (xr,yr) are specified to generate rotation. Rotation point is the point about which the object will rotate also known as pivot point. Positive values of rotation angle define counterclockwise rotations about the rotation point and Negative values indicate the rotation in clockwise direction. For the equation of rotation, r is the constant distance of the point form the origin, angle Φ is the original angular position and θ is the rotational angle. sinsincoscos)cos(' rrrx cossinsincos)sin(' rrry Original coordinates of the point are: cosrx sinry By equation 1 cossin' sincos' yxy yxx In column vector form rotation is defined as: P’=R.P Where rotation matrix is: cossin sincos R For positive rotation it becomes: For negative rotation: cossin sincos R cossin- sincos R 1
- 5. (x,y) (x’,y’) r r Φ θ Rotation is also a rigid body transformation, as image deformation doesn't occur. Since, every point on the object is rotated through the same angle. Scaling “This transformation alters the size and possibly the position of object.” Scaling operation is performed by multiplying the coordinate values (x,y) of each vertex by scaling factors sx and sy to produce the transformed coordinates (x’,y’). y x syy sxx .' .' sx scales the object in x direction and sy scales the object in y direction P’ = S.P Scaling factors(sx, sy)can be assigned any +ve numeric values. Values less than 1deduce the size of the objects Values greater than 1 produce an enlargement 1 value for both does not make any changes to the size of the object In Matrix form: y x s s y x y x . 0 0 ' ' The location of a scaled object can be controlled by choosing a position, called the fixed point, that is to remain unchanged after the scaling transformation.while,the Coordinates for the fixed point (xf, yf) can be chosen as one of the vertices, the object centroid, or any other position. Object is rotated at Rotational angle θ from Original angle Φ
- 6. Scale by (2, 2) P = (x, y) P’=(x’,y’) ; S = (2,2) P1=(2,3) P’1 =(2,3).(2,2)=(4,6) P2=(1,1) P’2=(1,1).(2,2) =(2,2) P3=(3,1) P’3=(3,1).(2,2)=(6,2) Composite Transformation Composite transformation is achieved by concatenation of two or more transformation matrices to obtained a combined transformation matrix. i. e; composite Transformation matrix.matrix Operations perform in Composite transformation: Rotate about an arbitrary point translate,rotate,translate Scale about an arbitrary point translate, scale,translate Change Co-ordinate system translate, rotate, scale y 0 1 1 2 2 3 4 5 6 7 8 9 1 3 4 5 6 (2, 3) (1, 1) (3,1) x’ = sx*x y’ = sy*y y 0 1 1 2 2 3 4 5 6 7 8 9 10 3 4 5 6 (2, 2) (6, 2) (4,6)
- 7. Since, the composite transformations are as follows: Composite Translation: The two successive translation vectors (tx1, tyl) and (tx2, ty2) are applied to a coordinate position P, the final transformed location P' is calculated as P’=[T{tx2,ty2}.T{tx1,ty1}].P In matrix form 100 tyty10 txtx01 100 ty10 tx01 . 100 ty10 tx01 12 12 1 1 2 2 Hence, two successive translations are additive. Composite Rotation Two successive rotations applied to point p product the transformed position P' = {R(θ2) . R(θ1) }. P By multiplying the two rotation matrices, we can verify that two successive rotations are additive PRP PRRP )..(' ).().(' 12 12 so that the final rotated coordinates with the composite rotation matrix is 100 0)21cos()21sin( 0)21sin(-)21cos( Composite Scaling Concatenating transformation matrices for two successive scaling operations produces the following composite scaling matrix: 100 0Sy.Sy0 00Sx.Sx 100 0Sy0 00Sx . 100 0Sy0 00Sx 1 1 2 2 OR ).,.().()..( 21211122 sysysxsxSsysxSsysxS The resulting matrix indicates that successive scaling operations are multiplicative. To triple the size of an object twice in succession, then the final size would be nine times of original object.
- 8. Window to Viewport Transformation “The process of transforming a two dimensional, world coordinate scene into device coordinates.” In particular, object inside world or clipping window are mapped to viewport. Clipping window Viewport Purpose: The purpose of this transformation is to convert the size of clipping window to size of viewport by using some mathematical computations. Steps for window to viewport transformation: 1. Translate window towards origin i.e; lower left corner and upper left corner of window become negative. 2. Resize window to viewport size applying following computation minmax minmax x XWXW XVXV S & minmax minmax ywyw yvyv Sy 3. Translate window I.e; position of window and viewport is same. i. if viewport is at(0,0) then don’t translate because window is already at origin. ii. If viewport is not at (0,0) take translation factor positive. World port/clipping window: The part of world coordinate to be displayed on screen. Viewport: Rectangular region selected for displaying image on computer screen. YW-max YW-min XW-min XW-max Viewport Coordinate YV-max YV-min XV-min XV-max Window in world coordinate Window translated to Origin Window scaled to size of viewport Translated by (xmin,ymin) to final position (Xmax,Ymax) (Xmin,Ymin) (Xmin,Ymin) (Xmax.Ymax)