How to Calculate the Distance Between Points on a Circle or Sphere

Calculating the distance between points on the circumference of a circle or on the surface of a sphere involves different methods depending on the shape and the information available about the points. In this comprehensive guide, we will explore the methods for calculating distances on both a circle and a sphere, along with practical examples to illustrate the process.

Distance Along the Circumference of a Circle

The distance between two points on the circumference of a circle can be calculated using different methods. Here, we provide two commonly used approaches:

Using the Radius and Angle

When you know the radius r of the circle and the central angle θ in radians between the two points, you can use the following formula:

d r middot; θ

If the angle is given in degrees, convert it to radians using the formula:

θ_{rad} θ_{deg} middot; frac{π}{180}

Using Coordinates

When you have the coordinates of the two points on the circle, e.g., (x?, y?) and (x?, y?), you can calculate the distance using the Euclidean distance formula:

d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}

This method is straightforward but requires precise coordinate values.

Distance Along the Surface of a Sphere

The shortest distance between two points on the surface of a sphere is along the great circle. The Haversine formula is commonly used for this purpose. Given two points with latitude and longitude, you can use the following formula:

Given Points 1: lat?, lon?

Given Points 2: lat?, lon?

Calculate the distance using the Haversine formula: a sin^2left(frac{lat_2 - lat_1}{2}right) times cos(lat_1) times cos(lat_2) times sin^2left(frac{lon_2 - lon_1}{2}right) c 2 cdot text{atan2}left(sqrt{a}, sqrt{1 - a}right) d R cdot c

Where R is the radius of the sphere (e.g., for Earth, R ≈ 6371 km). For points that are very close to each other on the sphere, you can approximate the distance using the Euclidean distance formula in 3D space after converting latitude and longitude to Cartesian coordinates.

Summary

To calculate the distance between two points on a circle, use the radius and angle or the coordinates of the points. To calculate the distance between two points on a sphere, use the great circle distance formula (Haversine for accurate results) based on latitude and longitude. For small distances, the Euclidean distance can be a useful approximation.

For a circle, the distance (d) between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) can be calculated using the Euclidean distance formula:

d^2 (x_2 - x_1)^2 (y_2 - y_1)^2

Then you need either the circumference (C) or the radius (r). If the center of the circle is at (0, 0), both circle points are (r) distant so:

r^2 (x_2 - 0)^2 (y_2 - 0)^2

or

r^2 (x_1 - 0)^2 (y_1 - 0)^2

Once you have the chord length, you can calculate the radius and the central angle. The central angle (theta) in radians can be found using trigonometric functions:

theta/2 arctanleft(frac{1/2d}{h}right)

Where (d) is the chord length and (h) is the height from the center to the chord at the midpoint. The arc length can then be calculated as:

text{arc length} C cdot frac{CA}{360}

where (C 2pi r).

By following these steps, you can accurately calculate the distance between points on circles and spheres, which is crucial in fields such as astronomy, navigation, and engineering.