Talk:Shoelace formula

Wording

The introduction talks about the "determinant" of a quantity. Since this quantity is just a simple arithemetic expression, and not a square matrix, shouldn't it just talk about the "absolute value"? Lehnekbn (talk) 09:08, 12 June 2013 (UTC)

Missing source

The first source, at http://staff.imsa.edu/math/journal/volume2/articles/Shoelace.pdf, appears to have moved. (I obtained a 404 Not Found error when I attempted to access it today). In fact, this applied to the entire journal, although http://staff.imsa.edu/ still worked. I'll wait and search for awhile, but if nothing turns up I'll have to remove the reference and use another.

I recreated this page that MZMcBride had previously deleted since he did so under WP:CSD G7 (author request), interpreting my blanking of my only edit to that page as a request for deletion. Actually, I just realized my comment was unnecessary and could be interpreted as negative criticism, though that was not its intent. Nat2 (talk) 00:03, 25 July 2012 (UTC)

Self-intersecting polygons

Some interesting things happen when the formula is (mis)used to find the areas of self-intersecting polygons. Double sharp (talk) 14:15, 18 February 2013 (UTC)

Proof

How to prove this formula? I think it's essential to prove formulas. — Preceding unsigned comment added by 149.78.251.255 (talk) 19:42, 13 December 2015 (UTC)

OH NO I just realized their IS no proof on this page. I'll add a proof as quickly as I can type it. :| — Preceding unsigned comment added by Vivek378 (talk • contribs) 17:40, 8 August 2016 (UTC)

"Surveyor's formula" misnamed?

Is there any evidence that this formula was used by surveyors? The article by Bart Braden offers none. On the contrary, they were more likely to use the simple trapezium rule,

${\begin{aligned}A&=-{\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i+1}-x_{i})(y_{i}+y_{i+1})\\&={\frac {1}{2}}\sum _{i=0}^{n-1}(y_{i+1}-y_{i})(x_{i}+x_{i+1}).\end{aligned}}$

See, for example, Flint (1808). The shoelace formula follows simply from these expressions, but is typically slower (2n multiplies instead of n) and, more seriously, subject to worse round-off error. To illustrate the second point apply the formulas to the rectangle with corners (100000001,100000000), (100000000,100000001), (99999999,100000000), (100000000,99999999). With the shoelace formula the result is 0, whereas the trapezium rule gives the correct result, 2.

False Statement about Sign of Determinants

I made an edit: The original text was "If the points are labeled sequentially in the counterclockwise direction, then the above determinants are positive and the absolute value signs can be omitted; if they are labeled in the clockwise direction, the determinants will be negative."

But this is not true, only the sum of the determinants will be positive, individual determinants can still be negative to account for excluded area between zero and the polygon. If the zero is inside the polygon, the problem might not appear, but this is nowhere stated.

Here is an example:

You can try yourself, take these points:

n xn yn

1 3 4

2 1 2

3 4 1

Which is in the spirit of this triangle: https://en.wikipedia.org/wiki/Shoelace_formula#/media/File:Triangle_area_from_coordinates_JCB.jpg

Now compute the determinants:

n det/2

1 1.0

2 -3.5

3 6.5

Total 4.0

Jan Burse (talk) 18:42, 1 August 2017 (UTC)

Good catch, I meant the expression as a whole when I wrote that, but obviously my intention did not come through. Thank you! Nat2 (talk) 18:52, 1 August 2017 (UTC)

Proofs given

There are about three different derivations/proofs mentioned, including starting with triangles and joining them together, considering triangles radiating outward from the origin, and invoking Green's theorem. I think they could all currently use some work, especially in terms of references and being clearer whether the derivations given are truly general or if some cases are being elided. Nat2 (talk) 20:52, 29 March 2018 (UTC)

Better name for "other formulas"?

The article is consistent in its use of the name "other formulas" for the xdy formula. Consistency is a great start. But now can we come up with a better name for this formula, since it's probably the most popular one?