3D Geometric Transformations
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The document discusses 3D geometric transformations including rotation and reflection. It explains that transformations move points in space and can be expressed through 4x4 matrices. Specifically, it covers rotating objects around axes, conventions for right-handed vs left-handed systems, and how to perform rotations around arbitrary axes through a series of translations and rotations. It also discusses reflecting objects across planes by treating it as a scaling by -1 along one axis and provides AutoCAD commands for rotations and reflections.
3D Geometric Transformations
- 1. 3D Geometric Transformations Ishan Parekh MBA(tech.) Manufacturing #315 Ishan Parekh MBA(tech.) Manufacturing #315 1
- 2. Kinds of Transformations Rotation Reflection • The process of moving points in space is called transformation • These transformations are an important component of computer graphics programming • Each transformation type can be expressed in a (4 x 4) matrix, called the Transformation Matrix NOTES: Ishan Parekh MBA(tech.) Manufacturing #315 2
- 3. Rotation Rotating the object about an axis Ishan Parekh MBA(tech.) Manufacturing #315 3
- 4. Rotation Rotation is the process of moving a point in space in a non-linear manner it involves moving the point from one position on a sphere whose center is at the origin to another position on the sphere Rotation a point requires: 1) The coordinates for the point. 2) The rotation angles. Ishan Parekh MBA(tech.) Manufacturing #315 4
- 5. 3D Rotation Convention Right-handed Cartesian coordinates x y z x y z Positive rotation goes counter-clockwise looking in this direction x y z Left-handed: Ishan Parekh MBA(tech.) Manufacturing #315 5
- 6. 3D Rotation Rotation about the z axis cos q - sin q 0 0 sin q cos q 0 0 0 0 1 0 0 0 0 1 x’ y’ z’ 1 x y z 1 = x y z Ishan Parekh MBA(tech.) Manufacturing #315 6
- 7. Rotation Rotation about x-axis (i.e. in yz plane): x′ = x y′ = y cosθ – z sinθ z′ = y sinθ + z cosθ Rotation about y-axis (i.e. in xz plane): x′ = z sinθ + x cosθ y ′ = y z′ = z cosθ – x sinθ Ishan Parekh MBA(tech.) Manufacturing #315 7
- 8. 3D rotation around axis parallel to coordinate axis Translate object so that rotation axis aligned with coordinate axis Rotate about that axis Translate back Ishan Parekh MBA(tech.) Manufacturing #315 8
- 9. 3D rotation around any axis Translate object so that rotation axis passes through coordinate origin Rotate object so that axis of rotation coincides with coordinate axis Perform rotation Inverse rotate so that rotation axis goes back to original orientation Inverse translate so that rotation axis goes back to original position x y z y’ x’ z’ (x0,y0,z0) 3D rotation around arbitrary axis: Given: (x0,y0,z0) and vector v Method: (1) Let v be z’ (2) Derive y’ and x’ (3) Translate by (-x0,-y0,-z0) (4) Rotate to line up x’y’z’ with xyz axes (see next page), call this the rotation R. (5) Rotate by q about the z axis (6) Rotate back (R-1) (7) Translate by (x0,y0,z0) Ishan Parekh MBA(tech.) Manufacturing #315 9
- 10. Rotations – Positive and Negative Ishan Parekh MBA(tech.) Manufacturing #315 10
- 11. Rotations – Positive about x,y,z Ishan Parekh MBA(tech.) Manufacturing #315 11
- 12. Multiple Rotations Ishan Parekh MBA(tech.) Manufacturing #315 12
- 13. Reflection Mirroring an object about a plane Ishan Parekh MBA(tech.) Manufacturing #315 13
- 14. Reflection A three-dimensional reflection can be performed relative to a selected reflection axis or with respect to a selected reflection plane. three dimensional reflection matrices are set up similarly to those for two dimensions. Reflections relative to a given axis are equivalent to 180 degree rotations. Ishan Parekh MBA(tech.) Manufacturing #315 14
- 15. 3D reflection Let z-plane be the reflection plane Comment: Reflection is like negative scaling Then, transformation matrix is: 1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 1 Ishan Parekh MBA(tech.) Manufacturing #315 15
- 16. AutoCAD Transformation Commands Geometric Transformation AutoCAD Command Rotation ROtate and ROTATE3D Reflection MIrror and MIRROR3D Ishan Parekh MBA(tech.) Manufacturing #315 16
- 17. 3D Transformations The End Ishan Parekh MBA(tech.) Manufacturing #315 17