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Difference between revisions of "User:J-y2009/sandbox"
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Let a simple polygon on the plane be defined by its vertices:
 
Let a simple polygon on the plane be defined by its vertices:
 
<math>\left(x_1,y_1\right),\left(x_2,y_2\right),...\left(x_n,y_n\right)</math>.  
 
<math>\left(x_1,y_1\right),\left(x_2,y_2\right),...\left(x_n,y_n\right)</math>.  
−
Then, the area of the polygon is  
+
Then, the area of the polygon is <cmath>A=\frac{1}{2}\left|\left(x_1y_2+x_1y_2+...x_{n-1}y_n+x_ny_1\right)-\left(y_1x_2+y_1x_2+...y_{n-1}x_n+y_nx_1\right)\right|</cmath>
  
 
The name of the theorem comes from the fact that when the coordinates are written out in a matrix, as shown:
 
The name of the theorem comes from the fact that when the coordinates are written out in a matrix, as shown:
  
−
<math>\left|\begin{matrix}x_1&y_1\\
+
<cmath>\left|\begin{matrix}x_1&y_1\\
 
x_2&y_2\\
 
x_2&y_2\\
 
\vdots&\vdots\\
 
\vdots&\vdots\\
 
x_n&y_n\\
 
x_n&y_n\\
−
x_1&y_1\end{matrix}\right|</math>
+
x_1&y_1\end{matrix}\right|</cmath>
  
 
==Other Forms==
 
==Other Forms==

Revision as of 13:43, 20 April 2023

The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices.

Theorem

Let a simple polygon on the plane be defined by its vertices: $\left(x_1,y_1\right),\left(x_2,y_2\right),...\left(x_n,y_n\right)$. Then, the area of the polygon is \[A=\frac{1}{2}\left|\left(x_1y_2+x_1y_2+...x_{n-1}y_n+x_ny_1\right)-\left(y_1x_2+y_1x_2+...y_{n-1}x_n+y_nx_1\right)\right|\]

The name of the theorem comes from the fact that when the coordinates are written out in a matrix, as shown:

\[\left|\begin{matrix}x_1&y_1\\ x_2&y_2\\ \vdots&\vdots\\ x_n&y_n\\ x_1&y_1\end{matrix}\right|\]

Other Forms

This can also be written in form of a summation \[A = \dfrac{1}{2} \left|\sum_{i=1}^n{(x_{i+1}+x_i)(y_{i+1}-y_i)}\right|\] or in terms of determinants as \[A = \dfrac{1}{2} \left|\sum_{i=1}^n{\det\begin{pmatrix}x_i&x_{i+1}\\y_i&y_{i+1}\end{pmatrix}}\right|\] which is useful in the $3D$ variant of the Shoelace theorem. Note here that $x_{n+1} = x_1$ and $y_{n+1} = y_1$.

The formula may also be considered a special case of Green's Theorem

\[\tilde{A}=\int \int \left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dxdy=\oint(Ldx+Mdy)\]

where $L=-y$ and $M=0$ so $\tilde{A}=A$.

Proof 1

Claim 1: The area of a triangle with coordinates $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ is $\frac{|x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2|}{2}$.

Proof of claim 1:

Writing the coordinates in 3D and translating $\triangle ABC$ so that $A=(0, 0, 0)$ we get the new coordinates $A'(0, 0, 0)$, $B(x_2-x_1, y_2-y_1, 0)$, and $C(x_3-x_1, y_3-y_1, 0)$. Now if we let $\vec{b}=(x_2-x_1 \quad y_2-y_1 \quad 0)$ and $\vec{c}=(x_3-x_1 \quad y_3-y_1 \quad 0)$ then by definition of the cross product $[ABC]=\frac{||\vec{b} \times \vec{c}||}{2}=\frac{1}{2}||(0 \quad 0 \quad x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)||=\frac{x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2}{2}$.

Proof:

We will proceed with induction.

By claim 1, the shoelace theorem holds for any triangle. We will show that if it is true for some polygon $A_1A_2A_3...A_n$ then it is also true for $A_1A_2A_3...A_nA_{n+1}$.

