Coordinate TransformationSuppose that we have 2 coordinate systems inthe plane. The first system is located at originO & has coordinate axes xy. The secondsystem is located at origin O’& has coordinateaxes x’y’. Now each point in the plane hastwo coordinate descriptions: (x,y) or (x’,y’),depending on which coordinate system isused. The second system x’y’ arises from atransformation applied to first system xywhich is called Coordinate transformation.
TranslationIf the xy coordinate system is displaced to anew position, where the direction & distanceof displacement is given by vector, v = txI + tyJthe coordinates of a point in both systems arerelated by the translation transformation T v. (x’, y’) = Tv (x, y)where x’ = x – tx and y’ = y – ty.The translation equation can be expressed as asingle matrix equation by using a column
vector to represent coordinate position &translation vector.P= x T = tx P’ = x’ y ty y’or P’ = P – TRotationA rotation is applied to the plane byrepositioning it along a circular path in xyplane. The xy system is rotated θº abt origin.
The coordinates of a point in both systems arerelated by rotation transformation Rθ. (x’, y’) = Rθ (x,y)where x’ = x cos θ + y sin θ y’ = -x sin θ + y cos θThe rotation equation in matrix form iswritten as P’ = Rθ .Pwhere P’ = x’ Rθ = cos θ sin θ P= x y’ -sin θ cos θ y
ScalingSuppose that a new coordinate system isformed by leaving the origin & coordinateaxes unchanged, but introducing differentunits of measurement along the x & y axes. Ifthe new units are obtained from the old unitsby a scaling of sx along the x axis & sy alongthe y axis, the coordinates in the new systemare related to coordinates in the old systemthrough the scaling transformation Ssx,sy.
(x’,y’) = S (x,y)The transformation equation in matrix form is: P’ = S.PP’ = x’ S = 1/sx 0 P= x y’ 0 1/sy y
ReflectionIf the new coordinate system is obtained byreflecting the old system about either x or yaxis, the relationship b/w coordinate is givenby mirror transformation Mx & My.(i) The mirror reflection transformation MxAbout the x-axis is given by P’ = Mx (P)where x’ = x & y’ = - y
Along X-axis Along Y-axis y y y’ P(1,1) P(1,1) 1 P’(1,-1)’ P’(-1,1)’ -1 x x’ x’ x 1 -1 y’
It can be represented in matrix form asP’ = x’ Mx = 1 0 P= x y’ 0 -1 y(ii) The mirror reflection transformation MyAbout y-axis is given by P’ = My (P)where x’ = -x & y’ = yIt can be represented in matrix form asP’ = x’ My = -1 0 P= x y’ 0 1 y
Inverse coordinate Transformation:Each coordinate transformation has an inversewhich can be found by applying the oppositeTransformation.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