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
Scale 4 in x, 3 in y, leaving point fixed. Let the object be a square as shown in the figure below.
Scale matrix =
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
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,
where T represents translation and S the scaling.