Distance from point to point is the length of the segment connecting these points on a given scale. Thus, when it comes to measuring distance, you need to know the scale (unit of length) in which the measurements will be taken. Therefore, the problem of finding the distance from point to point is usually considered either on a coordinate line or in a rectangular Cartesian coordinate system on a plane or in three-dimensional space. In other words, most often you have to calculate the distance between points using their coordinates.

In this article, we will firstly recall how the distance from point to point on a coordinate line is determined. Next, we obtain formulas for calculating the distance between two points of a plane or space according to given coordinates. In conclusion, we will consider in detail the solutions to typical examples and problems.

Page navigation.

The distance between two points on a coordinate line.

Let's first define the notation. We will denote the distance from point A to point B as .

From this we can conclude that the distance from point A with coordinate to point B with coordinate is equal to the modulus of the difference in coordinates, that is, for any location of points on the coordinate line.

Distance from point to point on a plane, formula.

We obtain a formula for calculating the distance between points and given in a rectangular Cartesian coordinate system on a plane.

Depending on the location of points A and B, the following options are possible.

If points A and B coincide, then the distance between them is zero.

If points A and B lie on a straight line perpendicular to the abscissa axis, then the points coincide, and the distance is equal to the distance . In the previous paragraph, we found out that the distance between two points on a coordinate line is equal to the modulus of the difference between their coordinates, therefore, . Hence, .

Similarly, if points A and B lie on a straight line perpendicular to the ordinate axis, then the distance from point A to point B is found as .

In this case, triangle ABC is rectangular in construction, and And . By Pythagorean theorem we can write down the equality, whence .

Let us summarize all the results obtained: the distance from a point to a point on a plane is found through the coordinates of the points using the formula .

The resulting formula for finding the distance between points can be used when points A and B coincide or lie on a straight line perpendicular to one of the coordinate axes. Indeed, if A and B coincide, then . If points A and B lie on a straight line perpendicular to the Ox axis, then. If A and B lie on a straight line perpendicular to the Oy axis, then .

Distance between points in space, formula.

Let us introduce a rectangular coordinate system Oxyz in space. Let's get a formula for finding the distance from a point to the point .

In general, points A and B do not lie in a plane parallel to one of the coordinate planes. Let us draw through points A and B planes perpendicular to the coordinate axes Ox, Oy and Oz. The intersection points of these planes with the coordinate axes will give us projections of points A and B onto these axes. We denote the projections .

The required distance between points A and B is the diagonal of the rectangular parallelepiped shown in the figure. By construction, the dimensions of this parallelepiped are equal And . In a high school geometry course, it was proven that the square of the diagonal of a cuboid is equal to the sum of the squares of its three dimensions, therefore, . Based on the information in the first section of this article, we can write the following equalities, therefore,

where do we get it from formula for finding the distance between points in space .

This formula is also valid if points A and B

• match up;
• belong to one of the coordinate axes or a line parallel to one of the coordinate axes;
• belong to one of the coordinate planes or a plane parallel to one of the coordinate planes.

Finding the distance from point to point, examples and solutions.

So, we have obtained formulas for finding the distance between two points on a coordinate line, plane and three-dimensional space. It's time to look at solutions to typical examples.

The number of problems in which the final step is to find the distance between two points according to their coordinates is truly enormous. A full review of such examples is beyond the scope of this article. Here we will limit ourselves to examples in which the coordinates of two points are known and it is necessary to calculate the distance between them.

Using coordinates, the location of an object on the globe is determined. Coordinates are indicated by latitude and longitude. Latitudes are measured from the equator line on both sides. In the Northern Hemisphere the latitudes are positive, in the Southern Hemisphere they are negative. Longitude is measured from the prime meridian either east or west, respectively, either eastern or western longitude is obtained.

According to the generally accepted position, the prime meridian is taken to be the one that passes through the old Greenwich Observatory in Greenwich. Geographic coordinates of the location can be obtained using a GPS navigator. This device receives satellite positioning system signals in the WGS-84 coordinate system, uniform for the whole world.

Navigator models differ in manufacturer, functionality and interface. Currently, built-in GPS navigators are also available in some cell phone models. But any model can record and save the coordinates of a point.

Distance between GPS coordinates

To solve practical and theoretical problems in some industries, it is necessary to be able to determine the distances between points by their coordinates. There are several ways you can do this. The canonical form of representing geographic coordinates: degrees, minutes, seconds.

