Free · Physics models
Spherical Horizon and Obscured Height
Find a geometric horizon and target obscuration on a fixed spherical Earth without atmospheric refraction.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
An observer at sea level has zero geometric horizon distance; at zero target distance no height is obscured.
H=sqrt(h(h+2R)); obscured=max(0,sqrt((distance−H)²+R²)−R) when distance exceeds H, R=6371000 meters.
Worked example
Enter these known values and leave the other values blank.
- Eyesight level
- 0 m
- Distance to the object
- 0 km
- Distance to horizon
- 0 km
- Obscured object part
- 0 m
Assumptions and limitations
- Distances use the captured straight tangent geometry on a sphere of radius 6371000 meters. This is not an ellipsoidal geodesic distance or a refracted optical horizon.
- Observer height and target distance are nonnegative. Beyond the horizon, an ideal spherical target requires the reported height to reach the observer line of sight.
- Atmospheric refraction, terrain, tidal height and ellipsoid variation are omitted.
- Clear the unknown numerical field before using a supported inverse; selection controls retain their captured branches.
Common questions
What assumptions matter for this model?
Distances use the captured straight tangent geometry on a sphere of radius 6371000 meters. This is not an ellipsoidal geodesic distance or a refracted optical horizon.
What does this model leave out?
Atmospheric refraction, terrain, tidal height and ellipsoid variation are omitted.
References
Model requirements: Observer height must be nonnegative. Target distance must be nonnegative. Port the exact pinned spherical-horizon helper to piecewise standalone arithmetic; preserve the zero-obscuration branch and editable horizon/height inverse. Permit the mathematically defined zero-height and zero-distance endpoints, which the captured positive-only input checks rejected despite describing sea-level geometry.