Understand the relationship
The reasoning behind the result
Sensor dimensions and focal length define an ideal angular field
θ = 2 atan(s/(2f))
A rectilinear ideal image maps a centred angular extent θ to focal-plane dimension s at effective focal length f. Applying the inverse tangent preserves the full geometric angle instead of assuming that every sensor-to-focal ratio is infinitesimal.
The camera width and height are reconstructed from native active pixels multiplied by native pixel pitch. Real optical distortion, field curvature, vignetting, tilt, cropping and a corrected image circle can change the usable footprint and are not inferred.
Plate scale is a local angular sampling interval
q = 206.264806247 p/f
At the optical axis, pixel pitch p in micrometres divided by focal length f in millimetres gives a small angle after the unit conversion. The constant is the number of arcseconds in a radian divided by one thousand.
Multiplying q by a symmetric binning factor gives arcseconds per binned output pixel. The full sensor field does not change because the same native detector area remains active.
Rotated object framing uses a declared rectangular bound
Wb = |W cos φ| + |H sin φ|; Hb = |W sin φ| + |H cos φ|
A rotated rectangle needs a larger axis-aligned box unless its orientation matches the sensor axes. Subtracting that bound from the exact camera field gives centred margins; a negative value exposes a geometric overrun.
This is deliberately a box model. It does not import an object catalogue, irregular sky outline, celestial position angle, coordinate epoch or mosaic footprint.
Seeing sampling retains the entered width
samples/FWHM = entered seeing FWHM / effective plate scale
The comparison mode expresses each entered seeing full width at half maximum in effective binned pixels. It keeps the atmospheric or measured width in the record instead of supplying a site value.
A numerical samples-per-FWHM ratio does not by itself establish image quality. Wavelength, diffraction, focus, guiding, reconstruction, optical blur and time variability remain outside this arithmetic.
Dither conversion preserves both signed components
Δxsky = Δxpx q; Δysky = Δypx q; r = hypot(Δxsky, Δysky)
Multiplying signed detector coordinates by the declared or derived local scale converts each component to arcseconds. The radial magnitude is supplemental and never replaces direction.
A real sky or mount command can require axis orientation, parity, declination handling, rotator angle, guider scale, distortion and control-system conventions. None is guessed from a detector-frame pixel path.
Follow the numbers
Reconcile a rotated target box on an entered camera
- Multiply 6,248 × 4,176 native pixels by 3.76 µm per pixel to reconstruct a 23.49248 × 15.70176 mm active sensor.
- Use 2 atan(sensor dimension/(2 × 500 mm)) to obtain the exact ideal horizontal and vertical fields.
- Use the local small-angle relation to obtain native arcseconds per pixel; 1 × 1 binning leaves that interval unchanged.
- Rotate the entered 120 × 60 arcminute rectangular target box by 20° and form its axis-aligned width and height.
- Subtract those rotated bounds from the corresponding camera fields and divide by two to retain centred per-side margins.
- Review the live footprint and ledger while keeping optical correction, target coordinates and observing feasibility outside the result.
The result is a transparent ideal framing and sampling record for the entered geometry, not a catalogue lookup or equipment recommendation.
Quick guide
How to use this calculator
- Choose framing for one camera and centred object box, inverse for a required field or local scale, comparison for named seeing-and-sampling records, or dither for signed detector offsets.
- Use effective focal length for the complete optical train and active native sensor dimensions. Enter object dimensions, seeing and dither coordinates only from the same declared scenario.
- Read exact full-field angles separately from the local small-angle plate scale. Binning changes the sampling interval but not the physical sensor footprint.
- Inspect the ledger and visual before using a result elsewhere. The page performs geometry and bookkeeping; equipment compatibility, optical correction and observing decisions remain external.
Calculation method
Calculation and interpretation
Turn entered sensor, focal-length, object-box, seeing and dither records into explicit ideal geometry while keeping local sampling separate from full-field projection and equipment suitability.
Sensor dimension = native pixels × pitch; field = 2 atan(sensor dimension/(2f)); local scale = 206.264806247 × pitch(µm)/f(mm) × binning; samples/FWHM = entered seeing/local scale; dither sky component = signed pixel component × local scale.
Worked example
Reconcile a rotated target box on an entered camera
The result is a transparent ideal framing and sampling record for the entered geometry, not a catalogue lookup or equipment recommendation.
Sensor dimension = native pixels × pitch; field = 2 atan(sensor dimension/(2f)); local scale = 206.264806247 × pitch(µm)/f(mm) × binning; samples/FWHM = entered seeing/local scale; dither sky component = signed pixel component × local scale.
Supported inputs
Precision and limits
Ideal central projection
Field angles use a centred rectilinear model. Distortion, field curvature, sensor tilt, reducer spacing, image-circle limits, vignetting and cropping are not modelled.
Entered object box only
Object dimensions and rotation are visitor-entered rectangular bounds. The workbench does not fetch a sky catalogue, ephemeris, orientation, coordinate epoch or irregular outline.
Sampling is descriptive
Entered seeing divided by effective plate scale is not labelled good, bad, under-sampled or over-sampled and does not establish focus, guiding, resolution or signal quality.
No mount or guider commands
Dither outputs remain signed detector-frame offsets. They do not convert to RA/Dec, pulse duration, guide-camera pixels, rotator coordinates or a safe motion command.
No equipment compatibility decision
Solved focal lengths, pitches and sensor dimensions are mathematical requirements. Availability, back focus, adapters, corrected field and mechanical clearance must be checked independently.
No exposure or noise model
This batch does not calculate signal-to-noise ratio, exposure time, well depth, storage, mosaic overlap or integration strategy; those remain separate reviewed workflows.
Continue calculating
Related calculators