For example, you can determine the distance between the following coordinates: point No. 1 - latitude 55°45′07″ N, longitude 37°36′56″ E; point No. 2 - latitude 58°00′02″ N, longitude 102°39′42″ E.

The easiest way is to use a calculator to calculate the length between two points. In the browser search engine, you must set the following search parameters: online - to calculate the distance between two coordinates. In the online calculator, latitude and longitude values ​​are entered into the query fields for the first and second coordinates. When calculating, the online calculator gave the result - 3,800,619 m.

The next method is more labor-intensive, but also more visual. You must use any available mapping or navigation program. Programs in which you can create points using coordinates and measure distances between them include the following applications: BaseCamp (a modern analogue of the MapSource program), Google Earth, SAS.Planet.

All of the above programs are available to any network user. For example, to calculate the distance between two coordinates in Google Earth, you need to create two labels indicating the coordinates of the first point and the second point. Then, using the “Ruler” tool, you need to connect the first and second marks with a line, the program will automatically display the measurement result and show the path on the satellite image of the Earth.

In the case of the example given above, the Google Earth program returned the result - the length of the distance between point No. 1 and point No. 2 is 3,817,353 m.

Why there is an error when determining the distance

All calculations of the extent between coordinates are based on the calculation of the arc length. The radius of the Earth is involved in calculating the length of the arc. But since the shape of the Earth is close to an oblate ellipsoid, the radius of the Earth varies at certain points. To calculate the distance between coordinates, the average value of the Earth's radius is taken, which gives an error in the measurement. The greater the distance being measured, the greater the error.

Mathematics

§2. Coordinates of a point on the plane

3. Distance between two points.

You and I can now talk about points in the language of numbers. For example, we no longer need to explain: take a point that is three units to the right of the axis and five units below the axis. Suffice it to say simply: take the point.

We have already said that this creates certain advantages. So, we can transmit a drawing made up of dots by telegraph, communicate it to a computer, which does not understand drawings at all, but understands numbers well.

In the previous paragraph, we defined some sets of points on the plane using relationships between numbers. Now let's try to consistently translate other geometric concepts and facts into the language of numbers.

We will start with a simple and common task.

Find the distance between two points on the plane.

Solution:
As always, we assume that the points are given by their coordinates, and then our task is to find a rule by which we can calculate the distance between points, knowing their coordinates. When deriving this rule, of course, it is allowed to resort to a drawing, but the rule itself should not contain any references to the drawing, but should only show what actions and in what order must be performed on the given numbers - the coordinates of the points - in order to obtain the desired number - the distance between dots.

Perhaps some readers will find this approach to solving the problem strange and far-fetched. What is simpler, they will say, the points are given, even by coordinates. Draw these points, take a ruler and measure the distance between them.

This method is sometimes not so bad. However, imagine again that you are dealing with a computer. She doesn’t have a ruler, and she doesn’t draw, but she can count so quickly that it’s not a problem for her at all. Note that our problem is formulated so that the rule for calculating the distance between two points consists of commands that can be executed by a machine.

It is better to first solve the problem posed for the special case when one of these points lies at the origin of coordinates. Start with a few numerical examples: find the distance from the origin of the points; And .

Note. Use the Pythagorean theorem.

Now write a general formula to calculate the distance of a point from the origin.

The distance of a point from the origin is determined by the formula:

Obviously, the rule expressed by this formula satisfies the conditions stated above. In particular, it can be used in calculations on machines that can multiply numbers, add them, and extract square roots.

Now let's solve the general problem

Given two points on a plane, find the distance between them.

Solution:
Let us denote by , , , the projections of points and on the coordinate axes.

Let us denote the point of intersection of the lines with the letter . From a right triangle using the Pythagorean theorem we obtain:

But the length of the segment is equal to the length of the segment. The points and , lie on the axis and have coordinates and , respectively. According to the formula obtained in paragraph 3 of paragraph 2, the distance between them is equal to .

Arguing similarly, we find that the length of the segment is equal to . Substituting the found values ​​and into the formula we get.

Calculating distances between points based on their coordinates on a plane is elementary; on the Earth’s surface it is a little more complicated: we will consider measuring the distance and initial azimuth between points without projection transformations. First, let's understand the terminology.

