Formula Used
For polar spherical coordinates, use radius r, polar angle θ, and azimuth φ.
x = r sin(θ) cos(φ)
y = r sin(θ) sin(φ)
z = r cos(θ)
cos(γ) = sin(θ1) sin(θ2) cos(φ1 - φ2) + cos(θ1) cos(θ2)
d = √(r1² + r2² - 2r1r2cos(γ))
Chord on reference sphere = 2R sin(γ / 2)
Arc on reference sphere = Rγ
How To Use This Calculator
Select the coordinate mode first. Use polar mode for radius, polar angle, and azimuth. Use latitude mode for radius, latitude, and longitude.
Choose degrees or radians. Enter both points. Enter a reference radius for surface arc and chord values. Press calculate. Results appear above the form and below the header.
Use CSV for spreadsheet work. Use PDF for a compact printable record.
Example Data Table
| Mode |
Radius 1 |
Angle 1A |
Angle 1B |
Radius 2 |
Angle 2A |
Angle 2B |
Reference Radius |
Expected Main Use |
| Latitude |
6371 |
40.7128 |
-74.0060 |
6371 |
51.5074 |
-0.1278 |
6371 |
Earth arc distance |
| Polar |
10 |
45 |
30 |
12 |
60 |
80 |
11 |
Vector distance |
| Polar |
5 |
1.2 |
0.8 |
5 |
1.5 |
1.1 |
5 |
Radian geometry |
Understanding Spherical Separation
Spherical coordinates describe a point by radius, polar angle, and azimuth. This form is useful when motion or shape is measured around a center. The calculator converts each input point into a three dimensional vector. It then compares those vectors with dot products and trigonometry.
Why The Method Matters
Straight distance is not always the same as surface travel. A satellite, atom, globe model, or sensor field may need different answers. The chord distance gives the direct line through space. The central angle shows the rotation between directions. The arc length shows travel along a chosen spherical surface. Seeing these results together reduces mistakes.
Coordinate Modes
Some users work with polar angle theta. Others use latitude and longitude. The polar mode treats theta as the angle measured down from the positive z axis. The latitude mode treats the first angle as north or south of the equator. The tool converts latitude to polar angle before solving. This keeps the formulas consistent.
Angle Units And Accuracy
Inputs may be written in degrees or radians. Degrees are common in maps and field notes. Radians are common in calculus and code. The calculator converts every angle to radians for computation. It also clamps the cosine value before inverse cosine is used. That step avoids tiny rounding errors near zero or one.
Reading The Results
Use the three dimensional distance when the two radii may differ. Use the chord on a shared sphere when both points lie on the same radius. Use the arc length when travel follows a curved surface. The average radius can help with approximate shells. A custom reference radius gives better control.
Practical Uses
This calculation appears in geodesy, astronomy, robotics, graphics, and multivariable calculus. It also helps compare two directions from a fixed origin. Engineers can test sensor coverage. Students can verify textbook examples. Developers can prepare geometry functions for apps. Always check the coordinate convention before trusting any numeric result.
Final Notes
It supports faster review during repeated design checks. Small angle differences can still produce large distances when radius is large. Large radius differences can dominate the answer even when angles match. Clean units, clear convention, and careful rounding make spherical distance work reliable.
FAQs
What distance does this calculator find?
It finds the direct three dimensional distance between two spherical coordinate points. It also reports central angle, reference chord, and reference arc distance for deeper comparison.
What is the difference between chord and arc distance?
Chord distance is the straight line across the sphere. Arc distance follows the curved surface. Use arc distance for surface travel and chord distance for direct spatial separation.
Can I use latitude and longitude?
Yes. Select the latitude and longitude mode. The first angle becomes latitude, and the second angle becomes longitude. The calculator converts them internally.
What is polar angle theta?
Theta is measured from the positive z axis. It is not the same as latitude. A latitude value can be converted with theta equal to ninety degrees minus latitude.
Why do I need a reference radius?
The reference radius controls the reported surface chord and arc. It is useful when both directions are projected onto a shared sphere.
What happens if the radii are different?
The direct distance uses both radii exactly. Arc and chord outputs use the reference radius, so they describe a chosen spherical surface instead.
Should I use degrees or radians?
Use degrees for map style inputs. Use radians for calculus, programming, or scientific data already expressed in radians. The calculator handles either unit.
Why is the cosine value clamped?
Floating point math can create tiny values slightly outside the valid inverse cosine range. Clamping keeps the result stable near identical or opposite directions.