BASIC TRANSFORMATION
Animation are produced by moving the 'camera' or the objects in a scene along animation paths. Changes in orientation, size and shape are accomplished with geometric transformations that alter the coordinate descriptions of the objects. The basic geometric transformations are translation, rotation, and scaling. Other transformations that are often applied to objects include reflection and shear.
Use of transformations in CAD
In mathematics, "Transformation" is the elementary term used for a variety of operation such as rotation, translation, scaling, reflection, shearing etc. CAD is used throughout the engineering process from conceptual design and layout, through detailed engineering and analysis of components to definition of manufacturing methods. Every aspect of modeling in CAD is dependent on the transformation to view model from different directions we need to perform rotation operation. To move an object to a different location translation operation is done. Similarly Scaling operation is done to resize the object.
Coordinate Systems
In CAD three types of coordinate systems are needed in order to input, store and display model geometry and graphics. These are the Model Coordinate System (MCS), the World Coordinate System (WCS) and the Screen Coordinate System (SCS).
Model Coordinate System
The MCS is defined as the reference space of the model with respect to which all the model geometrical data is stored. The origin of MCS can be arbitrary chosen by the user.
World Coordinate System
As discussed above every object have its own MCS relative to which its geometrical data is stored. Incase of multiple objects in the same working space then there is need of a World Coordinate System which relates each MCS to each other with respect to the orientation of the WCS. It can be seen by the picture shown below.
Screen Coordinate System
In contrast to the MCS and WCS the Screen Coordinate System is defined as a two dimensional device-dependent coordinate system whose origin is usually located at the lower left corner of the graphics display as shown in the picture below. A transformation operation from MCS coordinates to SCS coordinates is performed by the software before displaying the model views and graphics.
Viewing Transformations
As discussed that the objects are modeled in WCS, before these object descriptions can be projected to the view plane, they must be transferred to viewing coordinate system. The view plane or the projection plane, is set up perpendicular to the viewing zv axis. The World coordinate positions in the scene are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane.
The transformation sequence to align WCS with Viewing Coordinate System is.
1. Translate the view reference point to the origin of the world coordinate system.
2. Apply rotations to align xv, yv, and zv with the world xw, yw and zw axes, respectively.
 |  |
TRANSLATION
A translation is applied to an object by repositioning it along a straight line path from one coordinate location to another. We translate a two-dimensional point by adding translation distances, tx and ty, to the original coordinate position (x,y) to move the point to a new position (x',y')
The translation distance pair (tx, ty) is called translation vector or shift vector
Matrix representation of translation
This allows us to write the two-dimensional translation equations in the matrix form:
| |
ROTATION
A two-dimensional rotation is applied to an object by repositioning it along a circular path in the x-y plane. When we generate a rotation we get a rotation angle (θ) and the position about which the object is rotated (xr , yr) this is known as rotation point or pivot point. The transformation can also be described as a rotation about rotation axis that is perpendicular to x-yplane and passes through the pivot point. Positive values for the rotation angle define counter-clockwise rotations about the pivot point and the negative values rotate objects in the clockwise direction.
Suppose the pivot point be at origin, to understand the relationship between angular and coordinate points of original and transformed position lets look at the figure below:
Here, r - constant distance of the point from the origin.
Φ - original angular position of the point from the horizontal
θ - rotation angle
we can express the transformation by the following equations
we know the coordinate of x and y in polar form
on expanding and equating we get
The same equations we can write in matrix form as
Where the rotation matrix R is
Hence it is
| |
SCALING
Scaling is a kind of transformation in which the size of an object is changed. Remember the change is size does no mean any change in shape. This kind of transformation can be carried out for polygons by multiplying each coordinate of the polygon by the scaling factor. Sx and Sy which in turn produces new coordinate of (x,y) as (x',y'). The equation would look like
or
here S represents the scaling matrix.
NOTE: If the values of scaling factor are greater than 1 then the object is enlarged and if it is less that 1 it reduces the size of the object. Keeping value as 1 does not changes the object.
Uniform Scaling: To achieve uniform scaling the values of scaling factor must be kept equal.
