DZone
Thanks for visiting DZone today,
Edit Profile
  • Manage Email Subscriptions
  • How to Post to DZone
  • Article Submission Guidelines
Sign Out View Profile
  • Post an Article
  • Manage My Drafts
Over 2 million developers have joined DZone.
Log In / Join
Refcards Trend Reports Events Over 2 million developers have joined DZone. Join Today! Thanks for visiting DZone today,
Edit Profile Manage Email Subscriptions Moderation Admin Console How to Post to DZone Article Submission Guidelines
View Profile
Sign Out
Refcards
Trend Reports
Events
Zones
Culture and Methodologies Agile Career Development Methodologies Team Management
Data Engineering AI/ML Big Data Data Databases IoT
Software Design and Architecture Cloud Architecture Containers Integration Microservices Performance Security
Coding Frameworks Java JavaScript Languages Tools
Testing, Deployment, and Maintenance Deployment DevOps and CI/CD Maintenance Monitoring and Observability Testing, Tools, and Frameworks
Culture and Methodologies
Agile Career Development Methodologies Team Management
Data Engineering
AI/ML Big Data Data Databases IoT
Software Design and Architecture
Cloud Architecture Containers Integration Microservices Performance Security
Coding
Frameworks Java JavaScript Languages Tools
Testing, Deployment, and Maintenance
Deployment DevOps and CI/CD Maintenance Monitoring and Observability Testing, Tools, and Frameworks

Mathematica Geographic Functions and Spherical Triangles

We calculate the distance between three cities (one of them is where DZone is located!) using Mathematica and its geographic functions.

John Cook user avatar by
John Cook
·
Dec. 06, 18 · Tutorial
Like (3)
Save
Tweet
Share
5.46K Views

Join the DZone community and get the full member experience.

Join For Free

This post will look at the triangle behind North Carolina's Research Triangle using Mathematica's geographic functions.

Spherical Triangles

A spherical triangle is a triangle drawn on the surface of a sphere. It has three vertices, given by points on the sphere, and three sides. The sides of the triangle are portions of great circles running between two vertices. A great circle is a circle of maximum radius, a circle with the same center as the sphere.

An interesting aspect of spherical geometry is that both the sides and angles of a spherical triangle are angles. Because the sides of a spherical triangle are arcs, they have angular measure, the angle formed by connecting each vertex to the center of the sphere. The arc length of a side is its angular measure times the radius of the sphere.

Denote the three vertices by A, B, and C. Denote the side opposite A by a, etc. Denote the angles at A, B, and C by α, β, and γ respectively.

Research Triangle

Research Triangle is a (spherical!) triangle with vertices formed by Duke University, North Carolina State University, and University of North Carolina at Chapel Hill.

(That was the origin of the name, though it's now applied more loosely to the general area around these three universities.)

We'll take as our vertices

  • A = UNC Chapel Hill (35.9046 N, 79.0468 W)
  • B = Duke University in Durham (36.0011 N, 78.9389 W),
  • C = NCSU in Raleigh (35.7872 N, 78.6705 W)

Mathematica

We'll illustrate several features of Mathematica using the spherical triangle corresponding to Research Triangle.

Map

The map above was produced with the following Mathematica code.

    ptA = GeoPosition[{35.9046, -79.0468}]
    ptB = GeoPosition[{36.0011, -78.9389}]
    ptC = GeoPosition[{35.7872, -78.6705}]

    GeoGraphics[{Red, PointSize[Large], 
        Point[ptA], Point[ptB], Point[ptC]}, 
        GeoScaleBar -> "Imperial", 
        GeoRange -> 29000]

Note that longitude is in degrees east, so the longitudes above are negative.

Distance

To find the distance between two locations, you can use the function GeoDistance. For example:

    GeoDistance[ptA, ptB]

The above code tells me that the distance between UNC and Duke is 8.99185 miles. I assume it displays miles by default based on my location in the US, though the GeoRange option above defaults to meters. You can make the system of units explicit. For example:

    GeoDistance[ptA, ptB, UnitSystem -> "Metric"]

The above code returns 14.741 km.

If we want to find the length of the side c in degrees, we can use the Earth's radius.

    r = PlanetData["Earth", "Radius"]
    c = GeoDistance[ptA, ptB] 360 / (2 Pi r)

This shows that c is 0.13014°. Calculating the other sides similarly shows a = 0.30504° and b = 0.32739°.

Angles

Calling GeoDirection[ptA, ptB] returns 42.2432°, which says we need to head at an angle of about 42° to walk from UNC to Duke.

The code

    GeoDirection[ptA, ptB] - GeoDirection[ptA, ptC]

shows that the angle α is 68.6128° (the code returns the negative of this angle because the angle is clockwise). Similarly we find β = 87.9808° and γ = 23.4068.

The angles add up to 180° only because our triangle is small compared to the earth's total area. The actual sum should be slightly more than 180°, but we've not retained enough precision to detect the difference. In general the "spherical excess," i.e. the amount by which the sum of the angles exceed 180°, is proportional to the area of the triangle.

ANGLE (software)

Published at DZone with permission of John Cook, DZone MVB. See the original article here.

Opinions expressed by DZone contributors are their own.

Popular on DZone

  • Type Variance in Java and Kotlin
  • Unleashing the Power of JavaScript Modules: A Beginner’s Guide
  • Unlocking the Power of Polymorphism in JavaScript: A Deep Dive
  • Spring Boot Docker Best Practices

Comments

Partner Resources

X

ABOUT US

  • About DZone
  • Send feedback
  • Careers
  • Sitemap

ADVERTISE

  • Advertise with DZone

CONTRIBUTE ON DZONE

  • Article Submission Guidelines
  • Become a Contributor
  • Visit the Writers' Zone

LEGAL

  • Terms of Service
  • Privacy Policy

CONTACT US

  • 600 Park Offices Drive
  • Suite 300
  • Durham, NC 27709
  • support@dzone.com
  • +1 (919) 678-0300

Let's be friends: