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- 1. 3-D Geometric TransformationsGeometric Transformation : The object itself is moved relative to a stationary coordinate system or background.With respect to some 3-D coordinate system, an object Obj is considered as a set of points. Obj = { P(x,y,z)}If the Obj moves to a new position, the new object Obj’ is considered: Obj’ = { P’(x’,y’,z’)}
- 2. TranslationMoving an object is called a translation. We translate an object by translating each vertex in the object. x’ = x + tx y’ = y + ty z’ = z + tz
- 3. The translating distance pair( tx, ty, tz) iscalled a translation vector or shift vector.We can also write this equation in a singleMatrix using column vectors: x’ 1 0 0 tx x y’ = 0 1 0 ty y z’ 0 0 1 tz z 1 0 0 0 1 1
- 4. RotationIn 2-D, a rotation is prescribed by an angle θ & a center of rotation P. But in 3-D rotations require the prescription of an angle of rotation & an axis of rotation. Rotation about the z axis: R θ,K x’ = x cosθ – y sinθ y’ = x sinθ – y cosθ z’ = z
- 5. Rotation about the y axis: R θ,J x’ = x cosθ + z sinθ y’ = y z’ = - x sinθ + z cosθRotation about the x axis: R θ,I x’ = x y’ = y cosθ – z sinθ z’ = y sinθ + z cosθ
- 6. & the rotation matrix corresponding is cos θ -sin θ 0 R θ,K = sin θ cos θ 0 0 0 1 cos θ 0 sin θ R θ,J = 0 1 0 -sin θ 0 cos θ
- 7. 1 0 0R θ,I = 0 cos θ -sin θ 0 sin θ cos θ
- 8. Scaling Changing the size of an object is calledScaling . The scale factor s determineswhether the scaling is a magnification, s > 1,Or a reduction, s < 1. Scaling with respect tothe origin, where the origin remains fixed, x’ = x . sx Ssx,sy,sz y’ = y . sy z’ = z . sz
- 9. The transformation equations can be writtenin the matrix form: x’ sx 0 0 x y’ = 0 sy 0 . y z’ 0 0 sz z
- 10. Coordinate TransformationTranslationIf the xyz coordinate system is displaced to anew position, the coordinates of a point inboth systems are related by the translationTransformation:Tv (x’,y’,z’) = Tv (x,y,z)where x’ = x – tx, y’ = y – ty , z’ = z – tz
- 11. In matrix notation, 1 0 0 -tx Tv = 0 1 0 -ty 0 0 1 -tz 0 0 0 1Similarly, we can express the coordinate scaling & rotation transformations.
- 12. Composite TransformationMore complex geometric and coordinatetransformations are formed the process ofcomposition of functions.Rotation About an Arbitrary Axis in space:1.) Translate the object so that the rotation axis passes through the coordinate origin.2.) Rotate the object so that the axis of rotation coincides with one of the coordinate axes.
- 13. 3.) Perform the specific rotation about the coordinate axis.4.) Apply inverse rotations to bring the rotation axis back to its original orientation.5.) Apply the inverse translation to bring the rotation axis back to its original position.We can transform the rotation axis onto any of the three coordinate axes. For eg. We are taking rotation onto the z-axis.
- 14. y P2 P2 P1’ x P1 P1’z P2’’ P2 P2 P1’ P1’ P1 P2’’

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