Area of a Convex Polygon

This problem is a modification of Gilat $7\mathrm{.}16$ and $7\mathrm{.}17.$

The area of a convex polygon can be found by dividing the

polygon into triangles and summing the areas of the triangles.

The area of each triangle can be found by a cross product. For

example, for triangle ABC in the figure, its area is:

Area $=0\mathrm{.}5**|ABAC|$

Assume that a file exists giving the coordinates of the points of

the polygon. There are two columns, with the first column

being the $x$ coordinate and the second column the $y$ coordinate.

The points are given in the order that they are connected to form the polygon $($i$.$e$.,$ coordinates of $A$ in

the first row, $B$ in the second row, and so on$).$

Write a Matlab program to read the input file, calculate the area and write the area into an output file.

Prompt for the filenames.

Here is a sample input file. The columns have one or more spaces between them. There may be more

or fewer lines than shown here:

For the data above, the output file is just one number. The format is a field width of $12,$ with $6$ digits

after the decimal:

$116\mathrm{.}500000$

You must use fscanf to read the input and fprintf to write the output. You may not use a loop, as we

have not yet covered loops and they are also slow. Below are some suggestions on doing this using

Chapter $2$ material.

Read the data into a matrix:

data $=$

$11$

$72$

$105$

Page $2$ of $3$

$911$

$612$

$112$

$-38$

$-44$

Subtract the first row from all the other rows to form:

vecs $=$

$61$

$94$

$810$

$511$

$011$

$-47$

$-53$

Add a $3$rd column of zeros:

vecs $=$

$610$

$940$

$8100$

$5110$

$0110$

$-470$

$-530$

Tranpose vecs:

vecs $=$

$69850-4-5$

$1410111173$

$0000000$

Now create a matrix, first, that contains all the columns except for the last:

first $=$

$69850-4$

$141011117$

$000000$

Page $3$ of $3$

Create a matrix called second that contains all the columns of vec except for the first:

second $=$

$9850-4-5$

$410111173$

$000000$

Now take the cross product of first with second, then take the absolute value and divide by $2$:

areas $=$

$000000$

$000000$

$7\mathrm{.}500029\mathrm{.}000019\mathrm{.}000027\mathrm{.}500022\mathrm{.}000011\mathrm{.}5000$

The $3$rd row of matrix areas contains the areas of the triangles. Now just add them up and write the

output file.

One problem you$$ll face is that you don$$t know how many vertices will be in the input file. There are a

couple of solutions:

$1)$ Use the size function.

$2)$ Use end. This is not in your textbook, but here is the relevant section from the Matlab help.

You can also find examples on end by typing $$doc end$.$

end can also serve as the last index in an indexing expression. In

that context, end $=$ SIZE$($X$,$k$)$ when used as part of the k$-$th index.

Examples of this use are, X$(3$:end$)$ and X$(1,1$:$2$:end$-1).$ When using end

to grow an array, as in X$($end$+1)=5,$ make sure X exists first.