Shifting & Scaling Numbers
“Shifting and scaling” is another way to say adding and multiplying. But more, it describes the effect of adding and multiplying on ranges of numbers. In the tutorial a range of numbers is denoted “x:10->15” which means that the variable x goes from 10 to 15.
To shift a range of numbers means to add a number (positive or negative) to it. This moves the range, but doesn’t change the distance between the values.
To scale a range of numbers means to multiply it by a number. This does change the distance between the values, and also changes the end points unless they are at zero.
- Shift the source range to 0 by subtracting A
- Scale to the target range size.
The number to multiply by is (D-C)/(B-A) which is the ratio of the range sizes.
- Shift the range to the desired place by adding C.
The final form, then, of an equation that transforms x:A->B to y:C->D is:
y = (x – A) * (D – C)/(B – A) + C
Using Shifting & Scaling in Animation
Ranges of numbers are worked with often in animation. For example, the loop variable in a “repeat with” loop covers a range of numbers. The possible locations of an adjustable slider bar, the time elapsed since a certain start time, the possible mouse locations, etc, all cover ranges of numbers.
As an example of how shifting & scaling is used, say you’ve got a slider bar that moves horizontally and has a range loch:72->178. You want the slider to control the rotation of a sprite with the range rot:1125->-45. According to the steps above, you would:
- Subtract 72 to shift the range to zero.
loch – 72
- Scale the range, multiplying by targetRangeSize/sourceRangeSize
This would be (-45 – 1125)/(178 – 72) = -1170/106.0
(loch – 72) * -1170/106.0
- Shift the range by adding targetRangeStart
(loch – 72) * -1170/106.0 + 1125
The resulting expression is a linear function.
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