The Shoelace Formula

Transcription

1 Section The Shoelace ormula inding the area of simple polygons on a coordinate grid. simple polygon is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pair-wise end to end to form a closed path. If the sides intersect other at adjacent ends then the polygon is is not simple. lmost every formula involving polygons is based on a simple polygon. or this reason it is common to leave off the word simple. These are Simple Polygons These are not Simple Polygons Polygons are named by the letters that represent each vertex. The letters can be written in either counter clockwise direction or clockwise direction. We will choice to always use the counter clockwise direction for this unit. Polygon inding the area of simple polygons on a coordinate grid. The Shoelace ormula (also known as Gauss s area formula) is a mathematical method used to find the area of a simple polygon whose vertices are described by their x and y coordinates on a coordinate grid. The polygon has 6 vertex points. The coordinates of the 6 vertex are shown below. y (,8 ) ( 4,6 ) ( 6,5 ) x ( 4,3 ) (,1 ) (7,) Math 306 Shoelace rea ormula! Page 1! 019 itel

2 The Shoelace ormula Start with a polygon with it s vertex at points on the x, y coordinate grid. reate a vertical x, y table of the of coordinates of the points in counter clockwise order. nd the list with the same x, y pair you started with. ( x3, y3 ) (x y ) ( x1, y1 ) (x4, y4) x 1 y 1 x y x 3 y 3 x 4 y 4 x 1 y 1 1. List the 6 ordered pairs underneath each other in counter clock order.. List the first ordered pair you started with at the bottom of the list. ind the sum of the products of the Left to Right iagonals and Right to Left iagonals. ( x3, y3 ) (x y ) ( x1, y1 ) x 1 y 1 x y x 3 y 3 x 4 y 4 x 1 y 1 3. ind the sum of the products of the Left to Right iagonals 4. ind the sum of the products of the Right o Left iagonals (x4, y4) 3. The sum of the products of the Left to Right iagonals = ( x 1 )( y ) + ( x )( y 3 ) + ( x 3 )( y 4 ) + ( x 4 )( y 1 ) 4. The sum of the products of the Right to Left iagonals = y 1 ( )( x ) + ( y )( x 3 ) + ( y 3 )( x 4 ) + ( y 4 )( x 1 ) 5. The rea of the Polygon is the difference of the sums divided by. R = (Sum of the Left to Right products) (Sum of the Right to Left products) Math 306 Shoelace rea ormula! Page! 019 itel

3 Note: If the points are labeled sequentially in the counterclockwise direction, then the difference of the left to right total and right to left total will be positive. If the points are labeled sequentially in the clockwise direction the difference of the left to right total and right to left total will be negative. If you use the clockwise direction then the formula needs to contain an absolute value symbol to ensure the final answer positive. R = ( Sum of the Left to Right products) Sum of the Right to Left products xample 1 ( ) Start with a polygon with it s vertex at points on the x, y coordinate grid. reate a vertical x, y table of the of coordinates of the points in counter clockwise order. nd the list with the same x, y pair you started with. (,4 ) ( 4,3 ) ( 5,6 ) (, 3 ) List the 6 ordered pairs underneath each other in counter clock order.. List the first ordered pair you started with at the bottom of the list. ind the SUM O TH PROUTS of the LT to RIGHT diagonals for the listed numbers. ind the SUM O TH PROUTS of the RIGHT to LT diagonals for the listed numbers Left to Right (blue) = (5)(4) + ( )(3) + ( 4)( 3) + ()(6) Left to Right (blue) = = 3 Right to Left (red) = (6)( ) + (4)( 4) + (3)() + (3)(5) Right to Left (red) = = 7 R = (Sum of the Left to right products) (Sum of the Right to Left products) R = 3 ( 7) = = 39 sq. units Math 306 Shoelace rea ormula! Page 3! 019 itel

