2D Scaling

Theory of Computation & Computer Graphics — Two-Dimensional Transformation, NEC licence examination syllabus (Nepal Engineering Council).

2D Scaling: making a shape bigger or smaller

Scale factors above 1 grow it, below 1 shrink it — and the two axes don't have to match.

Scaling matrix (about the origin, factors sx, sy): [x'] [sx 0 0] [x] [y'] = [0 sy 0] [y] [1 ] [0 0 1] [1] Worked example: scale point (3,4) by sx=2, sy=3 x' = 2(3) = 6 y' = 3(4) = 12 New point: (6, 12)

x' = x·sx, y' = y·sy With sx = 1.5, sy = 1.5, x' = 3, area = 2.25.

Keep sx = sy for uniform scaling; make them differ and see why scaling is not the same as zooming.

Fixed-point scaling scales relative to some point OTHER than the origin — done by translating that fixed point to the origin, scaling normally, then translating back (this exact three-step pattern reappears constantly in the Composite Transformation topic).

💡 One-liner: "What happens if sx ≠ sy?" → The shape distorts (stretches/squashes unevenly) instead of scaling uniformly — sx=sy is required for proportional scaling.

Why scaling also moves the shape

Scaling multiplies coordinates, so it does not merely resize — it also pushes the shape away from the origin when factors exceed 1, and pulls it toward the origin when they are below 1. A point at x = 6 with sx = 2 moves to x = 12, six units further out, while a point at x = 1 moves only one unit.

So scaling about the origin changes both size and position. To scale a shape in place, use fixed-point scaling: translate the fixed point to the origin, scale, translate back. This is the same three-step sandwich as rotation about an arbitrary point, and recognising that these two problems are the same problem saves real time in an exam.

💡 Uniform scaling (sx = sy) preserves shape and angles — it is a genuine zoom. Differential scaling (sx ≠ sy) distorts: a circle becomes an ellipse and a square becomes a rectangle. Questions frequently hinge on which one is being asked for.

Properties worth stating

The determinant is sx·sy, and that is exactly the factor by which area multiplies — scale by 2 in both directions and the area quadruples, not doubles. A negative factor reflects as well as scales, which is why reflection is treated as a special case of scaling. The inverse is scaling by (1/sx, 1/sy), so a zero scale factor has no inverse: the shape collapses onto a line and the information is gone for good.

Syllabus points

Scaling matrix; fixed-point scaling

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