Homogeneous Representation: rotating, shearing
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The document discusses geometric transformations, focusing on 2D and 3D operations such as rotation, shearing, and reflection about arbitrary points and lines. It explains how transformations modify a graphic image through changes in the object's database and details the procedures for achieving combined transformations using matrices. Additionally, the document covers the mechanics of shearing in both two-dimensional and three-dimensional spaces.
![HOMOGENEOUS
REPRESENTATIONS
Many a times it becomes necessary to combine the
above mentioned individual transformations in order
to achieve the required results. In such cases the
combined transformation matrix can be obtained by
multiplying the respective transformation matrices.
However, care should to be taken that the order of the
matrix multiplication be done in the same way as that
of the transformations as follows.
[P*] = [Tn] [Tn – 1] [Tn – 2+ ……..*T3] [T2] [T1]](https://image.slidesharecdn.com/homogeneousrepresentmanthan-170403090521/85/Homogeneous-Representation-rotating-shearing-4-320.jpg)
Homogeneous Representation: rotating, shearing
- 2. CONTENT GEMETRIC TRANSFORMATION HOMOGENEOUS REPRESENTATION 2D AND 3D ROTATING AND SHEARING ROTATING ABOUT ARBITARY POINT AND LINE SHEARING ABOUT X(2D) ,XY(3D), AXIS
- 3. GEOMETRIC TRANSFORMATION All changes performed on the graphic image are done by changing the database of the original picture. These changes are called as transformations. Transformations allow the user to uniformly change the entire picture. An object created by the user is stored in the form of a database. If the database, which represents the object, is changed, the object also would change. This method is used to alter the orientation, scale, position of the drawing.
- 4. HOMOGENEOUS REPRESENTATIONS Many a times it becomes necessary to combine the above mentioned individual transformations in order to achieve the required results. In such cases the combined transformation matrix can be obtained by multiplying the respective transformation matrices. However, care should to be taken that the order of the matrix multiplication be done in the same way as that of the transformations as follows. [P*] = [Tn] [Tn – 1] [Tn – 2+ ……..*T3] [T2] [T1]
- 5. ROTATION ABOUT POINT The transformation given earlier for rotation is about the origin of the axes system. It may sometimes be necessary to get the rotation about any arbitrary base point as shown in Fig. 4.5. To derive the necessary transformation matrix, the following complex procedure comprising the followin1g three points would be required. Translate the point A to O, the origin of the axes system. Rotate the object by the given angle. Translate the point back to its original position. The transformation matrices for the above operations in the given sequence are the following.
- 6. ROTATION
- 7. Similar to the above, there are times when the reflection is to be taken about an arbitrary line as shown in Fig. 4.6. Translate the mirror line along Y axis such that line passes through the origin, O. Rotate the mirror line such that it coincides with the X axis. Mirror the object through the X axis. Rotate the mirror line back to the original angle with X axis. Translate the mirror line along the Y axis back to the original position. ROTATION ABOUT LINE
- 8. ROTATION
- 9. SHEARING A transformation that 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 shear.
- 10. 2D SHEARING Two common shearing transformations are those that shift coordinate x values and those that shift y values. An x-direction shear relative to the x axis is produced with the transformation matrix Any real number can be assigned to the shear parameter shx. A coordinate position (x, y) is then shifted horizontally by an amount proportional to its distance (y value) from the x axis (y = 0). Setting shx to 2, for example, changes the square in Fig. into a parallelogram. Negative values for shx shift coordinate positions to the left.
- 11. SHEARING ABOUT X AXIS
- 12. SHEARING ABOUT XY AXIS •We can generate x-direction shears relative to other reference lines with coordinate positions transformed as x’ = x + shx (y – yref), y’ = y ……..(eq) •An example of this shearing transformation is given in Fig. for a shear parameter value of 1 /2 relative to the line yref = –1.
- 13. 3D SHEARING In two 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.
- 14. SHEARING ABOUT XY AXIS 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.