Part7: Geometric Transformations Computer Graphics Part7: Geometric Transformations

April 23, 2017 2D Transformations Transformation means changes in orientation, size, and shape. The basic geometric transformations are translation, rotation, and scaling. Other transformations that are often applied to objects include reflection and shear ( جز).

Transformations Translation Rotation Scaling moving things April 23, 2017 Transformations Translation moving things Rotation moving about a point by a given angle rotation point may be origin or some other chosen point Scaling changing size relative to origin or some other chosen point

2D Translation Repositioning an object along a straight line path from one co-ordinate location to another (x,y) (x’,y’) To translate a 2D position, we add translation distances tx and ty to the original coordinates (x,y) to obtain the new coordinate position (x’,y’) x’= x + tx , y’= y + ty T x y Matrix form

April 23, 2017 2D TRANSLATION Example: Let P = (5, 9), T = (tx, ty) = ( 6, 8) , so P' = (11, 17) Note: To translate a line translate its endpoints. To translate polygon ( مضلع) translate its vertices. Circle or Ellipse: Translate boundary points.

2D Translation Moving a polygon from position (a) to position (b) with the translation vector (-5, 10), i.e. x y 5 10 15 20 x y 5 10 15 20 (a) (b)

April 23, 2017 2D TRANSLATION We think of a point as a column vector, written as a column of numbers between parentheses. We add vectors by adding corresponding coordinates.

2D Rotation Repositioning an object along a circular path in the xy-plane x y (x,y) (x’,y’) φ θ r The original coordinates are:

2D Rotation Substituting x y (x,y) (x’,y’) φ θ r Matrix form

2D Rotation Example Cos 90 = 0 Sin 90 = 1 Rotate the line endpoints (1,2) and ( 4,3) about origin by 90 degree. p(1,2) >>>>> p’(-2,1) x’=1*cos 90 – 2*sin 90 = -2 y’=1*sin 90 + 2*cos 90= 1 P(4,3) >>>>>>> p’(-3,4) x’=4*cos 90 – 3*sin 90= -3 y’=4*sin 90 + 3*cos 90= 4 Cos 90 = 0 Sin 90 = 1

2D Rotation about a Pivot position Rotating about pivot position (xr, yr) x y (x,y) (x’,y’) φ θ r xr yr

Example Rotate the line endpoints (2,1) about point (3,3) by 90 x’=3+(2-3) cos 90 – (1-3) sin 90 x’=3—2=5 y’=3+(2-3) sin 90 + (3-1) cos 90 = 3-1=2

2D Scaling Altering the size of an object. Sx and Sy are the scaling factors. If Sx = Sy then uniform scaling. Sx and Sy can be any positive value. – Values < 1 reduces the size of the object. – Values > 1 produces and enlargement. – If Sx and Sy is 1, then the size is unchanged. Matrix form

2D Scaling relative to Fixed point Scaling relative to fixed point (xf, yf) x y Sx = ¼ , Sy = ½ P1 P2 P3 P1’ P2’ P3’ (xf , yf)

Matrix Representation Use 3×3 matrices to combine transformations Translation Rotation Scaling

Inverse Transformations Translation Rotation Scaling

Example Consider the line with endpoints (10, 10) and (30, 25). Translate it by tx = -20, ty = -10 and then rotate it by θ = 90º. x y (10, 10) (30, 25)

Result x y x y x y T(-20, -10) R(90º) (30, 25) (-15, 10) (10, 10) (0, -10) (-15, 10) x y (10, 15) (-10, 0) x y (0, -10) (-15, 10) T(-20, -10) R(90º)

Exercises Consider the following object: Apply a rotation by 145º then scale it by Sx=2 and Sy=1.5 and then translate it by tx=20 and ty=-30. Scale it by Sx=½ and Sy=2 and then rotate it by 30º. Apply a rotation by 90º and then another rotation by 45º. Apply a rotation by 135º. x y 10 25 45

Exercises Consider the following object: Apply a rotation by 60° on the Pivot Point (-10, 10) and display it. Apply a rotation by 30° on the Pivot Point (45, 10) and display it. Apply a rotation by 270° on the Pivot Point (10, 0) and then translate it by tx = -20 and ty = 5. Display the final result. x y 10 25 45

Exercises Consider the following object: Scale it by sx = 2 and sy = ½ relative to the fixed point (140, 125) and display it. Apply a rotation by 90° on the Pivot Point (50, 60) and then scale it by sx = sy = 2 relative to the Fixed Point (0, 200). Display the result. Scale it sx = sy = ½ relative to the Fixed Point (50, 60) and then rotate it by 180° on the Pivot Point (50, 60). Display the final result. x y 60 50 125 220

(x,y,z) (x’,y’,z’) x’= x + tx , y’= y + ty , z’= z + tz 3D Translation (x,y,z) (x’,y’,z’) x’= x + tx , y’= y + ty , z’= z + tz Matrix form (4 × 4) T(tx, ty, tz) x y z

3D Rotation z-axis: x y z Matrix form x y z

3D Rotation x-axis: x y z Matrix form

3D Rotation y-axis: x y z Matrix form

3D Scaling x y z Matrix form