Original subtitles

There are several ways to round or repeat values, which among other things,

is useful when working with discrete and periodic textures, such as tiles.

Perhaps the most familiar of these operations is the Round function itself,

which behaves like a classic rounding operation,

simply outputting the integer closest to the input value, whether it is above or below the value.

On the other hand, the Floor operation always rounds down, to the closest integer below,

even if the integer above is closer to the input value.

And conversely, the Ceiling operation rounds up to the first integer above.

If we look at these operations on a larger range of values, it becomes clear that they

are all basically offset versions of each other. Of these operations,

Floor is probably the most common, as it has some convenient properties, that we'll see in a moment.

While the three rounding operations we looked at discard the decimal portion of the values in

one way or another, we also have operations to do the opposite. The Fraction operation is very

convenient for this. It outputs the distance to the input value from the first integer below.

That effectively discards the integer portion, and keeps the decimal,

though that is not exactly the case for negative numbers, as the Fraction operation always

counts up from the first integer below, while decimals count from the integer closest to zero.

The nice property of the Floor and Fraction operations, is that one outputs exactly what

the other discards, the integer and decimal portions of the value, respectively. So they

complement each other, and adding their outputs together always gives you back the original input.