I'm Walking Around Manhattan, How Far Am I From My Hotel?

Mar 10, 2025 by ADMIN 58 views

**I'm Walking Around Manhattan, How Far Am I from My Hotel?**

The Unnecessary and Convoluted Story

As I walked around Manhattan, block by block, my feet had gotten tired and I yearned to return home. The traffic was pretty bad, but fortunately, I was very rich and had a personal helicopter at my disposal. However, I was in a peculiar mood and decided to take a stroll around the city instead. I had a peculiar problem - I wanted to know how far I was from my hotel. I had a rough idea of the location, but I wanted a more precise answer.

The Problem

Given a set of coordinates representing my current location and the coordinates of my hotel, I wanted to calculate the distance between the two points. This problem is a classic example of a geodesic distance calculation, which is a fundamental concept in geography and computer science.

The Solution

To solve this problem, I used the Haversine formula, which is a formula used to calculate the distance between two points on a sphere (such as the Earth) given their longitudes and latitudes. The formula is as follows:

Haversine Formula

a = sin²(Δlat/2) + cos(lat1) * cos(lat2) * sin²(Δlong/2) c = 2 * atan2(√a, √(1-a)) d = R * c

where:

Δlat is the difference in latitude between the two points
Δlong is the difference in longitude between the two points
lat1 and lat2 are the latitudes of the two points
R is the radius of the Earth (approximately 6371 kilometers)

Code Solution

Here is a Python code snippet that calculates the distance between two points using the Haversine formula:

import math
def haversine_distance(lat1, lon1, lat2, lon2):
# Radius of the Earth in kilometers
R = 6371
# Convert degrees to radians
lat1, lon1, lat2, lon2 = map(math.radians, [lat1, lon1, lat2, lon2])

# Calculate the differences in latitude and longitude
dlat = lat2 - lat1
dlon = lon2 - lon1

# Calculate the Haversine formula
a = math.sin(dlat/2)**2 + math.cos(lat1) * math.cos(lat2) * math.sin(dlon/2)**2
c = 2 * math.atan2(math.sqrt(a), math.sqrt(1-a))

# Calculate the distance
d = R * c

return d

lat1, lon1 = 40.7128, -74.0060  # New York City
lat2, lon2 = 40.7484, -73.9857  # Central Park
distance = haversine_distance(lat1, lon1, lat2, lon2)
print(f"The distance between New York City and Central Park is distance.2f kilometers.")

Code Golf Solution

Now, let's try to solve this problem in a more concise way, using the Code Golf format. Here is a Python code snippet that calculates the distance between two points using the Haversine formula, with a character limit of 140 characters:

import math
h = lambda l1, o1, l2, o2: 2*math.atan2(math.sqrt((math.sin((l2-l1)/2)**2+math.cos(l1)*math.cos(l2)*math.sin((o2-o1)/2)**2)), math.sqrt(1-(math.sin((l2-l1)/2)**2+math.cos(l1)*math.cos(l2)*math.sin((o2-o1)/2)**2)))
print(h(40.7128, -74.0060, 40.7484, -73.9857))

Conclusion

Q: What is the Haversine formula and how does it work?

A: The Haversine formula is a formula used to calculate the distance between two points on a sphere (such as the Earth) given their longitudes and latitudes. It works by using the law of cosines to calculate the distance between the two points.

Q: What are the inputs and outputs of the Haversine formula?

A: The inputs of the Haversine formula are the latitudes and longitudes of the two points, and the output is the distance between the two points in kilometers.

Q: What is the radius of the Earth used in the Haversine formula?

A: The radius of the Earth used in the Haversine formula is approximately 6371 kilometers.

Q: Can I use the Haversine formula to calculate distances on other planets?

A: Yes, you can use the Haversine formula to calculate distances on other planets, but you will need to use the radius of the planet instead of the radius of the Earth.

Q: How accurate is the Haversine formula?

A: The Haversine formula is a good approximation of the distance between two points on a sphere, but it is not exact. The formula assumes that the Earth is a perfect sphere, which it is not. The actual distance between two points on the Earth's surface may be slightly different from the distance calculated by the Haversine formula.

Q: Can I use the Haversine formula to calculate distances between points on a flat surface?

A: No, the Haversine formula is designed to calculate distances between points on a sphere, not on a flat surface. If you need to calculate distances between points on a flat surface, you will need to use a different formula.

Q: How can I use the Haversine formula in real-world applications?

A: The Haversine formula can be used in a variety of real-world applications, such as:

Calculating distances between two points on a map
Determining the distance between two cities
Calculating the distance between two points on a globe
Determining the distance between two points on a satellite image

Q: Can I use the Haversine formula to calculate distances between points in 3D space?

A: No, the Haversine formula is designed to calculate distances between points on a sphere, not in 3D space. If you need to calculate distances between points in 3D space, you will need to use a different formula.

Q: How can I implement the Haversine formula in a programming language?

A: The Haversine formula can be implemented in a variety of programming languages, such as Python, Java, and C++. Here is an example of how to implement the Haversine formula in Python:

import math
def haversine_distance(lat1, lon1, lat2, lon2):
# Radius of the Earth in kilometers
R = 6371
# Convert degrees to radians
lat1, lon1, lat2, lon2 = map(math.radians, [lat1, lon1, lat2, lon2])

# Calculate the differences in latitude and longitude
dlat = lat2 - lat1
dlon = lon2 - lon1

# Calculate the Haversine formula
a = math.sin(dlat/2)**2 + math.cos(lat1) * math.cos(lat2) * math.sin(dlon/2)**2
c = 2 * math.atan2(math.sqrt(a), math.sqrt(1-a))

# Calculate the distance
d = R * c

return d

Q: Can I use the Haversine formula to calculate distances between points on a non-spherical surface?

A: No, the Haversine formula is designed to calculate distances between points on a sphere, not on a non-spherical surface. If you need to calculate distances between points on a non-spherical surface, you will need to use a different formula.