Distance from point to point: formulas, examples, solutions. Distance between two points on a plane Distance based on 2 points with coordinates

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Distance between two points on a line

Consider a coordinate line on which 2 points are marked: A A A And B B B. To find the distance between these points, you need to find the length of the segment A B AB A B. This is done using the following formula:

Distance between two points on a line

A B = ∣ a − b ∣ AB=|a-b|A B =∣ a −b∣,

Where a , b a, b a, b- coordinates of these points on a straight line (coordinate line).

Due to the fact that the formula contains a modulus, when solving it, it is not important which coordinate to subtract from which (since the absolute value of this difference is taken).

∣ a − b ∣ = ∣ b − a ∣ |a-b|=|b-a|∣ a −b ∣ =∣ b −a∣

Let's look at an example to better understand the solution to such problems.

Example 1

Points are marked on the coordinate line A A A, whose coordinate is equal to 9 9 9 and period B B B with coordinate − 1 -1 − 1 . We need to find the distance between these two points.

Solution

Here a = 9 , b = − 1 a=9, b=-1 a =9 , b =− 1

We use the formula and substitute the values:

A B = ∣ a − b ∣ = ∣ 9 − (− 1) ∣ = ∣ 10 ∣ = 10 AB=|a-b|=|9-(-1)|=|10|=10A B =∣ a −b ∣ =∣ 9 − (− 1 ) ∣ = ∣ 1 0 ∣ = 1 0

Answer

Distance between two points on a plane

Consider two points given on a plane. From each point marked on the plane, you need to lower two perpendiculars: To the axis O X OX O X and on the axle O Y OY OY. Then the triangle is considered A B C ABC A B C. Since it is rectangular ( B C BC B C perpendicular A C AC A C), then find the segment A B AB A B, which is also the distance between points, can be done using the Pythagorean theorem. We have:

A B 2 = A C 2 + B C 2 AB^2=AC^2+BC^2A B 2 = A C 2 + B C 2

But, based on the fact that the length A C AC A C equal to x B − x A x_B-x_A x B​ − x A​ , and the length B C BC B C equal to y B − y A y_B-y_A y B​ − y A​ , this formula can be rewritten as follows:

Distance between two points on a plane

A B = (x B − x A) 2 + (y B − y A) 2 AB=\sqrt((x_B-x_A)^2+(y_B-y_A)^2)A B =(x B​ − x A​ ) 2 + (y B​ − y A​ ) 2 ​ ,

Where x A , y A x_A, y_A x A​ , y A​ And x B , y B x_B, y_B x B​ , y B​ - coordinates of points A A A And B B B respectively.

Example 2

It is necessary to find the distance between points C C C And F F F, if the coordinates of the first (8 ; − 1) (8;-1) (8 ; − 1 ) , and second - (4 ; 2) (4;2) (4 ; 2 ) .

Solution

X C = 8 x_C=8 x C​ = 8
y C = − 1 y_C=-1 y C​ = − 1
x F = 4 x_F=4 x F​ = 4
y F = 2 y_F=2 y F​ = 2

C F = (x F − x C) 2 + (y F − y C) 2 = (4 − 8) 2 + (2 − (− 1)) 2 = 16 + 9 = 25 = 5 CF=\sqrt(( x_F-x_C)^2+(y_F-y_C)^2)=\sqrt((4-8)^2+(2-(-1))^2)=\sqrt(16+9)=\sqrt( 25)=5C F =(x F​ − x C​ ) 2 + (y F​ − y C​ ) 2 ​ = (4 − 8 ) 2 + (2 − (− 1 ) ) 2 ​ = 1 6 + 9 ​ = 2 5 ​ = 5

Answer

Distance between two points in space

Finding the distance between two points in this case is similar to the previous one, except that the coordinates of the point in space are specified by three numbers; accordingly, the coordinate of the applicate axis must also be added to the formula. The formula will look like this:

Distance between two points in space

A B = (x B − x A) 2 + (y B − y A) 2 + (z B − z A) 2 AB=\sqrt((x_B-x_A)^2+(y_B-y_A)^2+( z_B-z_A)^2)A B =(x B​ − x A​ ) 2 + (y B​ − y A​ ) 2 + (z B​ − zA​ ) 2 ​

Example 3

Find the length of the segment FK FKFK in space if the coordinates of the points at its ends are as follows: (− 1 ; − 1 ; 8) (-1;-1;8) (− 1 ; − 1 ; 8 ) And (− 3 ; 6 ; 0) (-3;6;0) (− 3 ; 6 ; 0 ) . Round the answer to the nearest whole number.

