What is concatenation of transformations? Explain the following considering 2D: i) Rotation about a fixed point ii) Scaling about a fixed point.

4.C] What is concatenation of transformations? Explain the following considering 2D:
i) Rotation about a fixed point
ii) Scaling about a fixed point.

Answer:

Concatenation of transformations

In computer graphics, concatenation of transformations refers to the process of combining multiple transformation operations into a single transformation. This is crucial for efficiently managing and applying transformations to objects in a 2D or 3D space.

i) Rotation about a fixed point

Rotation transformation of an object is generated by specifying a rotation axis and a rotation angle. All points of the object are then transformed to new positions by rotating the points through the specified angle about the rotation axis.

A two-dimensional rotation of an object is obtained by repositioning the object along a circular path in the xy plane. Parameters for the two-dimensional rotation are the rotation angle θ and a position(x_r,y_r), called the rotation point (or pivot point), about which the object is to be rotated.

A positive value for the angle θ defines a counterclockwise rotation about the pivot point. A negative value for the angle θ rotates objects in the clockwise direction.

Figure: Rotation of an object through angle θ about the pivot point (x_r,y_r )

1) Determination of the transformation equations for rotation of a point position P when the pivot point is at the coordinate origin.

The angular and coordinate relationships of the original and transformed point positions are shown in Figure.

r is the constant distance of the point from the origin, angle f is the original angular position of the point from the horizontal, and θ is the rotation angle.

Using standard trigonometric identities, transformed coordinates can be expressed in terms of angles of q and f

x’=rcos(f+θ) =rcosfcosθ-rsinfsinθ   —– (1)

y’=rsin(f+θ) =rcosfsinθ+rsinfcosθ

The original coordinates of the point in the polar coordinates are x=rcosf y=rsinf —— (2)

Substituting expressions (1) and (2) we obtain the transformation equations for rotating a point at position (x, y) through an angle θ about the origin.

x’=xcosq-ysinq

y’=xsinq+ycosq

The rotation equation in matrix form is written as P’=RP, where the rotation matrix is

2) Determination of the transformation equations for rotation of a point position P when the pivot point is at (x_r,y_r).

Rotation of a point about an arbitrary pivot position is illustrated in Figure.

Using the trigonometric relationships indicated by the two right triangles in this figure, we can generalize Equations 6 to obtain the transformation equations for rotation of a point about any specified rotation position (x_r,y_r ):

x¢= x_r +(x- x_r)cosq – (y- y_r)sinq

y¢= y_r +(x- x_r) sinq + (y- y_r)cosq

Two-dimensional rotation transformation equations about the coordinate origin can be expressed in the matrix form as

ii) Scaling about a fixed point.

To alter the size of an object, scaling transformation is used. A two dimensional scaling operation is performed by multiplying object positions (x,y) by scaling factors S_x and S_y to produce the transformed coordinates(x’,y’).

x’=x.S_x, y’=y.S_y

Scaling factor S_x  scales an object in the x direction, while scales S_y in the y direction. The basic two-dimensional scaling equations 10 can also be written in the following matrix form:

or

P’=S.P

where S is the 2 × 2 scaling matrix.

A scaling transformation relative to the coordinate origin can be expressed as the matrix multiplication.