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