Introduction

Great circle arc length– the shortest distance between any two points located on the surface of a sphere, measured along the line connecting these two points (such a line is called orthodromy) and passing along the surface of the sphere or other surface of rotation. Spherical geometry is different from normal Euclidean geometry and the distance equations also take a different form. In Euclidean geometry, the shortest distance between two points is a straight line. On a sphere, there are no straight lines. These lines on the sphere are part of great circles - circles whose centers coincide with the center of the sphere. Initial azimuth- azimuth, taking which when starting to move from point A, following a great circle for the shortest distance to point B, the end point will be point B. When moving from point A to point B along the great circle line, the azimuth from the current position to the end point B is constant is changing. The initial azimuth is different from a constant one, following which the azimuth from the current point to the final point does not change, but the route followed is not the shortest distance between two points.

Through any two points on the surface of a sphere, if they are not directly opposite to each other (that is, they are not antipodes), a unique great circle can be drawn. Two points divide a large circle into two arcs. The length of a short arc is the shortest distance between two points. An infinite number of large circles can be drawn between two antipodal points, but the distance between them will be the same on any circle and equal to half the circumference of the circle, or π*R, where R is the radius of the sphere.

On a plane (in a rectangular coordinate system), large circles and their fragments, as mentioned above, represent arcs in all projections except the gnomonic one, where large circles are straight lines. In practice, this means that airplanes and other air transport always use the route of the minimum distance between points to save fuel, that is, the flight is carried out along a great circle distance, on a plane it looks like an arc.

The shape of the Earth can be described as a sphere, so great circle distance equations are important for calculating the shortest distance between points on the Earth's surface and are often used in navigation. Calculating distance by this method is more efficient and in many cases more accurate than calculating it for projected coordinates (in rectangular coordinate systems), since, firstly, it does not require converting geographic coordinates to a rectangular coordinate system (carry out projection transformations) and, secondly, many projections, if incorrectly selected, can lead to significant length distortions due to the nature of projection distortions. It is known that it is not a sphere, but an ellipsoid that describes the shape of the Earth more accurately, however, this article discusses the calculation of distances specifically on a sphere; for calculations, a sphere with a radius of 6,372,795 meters is used, which can lead to an error in calculating distances of the order of 0.5%.

Formulas

There are three ways to calculate the great circle spherical distance. 1. Spherical cosine theorem In the case of small distances and small calculation depth (number of decimal places), the use of the formula can lead to significant rounding errors. φ1, λ1; φ2, λ2 - latitude and longitude of two points in radians Δλ - difference in coordinates in longitude Δδ - angular difference Δδ = arccos (sin φ1 sin φ2 + cos φ1 cos φ2 cos Δλ) To convert the angular distance to metric, you need to multiply the angular difference by the radius Earth (6372795 meters), the units of the final distance will be equal to the units in which the radius is expressed (in this case, meters). 2. Haversine formula Used to avoid problems with short distances. 3. Modification for the antipodes The previous formula is also subject to the problem of antipodal points; to solve it, the following modification is used.

My implementation on PHP

// Earth radius define("EARTH_RADIUS", 6372795); /* * Distance between two points * $φA, $λA - latitude, longitude of the 1st point, * $φB, $λB - latitude, longitude of the 2nd point * Written based on http://gis-lab.info/ qa/great-circles.html * Mikhail Kobzarev< >* */ function calculateTheDistance ($φA, $λA, $φB, $λB) ( // convert coordinates to radians $lat1 = $φA * M_PI / 180; $lat2 = $φB * M_PI / 180; $long1 = $λA * M_PI / 180; $long2 = $λB * M_PI / 180; // cosines and sines of latitudes and longitude differences $cl1 = cos($lat1); $cl2 = cos($lat2); $sl1 = sin($lat1) ; $sl2 = sin($lat2); $delta = $long2 - $long1; $cdelta = cos($delta); $sdelta = sin($delta); // great circle length calculations $y = sqrt(pow( $cl2 * $sdelta, 2) + pow($cl1 * $sl2 - $sl1 * $cl2 * $cdelta, 2)); $x = $sl1 * $sl2 + $cl1 * $cl2 * $cdelta; // $ad = atan2($y, $x); $dist = $ad * EARTH_RADIUS; return $dist; ) Example of a function call: $lat1 = 77.1539; $long1 = -139.398; $lat2 = -77.1804; $long2 = -139.55; echo calculateTheDistance($lat1, $long1, $lat2, $long2) . "meters"; // Return "17166029 meters"

Article taken from the site