We cut $A_1A_2A_3...A_nA_{n+1}$ into two polygons, $A_1A_2A_3...A_n$ and $A_1A_nA_{n+1}$. Let the coordinates of point $A_i$ be $(x_i, y_i)$. Then, applying the shoelace theorem on $A_1A_2A_3...A_n$ and $A_1A_nA_{n+1}$ we get

\[[A_1A_2A_3...A_n]=\frac{1}{2}\sum_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)\] \[[A_1A_nA_{n+1}]=\frac{1}{2}(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)\]

Hence

\[[A_1A_2A_3...A_nA_{n+1}]=[A_1A_2A_3...A_n]+[A_1A_nA_{n+1}]=\frac{1}{2}\sum_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)+\frac{1}{2}(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)\] \[=\frac{1}{2}((x_2y_1+x_3y_2+...+x_{n+1}y_n+x_1y_{n+1})-(x_1y_2+x_2y_3+...+x_ny_{n+1}+x_{n+1}y_1))=\boxed{\frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i)}\]

as claimed.

~ShreyJ

Proof 2

Let $\Omega$ be the set of points belonging to the polygon. We have that \[A=\int_{\Omega}\alpha,\] where $\alpha=dx\wedge dy$. The volume form $\alpha$ is an exact form since $d\omega=\alpha$, where \[\omega=\frac{x\,dy}{2}-\frac{y\,dx}{2}.\label{omega}\] Using this substitution, we have \[\int_{\Omega}\alpha=\int_{\Omega}d\omega.\] Next, we use the Theorem of Stokes to obtain \[\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.\] We can write $\partial \Omega=\bigcup A(i)$, where $A(i)$ is the line segment from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$. With this notation, we may write \[\int_{\partial\Omega}\omega=\sum_{i=1}^n\int_{A(i)}\omega.\] If we substitute for $\omega$, we obtain \[\sum_{i=1}^n\int_{A(i)}\omega=\frac{1}{2}\sum_{i=1}^n\int_{A(i)}{x\,dy}-{y\,dx}.\] If we parameterize, we get \[\frac{1}{2}\sum_{i=1}^n\int_0^1{(x_i+(x_{i+1}-x_i)t)(y_{i+1}-y_i)}-{(y_i+(y_{i+1}-y_i)t)(x_{i+1}-x_i)\,dt}.\] Performing the integration, we get \[\frac{1}{2}\sum_{i=1}^n\frac{1}{2}[(x_i+x_{i+1})(y_{i+1}-y_i)- (y_{i}+y_{i+1})(x_{i+1}-x_i)].\] More algebra yields the result \[\frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i).\]

Proof 3

This is a very nice approach that directly helps in understanding the sum as terms which are areas of trapezoids.

The proof is in this book: https://cses.fi/book/book.pdf#page=281

(The only thing that needs to be slightly modified is that one must shift the entire polygon up by k, until all the y coordinates are positive, but this term gets canceled in the resulting sum.)

Problems

Introductory

In right triangle $ABC$, we have $\angle ACB=90^{\circ}$, $AC=2$, and $BC=3$. Medians $AD$ and $BE$ are drawn to sides $BC$ and $AC$, respectively. $AD$ and $BE$ intersect at point $F$. Find the area of $\triangle ABF$.

Exploratory

Observe that \[\frac12\left|\det\begin{pmatrix} x_1 & y_1\\ x_2 & y_2 \end{pmatrix}\right|\] is the area of a triangle with vertices $(x_1,y_1),(x_2,y_2),(0,0)$ and \[\frac16\left|\det\begin{pmatrix} x_1 & y_1 & z_1\\ x_2 & y_2 & z_2\\ x_3 & y_3 & z_3 \end{pmatrix}\right|\] is the volume of a tetrahedron with vertices $(x_1, y_1, z_1), (x_2, y_2, z_2),(x_3, y_3, z_3),(0,0,0)$. Does a similar formula hold for $n$Dimensional triangles for any $n$? If so how can we use this to derive the $n$D Shoelace Formula?

External Links

A good explanation and exploration into why the theorem works by James Tanton: [1]

Nice geometric approach and discussion for proving the 3D Shoelace Theorem by Nicholas Patrick and Nadya Pramita: [2]

Nice integral approach for proving the 3D Shoelace Theorem (ignoring sign of volume) by @george2079: [3] AOPS

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