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CONTENTS

INTRODUCTION TO SCALING
  1. Two dimensional scaling
  2. Three dimensional scaling

INTRODUCTION TO 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.


Contents Introduction to scaling 2D scaling about arbitrary axis

TWO DIMENSIONAL SCALING

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.


Contents 2D Scaling 2D scaling about arbitrary axis

Inverse of 2D Scale

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.


Contents Inverse of 2D scale Example of 2D scaling

2D Scaling about an arbitrary fixed point

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

  1. Translate to
  2. Scale by
  3. Translate to
The result is


Contents 2D scaling about an arbitrary axis 3D scaling

Example

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

Scale matrix =

  1. Translate to . The matrix representation is

  2. Scale by

  3. 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,


Contents 2D 3D

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


Contents 3D

3D scaling about an arbitrary fixed point

Scaling about a fixed point is given by

To scale about an arbitrary axis,

  1. Translate the fixed point to
  2. Scale by
  3. Translate the fixed point to its original position. It can be represented as a composite matrix

    where T represents translation and S the scaling.


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Priya Asokarathinam /ADVISOR Shoaff