# Distance and Azimuth Calculation

• marlan_msr
In summary, to calculate the azimuth and distance between two points on earth in two-dimensional cartesian format using spherical trigonometry, you can use the sinus and cosine rules. First calculate the distance using the cosine rule and then calculate the azimuth using the sinus rule. In celestial navigation, there are similar calculations. An easier method is to use an inverse calculation, where you subtract the coordinates of the destination point from the starting point and use the square root and arctan functions to find the distance and azimuth.

#### marlan_msr

I hope you could help me on this.
Is there anyone can tell how to calculate azimuth and distance of two points on earth, the points is in two-dimensional cartesian format, using spherical trigonometri for more accurate calculation. If you have reference or you have sample of answered exercise how to calculate it, please let me know. Thx.

are you referring to aeronautical navigation?

no, what I mean is how to calculate distance and azimuth/bearing if you have a pair of site coodinates (in two-dimensional cartesian format) using spherical trigonometri (spherical triangels, not just a planar triangels) ?

you can use

x = p sin @ cos #

y = p sin @ sin #

z = p cos @

where p is the distance from the center of the sphere

@ is the angle made from the positive x-axis to the pprojection of the point onto the x y plane

and # is the angle from the positive z - axis to the point

All this is calculated by the sinus and cosine rule. In any spherical triangle call the corners A, B and C and the opposite sides a, b and c. Then the sinus rule is: sin(A)/sin(a)=sin(B)/sin(b)=sin(C)/sin(c) or in words: the ratio between the sinus of an angle and the sinus of its opposite side is equal for all three pairs.
The cosine rule: cos(a)=cos(b)cos(c)+sin(b)sin(c)cosA which may be applied for the other sides as well of course.
So in calculating distance and azimuth (or bearing/course) let distance be d, the latitude of place A is a and of place B is b and the angle between the meridians at the pole is the difference in longitude, which I'll call P, then applying the cosine rule gives you:
cos(d)=cos(90-a)cos(90-b)+sin(90-a)sin(90-b)cos(P) or
cos(d)=sin(a)sin(b)+cos(a)cos(b)cos(P)
Then calculate azimuth by the sinus rule:
sin(A)=sin(P)sin(90-b)/sin(d) from A to B or sin(B)=sin(P)sin(90-a)/sin(d) from B to A.
or sin(A)=sin(P)cod(b)/sin(d) and sin(B)=sin(P)cos(a)/sin(d) resp.
Please remember to "make sense" of your output, because for example invsin(0,5) gives you 030, 150, 210 and 330.

Last edited:
Inverse

The easiest way to get a distance and an azimuth between two known points is an inverse calculation. Subtract the x coordinate of the destination point from the x coord. of the starting point. Perform the same operation for the y coordinates. Call these values Dx and Dy respectively. Now, take the square root of Dx squared plus DY squared this will give you the distance between the two points. If you take the Arctan of Dx/Dy you will get the azimuth. Note, If Dx and Dy are positive your azimuth is good as is. If Dx is positive and DY is negative add 180 degress to your azimuth. If both Dx and Dy are negative add 180 degrees to your azimuth. If Dx is negative and Dy is Positive add 360 degrees to your azimuth.

Last edited:

## 1. What is the purpose of distance and azimuth calculation in science?

The purpose of distance and azimuth calculation in science is to determine the location and direction of an object or point in a given space. This information is crucial for various scientific disciplines such as geography, geology, astronomy, and navigation.

## 2. How is distance calculated?

Distance is typically calculated using the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two sides. In the context of distance and azimuth calculation, this formula is used to calculate the straight-line distance between two points on a map or in a three-dimensional space.

## 3. What is azimuth and how is it calculated?

Azimuth is the direction of a point relative to a fixed reference point, usually the north direction. It is measured in degrees clockwise from the north direction. Azimuth is calculated using trigonometric functions such as sine, cosine, and tangent, based on the coordinates of the starting and ending points.

## 4. Why is it important to use accurate measurements in distance and azimuth calculation?

Accurate measurements in distance and azimuth calculation are important because they directly impact the precision and reliability of scientific research and applications. Small errors in measurements can lead to significant discrepancies in results, which can have serious implications in fields such as navigation and mapping.

## 5. What are some common tools and methods used for distance and azimuth calculation?

Some common tools and methods used for distance and azimuth calculation include maps, compasses, GPS devices, and specialized software programs. Surveying and triangulation techniques are also commonly used to measure distances and angles, which are then used to calculate distance and azimuth.