Solution

$F=(-1;-1;8)K\; =\; (-\; 3\; ;\; 6\; ;\; 0)\; K=(-3;6;0)K=(-3;6;0)$

$FK=\sqrt{(x^{K-x^{F{)}^{2+(y^{K-y^{F{)}^{2+(z^{K-z^{F{)}^{2=\sqrt{(-3-(-1){)}^{2+(6-(-1){)}^{2+(0-8{)}^{2=\sqrt{117\approx 10.8}}}}}}}}}}}}}}}$

According to the conditions of the problem, we need to round the answer to a whole number.

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 revolution. 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 longitudes $cl1 = cos($lat1); ; $sl2 = sin($lat2); $delta = $long2 - $long1; $cdelta = cos($delta); $sdelta = sin($delta); $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

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.

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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.

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. Notice 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.

In this article we will look at ways to determine the distance from point to point theoretically and using the example of specific tasks. To begin with, let's introduce some definitions.

Definition 1

Distance between points is the length of the segment connecting them, on the existing scale. It is necessary to set a scale in order to have a unit of length for measurement. Therefore, basically the problem of finding the distance between points is solved by using their coordinates on a coordinate line, in a coordinate plane or three-dimensional space.

Initial data: coordinate line O x and an arbitrary point A lying on it. Any point on the line has one real number: let it be a certain number for point A x A, it is also the coordinate of point A.

In general, we can say that the length of a certain segment is assessed in comparison with a segment taken as a unit of length on a given scale.

If point A corresponds to an integer real number, by laying off sequentially from point O to point along the straight line O A segments - units of length, we can determine the length of the segment O A from the total number of set aside unit segments.

For example, point A corresponds to the number 3 - to get to it from point O, you will need to lay off three unit segments. If point A has coordinate - 4, unit segments are laid out in a similar way, but in a different, negative direction. Thus, in the first case, the distance O A is equal to 3; in the second case O A = 4.

If point A has a rational number as a coordinate, then from the origin (point O) we plot an integer number of unit segments, and then its necessary part. But geometrically it is not always possible to make a measurement. For example, it seems difficult to plot the fraction 4 111 on the coordinate line.

Using the above method, it is completely impossible to plot an irrational number on a straight line. For example, when the coordinate of point A is 11. In this case, it is possible to turn to abstraction: if the given coordinate of point A is greater than zero, then O A = x A (the number is taken as the distance); if the coordinate is less than zero, then O A = - x A . In general, these statements are true for any real number x A.

To summarize: the distance from the origin to the point that corresponds to a real number on the coordinate line is equal to:

• 0 if the point coincides with the origin;

• x A, if x A > 0;

• - x A if x A< 0 .

In this case, it is obvious that the length of the segment itself cannot be negative, therefore, using the modulus sign, we write the distance from point O to point A with the coordinate xA: O A = x A

The following statement will be true: the distance from one point to another will be equal to the modulus of the coordinate difference. Those. for points A and B lying on the same coordinate line for any location and having corresponding coordinates xA And x B: A B = x B - x A .

Initial data: points A and B lying on a plane in a rectangular coordinate system O x y with given coordinates: A (x A, y A) and B (x B, y B).

Let us draw perpendiculars through points A and B to the coordinate axes O x and O y and obtain as a result the projection points: A x, A y, B x, B y. Based on the location of points A and B, the following options are then 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 O x axis (abscissa axis), then the points coincide, and | A B | = | A y B y | . Since the distance between the points is equal to the modulus of the difference of their coordinates, then A y B y = y B - y A, and, therefore, A B = A y B y = y B - y A.

