Re^2: Seeing if two numbers have the same sign ($x / abs $x)

<obsMathRef>
lodin's observation and syphilis' cantrip come from the same underlying mathematical principles.

Triangle equality:
for vectors a, b,

||a+b|| <= ||a|| + ||b||, with equality iff a is a positive scalar multiple of b

(Note here || is "norm", not or.)

Angle between two vectors:
The angle theta between two vectors a, b satisfies

cos theta = (a dot b) / (||a|| ||b||)

(That the second theorem implies the first is left as an exercise for the reader. :-)

For one dimensional vectors ("numbers") the norm is the absolute value, and there are only two "directions" -- out along the positive axis and out along the negative axis. So syphilis is applying the triangle equality, while lodin is computing the cosine directly (note cos 0 = 1 -> "same sign" while cos 180 degrees = -1 -> "opposite sign").
</obsMathRef>

We now return to our regular interpretation of line noise as script.

--woody

Comment on Re^2: Seeing if two numbers have the same sign ($x / abs $x)

Replies are listed 'Best First'.
Re^3: Seeing if two numbers have the same sign ($x / abs $x) by lodin (Hermit) on Jan 12, 2008 at 05:28 UTC
lodin is computing the cosine directly (note cos 0 = 1 -> "same sign" while cos 180 degrees = -1 -> "opposite sign"). Maybe this is what you mean, but I don't actually calculate the angle (or cosine) between `$_[0]` and `$_[1]`. I calculate the respective angles of `$_[0]` and `$_[1]` against the positive axis, and compare them. Equivalently, you could also view it as I'm taking the norm of the two 1D vectors `$_[0]` and `$_[1]`, which gives me two unit vectors pointing in the direction of `$_[0]` and `$_[1]`, which I then test for equality. If they're equal, they point in the same direction, and in 1D that means they have the same sign. lodin	[reply] [d/l] [select]
Re^4: Seeing if two numbers have the same sign ($x / abs $x) by WoodyWeaver (Monk) on Jan 12, 2008 at 13:24 UTC
arg, eyes are getting old. You are absolutely right. I was thinking of the "trick" people had described about taking the product and checking if positive; you are indeed normalizing the vectors and checking for equality.	[reply]