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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.
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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
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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.
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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
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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.
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(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
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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
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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’
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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
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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
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