Differential Scaling: Unequal or Differential scaling is produce incases when values for scaling factor are not equal.
As per usual phenomenon of scaling an object moves closer to origin when the values of scaling factor are less than 1. To prevent object from moving or changing its position while is scaling we can use a point that is would be fixed to its position while scaling which is commonly referred as fixed point (xf yf).
| |
REFLECTION
Reflection is nothing more than a rotation of the object by 180o. In case of reflection the image formed is on the opposite side of the reflective medium with the same size. Therefore we use the identity matrix with positive and negative signs according to the situation respectively.
The reflection about the x-axis can be shown as:
The reflection about the y-axis can be shown as:
REFLECTION ABOUT A ORIGIN
When both the x and y coordinates are flipped then the reflection produced is relative to an axis that is perpendicular to x-yplane and that passes through the coordinate origin. This transformation is referred as a reflection relative to coordinate origin and can be represented using the matrix below.
REFLECTION ABOUT AN ARBITRARY LINE
Reflection about any line y= mx + c can be accomplished with a combination of translate-rotate-reflect transformations.
Steps are as follows
1. Translate the working coordinate system (WCS) so that the line passes through the origin.
2. Rotate the WCS such that one of the coordinate axis lies onto the line.
3. Reflect about the aligned axis
4. Restore the WCS back by using the inverse rotation and translation transformation.
REFLECTION ABOUT AN ARBITRARY POINT
As seen in the example above, to reflect any point about an arbitrary point P (x,y) can be accomplished by translate-reflect transformation i.e. the origin is first translated to the the arbitrary point and then the reflection is taken about the origin. And finally the origin is translated back to its original position.
The whole process can be visualized using the animation below.
| |
HOMOGENEOUS COORDINATES
We have seen that basic transformations can be expressed in matrix form. But many graphic application involve sequences of geometric transformations. Hence we need a general form of matrix to represent such transformations. This can be expressed as:
Where P and P' - represent the row vectors.
T1 - is a 2 by 2 array containing multiplicative factors.
T2 - is a 2 element row matrix containing translation terms.
We can combine multiplicative and translational terms for 2D geometric transformations into a single matrix representation by expanding the 2 by 2 matrix representations to 3 by 3 matrices. This allows us to express all transformation equations as matrix multiplications, providing that we also expand the matrix representations for coordinate positions. To express any 2D transformations as a matrix multiplication, we represent each Cartesian coordinate position (x,y) with the homogeneous coordinate triple (xh,yh,h),
such that
Thus, a general homogeneous coordinate representation can also be written as (h.x, h.y, h). For 2D geometric transformations, we can choose the homogeneous parameter h to any non-zero value. Thus, there is an infinite number of equivalent homogeneous representations for each coordinate point (x,y). A convenient choice is simply to h=1. Each 2D position is then represented with homogeneous coordinates (x,y,1). Other values for parameter h are needed, for eg, in matrix formulations of 3D viewing transformations.
Expressing positions in homogeneous coordinates allows us to represent all geometric transformation equations as matrix multiplications. Coordinates are represented with three element row vectors and transformation operations are written as 3 by 3 matrices.
For Translation, we have
or
Similarly for Rotation transformation, we have
or
Finally for Scaling transformation, we have
or
|
 | |
TRANSLATION
In translation, an object is displayed a given distance and direction from its original position. If the displacement is given by the vector the new object point P'(x',y') can be found by applying the transformation Tv to P(x,y)
where
and
| |
ROTATION
A generalized rotation about an arbitrary point (a,b) can be obtained by performing the following transformations
Translate the object so that the pivot-point position is moved to coordinate origin.
Rotate the object about the coordinate origin.
Translate the object so that the pivot point is returned to its original position.
The composite matrix for this sequence is obtained as:
where,
|
| |
SCALING
The animation illustrates a transformation sequence to produce scaling with respect to a selected fixed position (a,b) using a scaling function that can only scale relative to the coordinate origin.
Translate object so that the fixed point coincides with the coordinate origin.
Scale the object with respect to the coordinate origin.
Use the inverse translation of step 1 to return the object to its original position.
The composite matrix for this sequence is obtained as:
where,
|
|
|
|
|
|
|
