Angles and distances

The distance to a point, the angle to it and the difference between two angles — three operations everybody needs

mediumYou need: op: arithmetic

Almost everything a processor does with the world comes down to two questions: how far and in which direction. How far is len, in which direction is angle. Neither works with points but with an offset: how far right and how far up.

steptick 0

Set0

=

Set1

=

Operation2

= len

Operation3

= angle

Operation4

= anglediff

Operation5

= sin

@counter0number

dxnullnull

dynullnull

distancenullnull

anglenullnull

gapnullnull

heightnullnull

Operation	What it computes	Answer
len	the length of the offset, that is, the distance	50
angle	the angle of that offset, from 0 to 360	53.130008697509766
angleDiff	the shortest difference of two angles, from 0 to 180	20
sin	sine — and it is in degrees	0.49999999999999994

Two of the four answers have a long tail, and that is as it should be; why, is explained below.

The offset is computed by subtraction: op sub dx toX fromX and the same for y. The processor gets coordinates from the world — how exactly is in the lesson “Properties of buildings”.

Angles count counter-clockwise from pointing right

Zero is “right”, 90 is “up”, 180 is “left”, 270 is “down”. The same order as block rotation in the game, and the same system sensor @rotation answers in.

angleDiff answers “how far apart are these two angles”, and the answer is always non-negative and no more than 180: between 350 and 10 the difference is 20, not 340. It counts correctly across zero — that is what makes it useful.

Note

All three operations take degrees, not radians. That goes for trigonometry too: the sine of thirty is a half, not −0.988 as it would be in radians. Neither @pi nor any conversion is needed for them.

The answers are slightly off from the textbook

steptick 0

Operation0

= len

Operation1

= angle

@counter0number

diagonalnullnull

behindnullnull

The diagonal of a unit square is √2, that is 1.4142135623730951. mlog answers 1.4142135381698608. The angle of “left” is 180, and mlog answers 179.99989.

This is not a bug but precision, and there are two reasons for it.

len, angle and angleDiff are computed in the game with single-precision numbers, half the usual length, and the angle itself comes from an approximate formula — hence both the truncated root and 179.99989 instead of 180. The rest of the arithmetic runs at full precision: only these three diverge.

The sine is computed at full precision, and its tail has another origin — the one from the lesson «Whole numbers and precision»: degrees are converted to radians by division, and a half does not come out even. Nought point four nines and ninety-nine is a half, simply written down honestly.

Silent failure

Hence the rule: angles and distances are not compared for exact equality. “The unit is exactly ten tiles away” will never happen; you check “closer than ten” with lessThan, and a rotation with angleDiff and “less than five degrees”.

The rest of the trigonometry

cos and tan are like sin, in degrees. The inverses — asin, acos, atan — give degrees back. None of the five has a second field: they compute one number.

angle stands slightly apart: it does what atan2 does in other languages, and the game still understands the old name atan2 when reading a program.

Remember

• len is the distance of an offset, angle its direction, angleDiff the difference of angles.
• Zero degrees is right, and it goes counter-clockwise from there.
• angleDiff is never negative and never above 180.
• Trigonometry is in degrees.
• These operations have half the usual precision: do not compare them for equality.

Task

Compute the distance between two points — (12, 5) and (40, 45):

op sub dx 40 12 gives 28, op sub dy 45 5 gives 40, and op len path dx dy gives 48.8.

Then swap the points in the subtraction: op sub dx 12 40. The distance does not change — len computes a length and the sign means nothing to it — but angle changes by 180 degrees. That is the difference between “how far” and “in which direction”.