4 xample ( 6,5 ) ( 4,3 ) (,8 ) (,1 ) ( 4,6 ) (7,) List the 6 ordered pairs underneath each other in counter clock order.. List the first ordered pair you started with at the bottom of the list. ind the SUM O TH PROUTS of the LT to RIGHT diagonals for the listed numbers Left to Right = ()() + (7)(6) + (4)(8) + ( )(5) + ( 6)(3) + ( 4)(1) Left to Right = = 46 ind the SUM O TH PROUTS of the RIGHT to LT diagonals for the listed numbers Right to Left = (1)(7) + ()(4) + (6)( ) + (8)( 6) + (5)( 4) + (3)() Right to Left = = 59 R = (Sum of the Left to right products) (Sum of the Right to Left products) R = 46 ( 59) = = 105 sq. units Math 306 Shoelace rea ormula! Page 4! 019 itel

5 xample 3 ( 6,6 ) ( 4,4 ) (.6 ) (, 5) (3,) G ( 5, 8 ) (6,3) G List the 6 ordered pairs underneath each other in counter clock order.. List the first ordered pair you started with at the bottom of the list. ind the SUM O TH PROUTS of the LT to RIGHT diagonals for the listed numbers Left to Right = (5)(6) + ( )(5) + ()(6) + ( 6)(4) + ( 4)() + (3)(3) + (6)(8) Left to Right = = 57 ind the SUM O TH PROUTS of the RIGHT to LT diagonals for the listed numbers Right to Left = (1)(7) + ()(4) + (6)( ) + (8)( 6) + (5)( 4) + (3)() Right to Left = = 59 R = 57 ( 65) = = 1 = 61 sq. units Solve for the unknown x coordinate Math 306 Shoelace rea ormula! Page 5! 019 itel

6 ind the value for the unknown x coordinate x if the area of the polygon is 8. (.6 ) ( 4,4 ) ( 5, 8 ) (x, 3) (3, ) x 3 Left to Right = (5)(6) + ( )(4) + ( 4)( ) + (3)(3) + (x)(8) Left to Right = x = 8x + 39 Right to Left = (8)( ) + (6)( 4) + (4)(3) + ( )(x) + (3)(5) Right to Left = x +15 = x = 8x + 39 ( x + 43 ) ( ) 56 = 8x + 39 x = 8x x = 10x 4 60 = 10x x = 10 Math 306 Shoelace rea ormula! Page 6! 019 itel

7 The shoelace formula or shoelace algorithm (also known as Gauss's area formula is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their artesian coordinates. The user cross multiplies numbers corresponding coordinates to find the area encompassing the polygon, and subtracts it from the surrounding polygon to find the area of the polygon within. It is called the shoelace formula because of the patern of products creeats a pattern that looks like shoelaces. It has applications in surveying and forestry, among other areas. The formula was described by Meister in 1769 and by Gauss in It can be verified by dividing the polygon into triangles, and can be considered to be a special case of Green s Theorem. The formula can be represented by the expression where is the area of the polygon, n is the number of sides of the polygon, and (xi, yi), i = 1,,..., n are the vertices (or "corners") of the polygon. Math 306 Shoelace rea ormula! Page 7! 019 itel

8 Pythagorean Theorem from the Shoelace ormula The Pythagorean theorem follows from Gauss' Shoelace ormula. The formula gives the area of a polygon 1...n in the artesian plane, where vertices are defined by their coordinates, j = (xj, yj), j = 1,..., n: where the sum is taken for j = 1,..., n and the indices are defined cyclically: xn+1 = x1 and x0 = xn and similarly for y's. or the square with vertices 1(b, 0), (a+b, b), 3(a, a+b), 4(0, a), the formula gives On the other hand, if c is the hypotenuse of the right triangle with vertices (0, 0), (b, 0), and (0, a) then the area of the square formed on the hypotenuse is c², showing that indeed a² + b² = c². Number of paths for a Shoelace While we are talking about shoelaces, they also showed up in an interesting book by ustralian Mathematician urkard Polster. In "The Shoelace ook", Polster examines various approaches to methods of lacing a pair of shoes, and "mathematizes" lacings formally enough to enumerate the possible lacing paths.. or a shoe with six eyelets on each side there turns out to be 43,00 different paths for a shoelace to pass through every eyelet, even with the added condition that each eyelet must contribute to the essential purpose of pulling the two halves of the shoe together. He describes lacings by such names as risscross, zigzag, bowtie, devil, angel, and star. urkard Polster is a German mathematician who runs the "Mathologer" channel on Youtube. Math 306 Shoelace rea ormula! Page 8! 019 itel