If points A and B lie on a straight line perpendicular to the O y axis (ordinate axis) - by analogy with the previous paragraph: A B = A x B x = x B - x A

If points A and B do not lie on a straight line perpendicular to one of the coordinate axes, we will find the distance between them by deriving the calculation formula:

We see that triangle A B C is rectangular in construction. In this case, A C = A x B x and B C = A y B y. Using the Pythagorean theorem, we create the equality: A B 2 = A C 2 + B C 2 ⇔ A B 2 = A x B x 2 + A y B y 2 , and then transform it: A B = A x B x 2 + A y B y 2 = x B - x A 2 + y B - y A 2 = (x B - x A) 2 + (y B - y A) 2

Let's draw a conclusion from the result obtained: the distance from point A to point B on the plane is determined by calculation using the formula using the coordinates of these points

A B = (x B - x A) 2 + (y B - y A) 2

The resulting formula also confirms previously formed statements for cases of coincidence of points or situations when the points lie on straight lines perpendicular to the axes. So, if points A and B coincide, the following equality will be true: A B = (x B - x A) 2 + (y B - y A) 2 = 0 2 + 0 2 = 0

For a situation where points A and B lie on a straight line perpendicular to the x-axis:

A B = (x B - x A) 2 + (y B - y A) 2 = 0 2 + (y B - y A) 2 = y B - y A

For the case when points A and B lie on a straight line perpendicular to the ordinate axis:

A B = (x B - x A) 2 + (y B - y A) 2 = (x B - x A) 2 + 0 2 = x B - x A

Initial data: a rectangular coordinate system O x y z with arbitrary points lying on it with given coordinates A (x A, y A, z A) and B (x B, y B, z B). It is necessary to determine the distance between these points.

Let's consider the general case when points A and B do not lie in a plane parallel to one of the coordinate planes. Let us draw planes perpendicular to the coordinate axes through points A and B and obtain the corresponding projection points: A x , A y , A z , B x , B y , B z

The distance between points A and B is the diagonal of the resulting parallelepiped. According to the construction of the measurements of this parallelepiped: A x B x , A y B y and A z B z

From the geometry course we know that the square of the diagonal of a parallelepiped is equal to the sum of the squares of its dimensions. Based on this statement, we obtain the equality: A B 2 = A x B x 2 + A y B y 2 + A z B z 2

Using the conclusions obtained earlier, we write the following:

A x B x = x B - x A , A y B y = y B - y A , A z B z = z B - z A

Let's transform the expression:

A B 2 = A x B x 2 + A y B y 2 + A z B z 2 = x B - x A 2 + y B - y A 2 + z B - z A 2 = = (x B - x A) 2 + (y B - y A) 2 + z B - z A 2

Final formula for determining the distance between points in space will look like this:

A B = x B - x A 2 + y B - y A 2 + (z B - z A) 2

The resulting formula is also valid for cases when:

The points coincide;

They lie on one coordinate axis or a straight line parallel to one of the coordinate axes.

Examples of solving problems on finding the distance between points

Example 1

Initial data: a coordinate line and points lying on it with given coordinates A (1 - 2) and B (11 + 2) are given. It is necessary to find the distance from the origin point O to point A and between points A and B.

Solution

1. The distance from the reference point to the point is equal to the modulus of the coordinate of this point, respectively O A = 1 - 2 = 2 - 1
2. We define the distance between points A and B as the modulus of the difference between the coordinates of these points: A B = 11 + 2 - (1 - 2) = 10 + 2 2

Answer: O A = 2 - 1, A B = 10 + 2 2

Example 2

Initial data: given a rectangular coordinate system and two points lying on it A (1, - 1) and B (λ + 1, 3). λ is some real number. It is necessary to find all values ​​of this number at which the distance A B will be equal to 5.

Solution

To find the distance between points A and B, you must use the formula A B = (x B - x A) 2 + y B - y A 2

Substituting the real coordinate values, we get: A B = (λ + 1 - 1) 2 + (3 - (- 1)) 2 = λ 2 + 16

We also use the existing condition that A B = 5 and then the equality will be true:

λ 2 + 16 = 5 λ 2 + 16 = 25 λ = ± 3

Answer: A B = 5 if λ = ± 3.

Example 3

Initial data: a three-dimensional space is specified in the rectangular coordinate system O x y z and the points A (1, 2, 3) and B - 7, - 2, 4 lying in it.

Solution

To solve the problem, we use the formula A B = x B - x A 2 + y B - y A 2 + (z B - z A) 2

Substituting real values, we get: A B = (- 7 - 1) 2 + (- 2 - 2) 2 + (4 - 3) 2 = 81 = 9

Answer: | A B | = 9

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