INTRODUCTION TO SCALING
- Two dimensional scaling
- Three dimensional scaling

The process of altering the dimensions of an object,
*i. e. * enlarging or reducing the size of an
object is called scaling. The object is scaled by
multiplying the coordinates of each endpoint
by the scaling factor. The lengths and the distances from the
origin are scaled. For enlarging an object the
scaling constant should be greater than 1, whereas, for
compressing it is less than 1.

Let be the scale in the positive **x** and **y**
directions respectively.
Then the scaled vertex is given by

If , then it is said to be *homogenous*
or *uniform* scaling transformation that mainatains
relative propoortions of the scaled objects.
The magnification factor is **|s|**. All points move **s**
times away from the origin. If **|s| < 1**, all the points
move towards the origin, or demagnified.

If **s** is negative, reflections occur. Pure reflections
occur when both the scaling factors have the value
**+1** or **-1**. For example, if ,
the image is flipped about the y-axis, replacing
each point on **x** to **-x** and scaled by **2** in
the y direction.

If , then the scaling is called
*differential scaling*. When either
or equals one, simplest differential scaling
called strain occurs.
2-Dimensional scaling in the matrix form is given by

Lengths and distances from the origin are scaled by
multiplying the coordinates of each endpoint by
the scaling factors.

The scaling discussed so far has the origin
as the fixed point and scaling is about the origin.
When an object is scaled, it is moved
times away from the origin. It is also possible to
have any arbitrary point as a fixed point, and
scale about that point.

Usually the origin is fixed under scaling.
However, any arbitrary fixed point can be
selected for scaling. To scale around the arbitrary point

- Translate to
- Scale by
- Translate to

The result is

Scale **4** in **x**, **3** in **y**, leaving point
fixed.
Let the object be a square as shown in the figure below.

Scale matrix =

- Translate to . The matrix representation is

- Scale by

- Translate back to

The resultant matrix is a composite matrix.

To check if is fixed even after the
transformation, add a homogenous coordinate to it
and multiply it with the composite matrix.

proves that the arbitrary point is fixed.
Whereas, for any other point say

shows that the point does not remain fixed.

It can also be proved algebraically

Therefore,

### 3D scaling

Given input vertex , and , , ,
the scale in **x, y, z** directions, respectively, the
scaled vertex is given by

Scaling in matrix form is given by

The inverse scaling in the matrix form is given by

Scaling about a fixed point is given by

To scale about an arbitrary axis,

- Translate the fixed point to
- Scale by
- Translate the fixed point to its original position.
It can be represented as a composite matrix
where **T** represents translation and **S** the scaling.

*Priya Asokarathinam /ADVISOR Shoaff *