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Antti H S Laaksonen 2017-05-31 23:23:04 +03:00
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chapter29.tex
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@ -85,7 +85,7 @@ It is clear for a human which of the lines is the correct
choice, but the situation is difficult for a computer. choice, but the situation is difficult for a computer.
However, it turns out that we can solve the problem using However, it turns out that we can solve the problem using
another method that is much easier to use. another method that is more convenient to a programmer.
Namely, there is a general formula Namely, there is a general formula
\[x_1y_2-x_2y_1+x_2y_3-x_3y_2+x_3y_4-x_4y_3+x_4y_1-x_1y_4,\] \[x_1y_2-x_2y_1+x_2y_3-x_3y_2+x_3y_4-x_4y_3+x_4y_1-x_1y_4,\]
that calculates the area of a quadrilateral that calculates the area of a quadrilateral
@ -209,7 +209,7 @@ counterclockwise.
The function $\texttt{polar}(s,a)$ constructs a vector The function $\texttt{polar}(s,a)$ constructs a vector
whose length is $s$ and that points to an angle $a$. whose length is $s$ and that points to an angle $a$.
Moreover, a vector can be rotated by an angle $a$ A vector can be rotated by an angle $a$
by multiplying it by a vector with length 1 and angle $a$. by multiplying it by a vector with length 1 and angle $a$.
The following code calculates the angle of The following code calculates the angle of
@ -320,7 +320,7 @@ points $s_1$, $s_2$ and $p$ are on the same line.
\index{line segment intersection} \index{line segment intersection}
Consider the problem of checking Next we consider the problem of testing
whether two line segments whether two line segments
$ab$ and $cd$ intersect. The possible cases are: $ab$ and $cd$ intersect. The possible cases are:
@ -404,7 +404,7 @@ exactly when both points $c$ and $d$ are
on different sides of a line through $a$ and $b$, on different sides of a line through $a$ and $b$,
and points $a$ and $b$ are on different and points $a$ and $b$ are on different
sides of a line through $c$ and $d$. sides of a line through $c$ and $d$.
Hence, we can use cross products to check this. We can use cross products to check this.
\subsubsection{Point distance from a line} \subsubsection{Point distance from a line}
@ -413,9 +413,8 @@ the area of a triangle can be calculated
using the formula using the formula
\[\frac{| (a-c) \times (b-c) |}{2},\] \[\frac{| (a-c) \times (b-c) |}{2},\]
where $a$, $b$ and $c$ are the vertices of the triangle. where $a$, $b$ and $c$ are the vertices of the triangle.
Using this fact, we can derive a formula
Using this formula, it is possible to calculate the for calculating the shortest distance between a point and a line.
shortest distance between a point and a line.
For example, in the following picture $d$ is the For example, in the following picture $d$ is the
shortest distance between the point $p$ and the line shortest distance between the point $p$ and the line
that is defined by the points $s_1$ and $s_2$: that is defined by the points $s_1$ and $s_2$:
@ -678,8 +677,8 @@ from the center point, using the Euclidean and Manhattan distances:
\subsubsection{Rotating coordinates} \subsubsection{Rotating coordinates}
Some problems are easier to solve if the Some problems are easier to solve if
Manhattan distance is used instead of the Euclidean distance. Manhattan distances are used instead of Euclidean distances.
As an example, consider a problem where we are given As an example, consider a problem where we are given
$n$ points in the two-dimensional plane $n$ points in the two-dimensional plane
and our task is to calculate the maximum Manhattan and our task is to calculate the maximum Manhattan
@ -721,7 +720,7 @@ between points $B$ and $C$:
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
A useful technique related to the Manhattan distance A useful technique related to Manhattan distances
is to rotate all coordinates 45 degrees so that is to rotate all coordinates 45 degrees so that
a point $(x,y)$ becomes $(x+y,y-x)$. a point $(x,y)$ becomes $(x+y,y-x)$.
For example, after rotating the above points, For example, after rotating the above points,
@ -773,7 +772,7 @@ and the Manhattan distance is
\[|1-3|+|0-3| = \max(|1-6|,|-1-0|) = 5.\] \[|1-3|+|0-3| = \max(|1-6|,|-1-0|) = 5.\]
The rotated coordinates provide a simple way The rotated coordinates provide a simple way
to operate with the Manhattan distance, because we can to operate with Manhattan distances, because we can
consider x and y coordinates separately. consider x and y coordinates separately.
To maximize the Manhattan distance between two points, To maximize the Manhattan distance between two points,
we should find two points whose we should find two points whose