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10. 3D Coordinate Operations e-mail: Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gyte.edu.trazsenalp@gyte.edu.tr Makine Mühendisliği Bölümü Gebze Yüksek Teknoloji Enstitüsü ME 521 Computer Aided Design

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Introduction Most of the problems needs 3D computer graphics for descrition. For this purpose 3D Trasnformations or coordinate ıperations are used. 3-D Transformations: Translation Rotation Scaling Dr. Ahmet Zafer Şenalp ME 521 2 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

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T x, T y, T z are the components of the translation in x, y, z, in object coordinates Dr. Ahmet Zafer Şenalp ME 521 3 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

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Through point (0, 0, 0) and about x axis, by the angle clockwise when looked from +x: Below figure shows +X and –X rotation directions (+y rotation direction) Dr. Ahmet Zafer Şenalp ME 521 4 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations İnitial state +X rotation -X rotation

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Below figure shows +X +Y, and +Z rotation directions. Dr. Ahmet Zafer Şenalp ME 521 5 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations İnitial state +X rotation +Y rotation +Z rotation

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Transformation matrices for rotation: Rotation about X axis: Rotation about Y axis: Rotation about Z axis: Dr. Ahmet Zafer Şenalp ME 521 6 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

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Transformation matrix for scaling: In below figure mirror images of two objects are onbtained by scaling. Dr. Ahmet Zafer Şenalp ME 521 7 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

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Inverse transforms are obtained by placing negative values: Inverse of Translation: Inverse of Rotation abou X: Inverse of Rotation about Y: Inverse of Rotation about Z: Inverse transform of T matrix is represented by T -1. Dr. Ahmet Zafer Şenalp ME 521 8 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

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or Concatenation of operations: Dr. Ahmet Zafer Şenalp ME 521 9 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations

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Performing the procedure must comply with the order of concatenation operations. Below example shows the difference between rotations about first +X and then +Y and the reverse. Dr. Ahmet Zafer Şenalp ME 521 10 GYTE-Makine Mühendisliği Bölümü 10. 3D Coordinate Operations İnitial state +X rotation İnitial state +Y rotation +X rotation

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Elementary 3D Transformations - a "Graphics Engine" Transformation procedures Transformations of coordinate systems Translation Scaling Rotation.

Elementary 3D Transformations - a "Graphics Engine" Transformation procedures Transformations of coordinate systems Translation Scaling Rotation.

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