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2-D Geometric Transformations In order to manipulate an object in 2-Dspace, we must apply various transformationfunctions to the object. This allows us to changethe position, size, and orientation of the objects.There are two complementary points of view for describing object movement.1.) Geometric Transformation : The object itself is moved relative to a stationary coordinate system or background.
2.) Coordinate transformation : The object is held stationary while the coordinate system is moved relative to the object.The Basic geometric transformations are: Translation Rotation Scaling Reflection Shear
TranslationMoving an object is called a translation. Wetranslate point by moving to the x and ycoordinates, the amount the point should beshifted in the x and y directions. We translatean object by translating each vertex in theobject. x’ = x + tx y’ = y + ty
The translating distance pair( tx, ty) is called atranslation vector or shift vector.We can also write this equation in a singleMatrix using column vectors: P = x1 P’ = x1’ T = tx x2 x2’ ty or, P’ = P + TThat is, every point on the object is translatedby the same amount.
RotationAn object can be rotated about the origin by aspecific rotation angle θ & the position (xr,yr)of the rotation point about which the object isto be rotated.Positive values for the rotation angle definecounterclockwise rotations & -ve definesclockwise direction. This transformation canalso be described as a rotation about therotation axis that is perpendicular to the xyplane.
In the fig., r is the constant distance of thepoint from the origin, angle Φ is the originalangular position of the point from thehorizontal, & θ is the rotation angle.We can express the coordinates as:x2 = r cos(Φ+θ) = r cosΦ cosθ – r sinΦ sinθy2 = r sin(Φ+θ) = r cosΦ sinθ + r sinΦ cosθThe original coordinates of the points in thepolar coordinates are x = r cosΦ , y = r sinΦ
We obtain the transformation equations forrotating a point (x,y) through an angle θ aboutthe origin is: x2 = x cos θ – y sin θ y2 = x sin θ + y cos θWe can write the rotation equations in thematrix form: P’ = R . P& the rotation matrix is R = cos θ -sin θ sin θ cos θ
Scaling Changing the size of an object is calledScaling . We scale an object by scaling the xand y coordinates of each vertex in the object.
Positive scaling constraints sx & sy which arethe scaling factors are used to produce thetransformed coordinates (x’, y’). x’ = x . sx , y’ = y . syScaling factor sx scales objects in the xdirection, while sy scales objects in the ydirection. The transformation equations can bewritten in the matrix form: x’ = sx 0 . x y’ = 0 sy y
or P’ = S . PThere are three scaling factors:(i) A scaling constant > 1 indicates expansion of length ie. Magnification(ii) A scaling constant < 1 indicates compression of length ie. reduction(iii) A scaling constant = 1 leaves the size of object unchanged.When assigned the same value, a uniformscaling is produced & for unequal valuesdifferential scaling is produced.
ReflectionA reflection is a transformation that producesa mirror image of an object. Since thereflection P’ of an object point P is located thesame distance from the mirror as P.(i) The mirror reflection transformation Mx about the x-axis is given by: P’ = Mx (P)where, x’ = x & y’ = -yIt can be represented in matrix form as:
P’ = x’ Mx = 1 0 P= x y’ 0 -1 y y P’(-x, y) P(x, y) x P’(x, -y)
(ii) The mirror reflection transformation My about y-axis is given by: P’ = My(P) where, x’ = -x & y’ = yIt can be represented in matrix form as: P’ = x’ My = -1 0 P= x y’ 0 1 y
ShearThe shear transformation distorts an object by scaling one coordinate using the otherOriginal Data Y Shear X Shear
An x-direction shear relative to the x axis is produced with the transformation matrix 1 Shx 0 1which transforms coordinate position as x’ = x + Shx . y , y’ = ySimilarly, a y-direction shear relative to the yaxis is produced with the transformationmatrix
1 0 Shy 1 which transforms coordinate position as y’ = x . Shy + y , x’ = xExample: Take (x,y) = (1,1) & Shx = 2X’ = x + Shx . Y y’ = y =1+2.1 y’ = 1 =1+2 =3(x’, y’) = (3,1)
Inverse Geometric TransformationsEach geometric transformation has an inversewhich is described by the opposite operationperformed by the transformation:Translation: Tv-1 = T-v, translation inopposite directionRotation: Rθ-1 = R-θ, rotation in oppositedirectionScaling: Ssx,sy-1 = S1/sx,1/syReflection: Mx-1 = Mx & My-1 = My