Shoelace scheme for determining the area of a polygon with point coordinates
The shoelace formula, also known as Gauss's area formula and the surveyor's formula,[1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane.[2] It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like threading shoelaces.[2] It has applications in surveying and forestry,[3] among other areas.
The formula was described by Albrecht Ludwig Friedrich Meister (1724–1788) in 1769[4] and is based on the trapezoid formula which was described by Carl Friedrich Gauss and C.G.J. Jacobi.[5] The triangle form of the area formula can be considered to be a special case of Green's theorem.
The area formula can also be applied to self-overlapping polygons since the meaning of area is still clear even though self-overlapping polygons are not generally simple.[6] Furthermore, a self-overlapping polygon can have multiple "interpretations" but the Shoelace formula can be used to show that the polygon's area is the same regardless of the interpretation.[7]
^Bart Braden (1986). "The Surveyor's Area Formula" (PDF). The College Mathematics Journal. 17 (4): 326–337. doi:10.2307/2686282. JSTOR 2686282. Archived from the original (PDF) on 29 June 2014.
^ abDahlke, Karl. "Shoelace Formula". Retrieved 9 June 2008.
^Hans Pretzsch, Forest Dynamics, Growth and Yield: From Measurement to Model, Springer, 2009, ISBN 3-540-88306-1, p. 232.
^Meister, A. L. F. (1769), "Generalia de genesi figurarum planarum et inde pendentibus earum affectionibus", Nov. Com. Gött. (in Latin), 1: 144.
^Max Koecher, Aloys Krieg: Ebene Geometrie, Springer-Verlag, 2013, ISBN 3662068095, 9783662068090, p. 116
^P.W. Shor; C.J. Van Wyk (1992), "Detecting and decomposing self-overlapping curves", Comput. Geom. Theory Appl., 2 (1): 31–50, doi:10.1016/0925-7721(92)90019-O
^Ralph P. Boland; Jorge Urrutia (2000). Polygon Area Problems. 12th Canadian Conference on Computational Geometry. pp. 159–162.
The shoelaceformula, also known as Gauss's area formula and the surveyor's formula, is a mathematical algorithm to determine the area of a simple polygon...
absolute value signs can be omitted. The above formula is known as the shoelaceformula or the surveyor's formula. If we locate the vertices in the complex...
_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3})} . This formula is also known as the shoelaceformula and is an easy way to solve for the area of a coordinate...
{\displaystyle A} is the polygon's signed area, as described by the shoelaceformula: A = 1 2 ∑ i = 0 n − 1 ( x i y i + 1 − x i + 1 y i ) . {\displaystyle...
after dividing the area into a set of triangles. This formula is related to the shoelaceformula and can be considered a special case of Green's theorem...
triangularization of the surface. This is the generalization of the Shoelaceformula to three dimensions. Using Stokes' theorem applied to an appropriately...
either case, the area formula is correct in absolute value. This is commonly called the shoelaceformula or surveyor's formula. The area A of a simple...
given a Cartesian coordinate system, this method is the 18th century shoelaceformula. The ancient Greeks had no general method for computing areas of shapes...
computing the area of the interior of a polygon. These include the shoelaceformula for arbitrary polygons, and Pick's theorem for polygons with integer...
rear of the vehicle and asked her to close her eyes and say how to tie a shoelace without opening her eyes or moving her hands. She agreed to this challenge...
and Kuryakin escape from their cell using explosives hidden in Solo's shoelaces. They sabotage the plant and destroy the gold-making machinery. Randolph...
the last act. And Jeremy Irons' villain isn't fit to tie Alan Rickman's shoelaces." In the Crime Time Filmbook, which archives various UK film reviews,...
'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding. If a torus...
concert at the 2009 Sziget Festival in Budapest, Hungary, Patton swallowed a shoelace from a shoe thrown at the stage, before loudly regurgitating it and throwing...
sense of fashion, which includes an over-sized shirt and one-meter long shoelaces. Saiji is an expert in Amakusa-style combat and is equipped with a 180-centimeter-long...
Telescope. The main shot shows a street scene with a young man tying the shoelace and then caressing the foot of his girlfriend, while an old man observes...
editors had criticized him over art details, such as not drawing "the shoelaces on a cavalryman's boots" and showing a Native American "mounting his horse...
additional education and practice (e.g., practiced instruction on tying shoelaces), giving reminders, or making accommodations (e.g., using slip-on shoes)...
defied convention in a manner similar to the 21st century fad for untied shoelaces.[page needed][page needed] Another suggestion to the origin of the term...
merchandise at Hot Topic store at Park City Center. T-shirts, pillows, shoelaces, pins, sunglasses, and air fresheners were sold "extremely well if they...
antipodal pairs of points and vertices on a convex polygon or convex hull. Shoelace algorithm: determine the area of a polygon whose vertices are described...
entails crawling under the bench in the dugout and lighting a teammate's shoelaces on fire. The team introduced T.C. Bear, its costumed mascot, on April...
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