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3D Transformation in Computer Graphics
The document provides an introduction to 3D transformations, including translation, rotation, scaling, reflection, and shearing, essential for modeling and viewing objects in computer graphics. It explains the mathematical foundations such as homogeneous coordinates and transformation matrices used to manipulate 3D objects. Key transformations and their matrix representations are discussed, highlighting their significance in object manipulation within a three-dimensional space.
In this document
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Overview of 3D transformation concepts including translation, rotation, scaling, reflections, and shearing.
Transformations are essential in computer graphics for positioning, shaping, and viewing objects.
Details on 3D transformations, including the use of the z-coordinate and the expansion from 2D.
Representation of 3D points as column vectors and the benefits of homogeneous coordinates.
Explains 3D translation by defining points in homogeneous coordinates and its matrix representation.
Covers specifications for 3D rotation around axes, including methods for determining angles.
Describes how to change object size using scaling and the process of scaling about fixed points.
Discusses 3D reflections, including techniques for reflecting across axes and their matrix forms.
Explains shearing transformations, methods for distorting shapes, and their application in perspectives.
Concludes the presentation, thanking the audience for their participation.
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3D Transformation in Computer Graphics
- 1. Welcome To The Presentation WorldUniversity Of Bangladesh 3D Transformation
- 2. INTRODUCTION Here we introduceto about 3D Transformation TRANSLATION ROTATION SCALINGREFLECTIONS SHEARING
- 3. OBJECTIVE To understand basic conventionsfor object transformations in 3D To understand basic transformations in 3D including Translation, Rotation, Scaling To understand other transformations like Reflection, Shear
- 4. Transformations are afundamental part of the computer graphics. Transformations are the movement of the object in Cartesian plane . Transformation
- 5. • Transformation areused 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 Why we use transformation
- 6. Three Dimensional Transformations Whenthe transformation takes place on a 3D plane , it is called 3D transformation. Methods for object modeling transformation in three dimensions are extended from two dimensional methods by including consideration for the z coordinate.
- 7. Three Dimensional Modeling Transformations •Generalize from 2D by including z coordinate • Straightforward for translation and scale, rotation more difficult • Homogeneous coordinates: 4 components • Transformation matrices: 4×4 elements
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- 9. 3D Point • Wewill consider points as column vectors. Thus, a typical point with coordinates (x, y, z) is represented as: z y x
- 10. 3D Point HomogenousCoordinate • We don't lose anything • The main advantage: it is easier to compose translation and rotation • Everything is matrix multiplication 1 z y x
- 11. 3D Coordinate Systems RightHand coordinate system: Left Hand coordinate system:
- 12. 3D Transformation In homogeneouscoordinates, 3D transformations are represented by 4×4 matrixes: 1000 z y x tihg tfed tcba
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- 14. 3D translation • Anobject is translated in 3D dimensional by transforming each of the defining points of the objects. • 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
- 15. 3D translation • Thematrix 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. • Translate an object by translating each vertex in the object.
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- 17. 3D Rotation In general,rotations are specified by a rotation axis and an angle. In two- dimensions there is only one choice of a rotation axis that leaves points in the plane.
- 18. 3D Rotation Theeasiest rotation axes are those that parallel to the coordinate axis. Positive rotation angles produce counterclockwise rotations about a coordinate axix, if we are looking along the positive half of the axis toward the coordinate origin. fig: 3D rotation
- 19. Coordinate Axis Rotations Obtainrotations around other axes through cyclic permutation of coordinate parameters: xzyx Fig:Coordinate Axis Rotations
- 20. Coordinate Axis Rotations Z-axisrotation: For z axis same as 2D rotation: x’=x*cos θ-y*sin θ Y’=x*sin θ +y*cos θ Z’=z 11000 0100 00cossin 00sincos 1 ' ' ' z y x z y x PRP )(z Fig : Z-axis rotation
- 21. Coordinate Axis Rotations 11000 0cossin0 0sincos0 0001 1 ' ' ' z y x z y x X-axisrotation: Y’=y*cos θ -z*sin θ Z’=z*sin θ +x*cos θ X’=x PRP )(x Fig : X-axis rotation
- 22. Coordinate Axis Rotations 11000 0cos0sin 0010 0sin0cos 1 ' ' ' z y x z y x PRP )(y Y-axis rotation: Z’=z*cos θ -x*sin θ X’=z*sin θ +x*cos θ Y’=y Fig : Y-axis rotation
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- 24. 3D Scaling You canchange the size of an object using scaling transformation . In the scaling process , you either expand or compress the dimensions of the object . Scaling can be achieved by multiplying the original coordinates of the object with scaling factor to get the desired result.
- 25. 3D Scaling About origin:Changes the size of the object and repositions the object relative to the coordinate origin. where Sx = scale factor in the x direction, Sy = scale factor in the y direction, and Sz = scale factor in the z direction. Fig: Scaling 11000 000 000 000 1 z y x s s s z y x z y x PSP
- 26. 3D Scaling About anyfixed point: Scaling with respect to an arbitrary fixed point is not as simple as scaling with respect to the origin . The procedure of scaling with respect to an arbitrary fixed point is: Translate the object so that the fixed point coincides with the origin. Scale the object with respect to the origin. Use the inverse translation of step 1 to return the objects to its original position.
- 27. 3D Scaling About anyfixed point: fig : fixed point scaling 1000 )1(00 )1(00 )1(00 ),,(),,(),,( fzz fyy fxx fffzyxfff zss yss xss zyxssszyx TST The corresponding composite transformation matrix is:
- 28. 3d scaling • Theequations for scaling : x’ = x . sx Ssx,sy,sz y’ = y . sy z’ = z . sz fig name: After scaling
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- 30. 3D Reflections About anaxis:equivalent to 180˚rotation about that axis.
- 31. 3D reflection • Reflectionin 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 . Fig: reflection
- 32. 3d reflection Reflection aboutx-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 Fig: X axis reflection Fig:Y axis reflection
- 33. 3D reflection • Thematrix 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 Fig: Z axis reflection
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- 35. 3D Shearing A transformationthat distorts the shape of an object such that the transformed shape appears as if the object were composed of internal layers that had been caused to slide over each other is called a shearing.
- 36. 3D Shearing • Intwo dimensions, transformations relative to the x or y axes to produce distortions in the shapes of objects. In three dimensions, we can also generate shears relative to the z axis. fig: before shearing fig: after shearing
- 37. 3D Shearing Modifyobject shapes Useful for perspective projections: 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
- 38. SHEARING ABOUT XYAXIS • Parameters a and b can be assigned any real values. The effect of this transformation matrix is to alter x- and y-coordinate values by an amount that is proportional to the z value, while leaving the z coordinate unchanged. • Boundaries of planes that are perpendicular to the z axis are thus shifted by an amount proportional to z. An example of the effect of this shearing matrix on a unit cube is shown in Fig., for shearing values a=b=1. Shearing matrices for the x axis and y axis are defined similarly.
- 39. In space, wedivide shear transformation according to the direction of the surfaces xy,xz and yz. Values of Sx,Sy and Sz determine shear transformation sizes for all the directions. A shear transformation about the xy plane : | 1 0 0 0 | Axy = | 0 1 0 0 | | Sx Sy 0 0| | 0 0 0 1 | A shear matrix about the xz plane : | 1 0 0 0 | Axz = | Sx 1 Sz 0| | 0 1 1 0 | | 0 0 0 1 | A shear matrix about the yz plane : | 1 Sy Sz 0 | | 0 1 0 0 | Ayz = | 0 0 1 0 | | 0 0 0 1 |
- 41. Thank you somuch for being with us up to now