In geometry, $d(E, F)$ represents the distance between two points $E$ and $F$. Calculating this distance can vary depending on the type of geometry you are dealing with—Euclidean, Manhattan, or others. Let’s focus on the most common type: Euclidean geometry.
Euclidean Distance
Euclidean distance is the straight-line distance between two points in Euclidean space. It is the most familiar and widely used form of distance. To calculate the Euclidean distance between two points $E(x_1, y_1)$ and $F(x_2, y_2)$ in a 2-dimensional plane, you can use the distance formula:
$d(E, F) = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$
Example
Suppose we have two points $E(1, 2)$ and $F(4, 6)$. Plugging these coordinates into the formula, we get:
$d(E, F) = sqrt{(4 – 1)^2 + (6 – 2)^2} = sqrt{3^2 + 4^2} = sqrt{9 + 16} = sqrt{25} = 5$
So, the distance between points $E$ and $F$ is 5 units.
Distance in 3D Space
If you are working in 3-dimensional space, the formula expands to include the z-coordinates of the points $E(x_1, y_1, z_1)$ and $F(x_2, y_2, z_2)$:
$d(E, F) = sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}$
Example
For points $E(1, 2, 3)$ and $F(4, 6, 8)$, the distance is calculated as follows:
$d(E, F) = sqrt{(4 – 1)^2 + (6 – 2)^2 + (8 – 3)^2} = sqrt{3^2 + 4^2 + 5^2} = sqrt{9 + 16 + 25} = sqrt{50} approx 7.07$
So, the distance between these points in 3D space is approximately 7.07 units.
Manhattan Distance
Another way to measure distance is the Manhattan distance, which is the sum of the absolute differences of their coordinates. For points $E(x_1, y_1)$ and $F(x_2, y_2)$, the Manhattan distance is given by:
$d(E, F) = |x_2 – x_1| + |y_2 – y_1|$
Example
Using the same points $E(1, 2)$ and $F(4, 6)$, the Manhattan distance is:
$d(E, F) = |4 – 1| + |6 – 2| = 3 + 4 = 7$
Conclusion
Understanding how to calculate the distance between two points is fundamental in geometry. Whether using Euclidean or Manhattan distance, these calculations are essential for solving various geometric problems and have practical applications in fields like physics, engineering, and computer science.
3. Wikipedia – Euclidean Distance