3d transformation computer graphics
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This document provides an overview of 3D transformations, including translation, rotation, scaling, reflection, and shearing. It explains that 3D transformations generalize 2D transformations by including a z-coordinate and using homogeneous coordinates and 4x4 transformation matrices. Each type of 3D transformation is defined using matrix representations and equations. Rotation is described for each coordinate axis, and reflection is explained for each axis plane. Shearing is introduced as a way to modify object shapes, especially for perspective projections.
3d transformation computer graphics
- 2. CONTENTS Transformation Types of transformation Why we use transformation 3D Transformation 3D Translation 3D Rotation 3D Scaling 3D Reflection 3D Shearing
- 3. TRANSFORMATION Transformations are a fundamental part of the computer graphics. Transformations are the movement of the object in Cartesian plane .
- 4. TYPES OF TRANSFORMATION There are two types of transformation in computer graphics. 1) 2D transformation 2) 3D transformation Types of 2D and 3D transformation 1) Translation 2) Rotation 3) Scaling 4) Shearing 5) Mirror reflection
- 5. WHY WE USE TRANSFORMATION Transformation are used to position objects , to shape object , to change viewing positions , and even how something is viewed. In simple words transformation is used for 1) Modeling 2) viewing
- 6. 3D TRANSFORMATION When the transformation takes place on a 3D plane .it is called 3D transformation. Generalize from 2D by including z coordinate Straight forward for translation and scale, rotation more difficult Homogeneous coordinates: 4 components Transformation matrices: 4×4 elements 1000 z y x tihg tfed tcba
- 7. 3D TRANSLATION Moving of object is called translation. In 3 dimensional homogeneous coordinate representation , a point is transformed from position P = ( x, y , z) to P’=( x’, y’, z’) This can be written as:- Using P’ = T . P 11000 100 010 001 1 z y x t t t z y x z y x
- 8. 3D TRANSLATION The matrix representation is equivalent to the three equation. x’=x+ tx , y’=y+ ty , z’=z+ tz Where parameter tx , ty , tz are specifying translation distance for the coordinate direction x , y , z are assigned any real value.
- 9. 3D ROTATION Where an object is to be rotated about an axis that is parallel to one of the coordinate axis, we can obtain the desired rotation with the following transformation sequence. Coordinate axis rotation Z- axis Rotation(Roll) Y-axis Rotation(Yaw) X-axis Rotation(Pitch)
- 10. COORDINATE AXIS ROTATION Obtain rotations around other axes through cyclic permutation of coordinate parameters: xzyx
- 11. X-AXIS ROTATION The equation for X-axis rotation x’ = x y’ = y cosθ – z sinθ z’ = y sinθ + z cosθ 11000 0cossin0 0sincos0 0001 1 ' ' ' z y x z y x
- 12. Y-AXIS ROTATION The equation for Y-axis rotaion x’ = x cosθ + z sinθ y’ = y z’ = z cosθ - x sinθ 11000 0cos0sin 0010 0sin0cos 1 ' ' ' z y x z y x
- 13. The equation for Y-axis rotaion x’ = x cosθ – y sinθ y’ = x sinθ + y cosθ z’ = z Z-AXIS ROTATION 11000 0100 00cossin 00sincos 1 ' ' ' z y x z y x
- 14. 3D SCALING Changes the size of the object and repositions the object relative to the coordinate origin. 11000 000 000 000 1 z y x s s s z y x z y x
- 15. 3D SCALING The equations for scaling x’ = x . sx Ssx,sy,sz y’ = y . sy z’ = z . sz
- 16. 3D REFLECTION Reflection in computer graphics is used to emulate reflective objects like mirrors and shiny surfaces Reflection may be an x-axis y-axis , z-axis. and also in the planes xy-plane,yz-plane , and zx-plane. Reflection relative to a given Axis are equivalent to 180 Degree rotations
- 17. 3D REFLECTION Reflection about x-axis:- x’=x y’=-y z’=-z 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 1 Reflection about y-axis:- y’=y x’=-x z’=-z
- 18. 3D REFLECTION The matrix for reflection about y-axis:- -1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 1 Reflection about z-axis:- x’=-x y’=-y z’=z -1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1
- 19. 3D SHEARING Modify object shapes Useful for perspective projections When an object is viewed from different directions and at different distances, the appearance of the object will be different. Such view is called perspective view. Perspective projections mimic what the human eyes see.
- 20. 3D SHEARING E.g. draw a cube (3D) on a screen (2D) Alter the values for x and y by an amount proportional to the distance from zref
- 21. 3D SHEARING Matrix for 3d shearing Where a and b can Be assigned any real Value.
- 22. Thank You