Find equidistant points between two coordinates

Finding Equidistant Points Between Two Coordinates

Learn how to calculate and generate a series of equidistant points along a straight line connecting two given geographical coordinates using Python.

Calculating equidistant points between two geographical coordinates is a common task in various fields, including mapping, navigation, and data visualization. Whether you're plotting a route, distributing sensors, or analyzing spatial data, the ability to accurately determine intermediate points is crucial. This article will guide you through the mathematical principles and provide practical Python implementations to achieve this.

Understanding the Problem: Linear Interpolation

At its core, finding equidistant points between two coordinates involves linear interpolation. Given two points, P1 (x1, y1) and P2 (x2, y2), and a desired number of intermediate points, we want to find points P_i such that the distance between P1 and P_i, and P_i and P_j (for consecutive points), and P_j and P2 are all equal or proportional. For geographical coordinates, we typically deal with latitude and longitude, which are angles on a sphere. However, for relatively short distances, a straight-line approximation (Euclidean distance) on a 2D plane is often sufficient and computationally simpler. For longer distances, spherical geometry might be required, but we'll focus on the simpler case first.

flowchart TD
    A[Start with P1(lat1, lon1) and P2(lat2, lon2)] --> B{Determine Number of Points (N)};
    B --> C{Calculate Latitude Step: (lat2 - lat1) / (N+1)};
    C --> D{Calculate Longitude Step: (lon2 - lon1) / (N+1)};
    D --> E{Loop from 1 to N};
    E --> F{Current Lat = lat1 + i * Lat Step};
    F --> G{Current Lon = lon1 + i * Lon Step};
    G --> H[Store (Current Lat, Current Lon)];
    H --> E;
    E -- All points generated --> I[End];

Flowchart for generating equidistant points using linear interpolation.

Mathematical Approach

Let's denote our two coordinates as (lat1, lon1) and (lat2, lon2). If we want to find N equidistant points between these two, we will effectively have N + 1 segments. The step size for latitude and longitude can be calculated as follows:

delta_lat = (lat2 - lat1) / (N + 1) delta_lon = (lon2 - lon1) / (N + 1)

Then, for each point i from 1 to N:

point_lat_i = lat1 + i * delta_lat point_lon_i = lon1 + i * delta_lon

This method assumes a straight line in a Cartesian coordinate system, which is a reasonable approximation for short distances on Earth. For more precise calculations over longer distances, especially when crossing significant longitudinal spans, Haversine or Vincenty formulas for spherical geometry would be more appropriate, but they add considerable complexity.

import math

def find_equidistant_points(coord1, coord2, num_points):
    """
    Finds a specified number of equidistant points between two coordinates.

    Args:
        coord1 (tuple): The first coordinate (latitude, longitude).
        coord2 (tuple): The second coordinate (latitude, longitude).
        num_points (int): The number of equidistant points to generate
                          BETWEEN coord1 and coord2.

    Returns:
        list: A list of tuples, where each tuple is an equidistant point
              (latitude, longitude).
    """
    if num_points < 0:
        raise ValueError("Number of points cannot be negative.")
    if num_points == 0:
        return []

    lat1, lon1 = coord1
    lat2, lon2 = coord2

    equidistant_points = []

    # Calculate the step size for latitude and longitude
    # We divide by (num_points + 1) because we are creating num_points
    # intermediate points, which creates num_points + 1 segments.
    delta_lat = (lat2 - lat1) / (num_points + 1)
    delta_lon = (lon2 - lon1) / (num_points + 1)

    for i in range(1, num_points + 1):
        current_lat = lat1 + i * delta_lat
        current_lon = lon1 + i * delta_lon
        equidistant_points.append((current_lat, current_lon))

    return equidistant_points

# Example Usage:
point_a = (34.0522, -118.2437)  # Los Angeles
point_b = (36.7783, -119.4179)  # Fresno

num_intermediate_points = 3

intermediate_coords = find_equidistant_points(point_a, point_b, num_intermediate_points)

print(f"Start Point: {point_a}")
for i, coord in enumerate(intermediate_coords):
    print(f"Intermediate Point {i+1}: {coord}")
print(f"End Point: {point_b}")

# Example with 0 points
print("\nExample with 0 intermediate points:")
print(find_equidistant_points(point_a, point_b, 0))

# Example with 1 point
print("\nExample with 1 intermediate point:")
print(find_equidistant_points(point_a, point_b, 1))

Considerations for Global Distances

For scenarios involving long distances, especially across different meridians or parallels, the simple linear interpolation described above will introduce significant errors because it doesn't account for the Earth's curvature. In such cases, you would need to use geodesic calculations. Libraries like geopy in Python provide functions for calculating distances and intermediate points on a sphere or ellipsoid using methods like the Haversine formula or Vincenty's formulae. These methods are more complex but offer much higher accuracy for global-scale applications.

1. Define Your Start and End Points

Identify the latitude and longitude of your two extreme points. Ensure they are in a consistent format (e.g., decimal degrees).

2. Determine the Number of Intermediate Points

Decide how many points you need to generate between your start and end points. This number directly influences the density of the generated points.

3. Apply the Interpolation Formula

Use the provided Python function or implement the linear interpolation logic. Calculate the delta_lat and delta_lon based on the total number of segments (num_points + 1).

4. Iterate and Generate Points

Loop from 1 to num_points, calculating each intermediate point's latitude and longitude by adding i times the respective delta values to the start point's coordinates.

5. Store and Utilize Results

Collect the generated points in a list or array. These points can then be used for plotting on a map, further calculations, or any other application requiring evenly spaced coordinates.