Understand the relationship
The reasoning behind the result
A mosaic is a grid of repeated field footprints
C(n) = F + (n − 1)F(1 − o)
One panel contributes one usable field width F. Every additional panel adds only the centre step F(1 − o), because fraction o overlaps the previous panel.
Rows and columns use their own fields and overlaps. Multiplying the two minimum whole-number counts gives the rectangular panel total; neither percentage is silently copied to the other axis.
Rotation enlarges the detector-axis bound
Wb = |W cos φ| + |H sin φ|; Hb = |W sin φ| + |H cos φ|
The entered target remains a rectangle, but a rotated rectangle projects onto both detector axes. The axis-aligned bounds are expanded by separately entered per-side margins before panel counts or inverse dimensions are solved.
This bounding rectangle is conservative for the entered box and can be larger than the actual target outline. No catalogue, irregular footprint or sky position angle is inferred.
Minimum counts retain the discrete boundary
n = 1 when R ≤ F; otherwise n = 1 + ceil((R − F)/(F(1 − o)))
Panel counts are discrete. A fractional requirement must round upward, while an exact mathematical boundary must not gain a spurious extra panel from floating-point rounding.
This interface supports up to 50 panels on an axis and 400 total so that the numbered visual and centre ledger remain inspectable.
Fixed-grid inverse field is an algebraic requirement
F = R/[1 + (n − 1)(1 − o)]
For a fixed count and overlap, the coverage equation can be rearranged to find the minimum ideal field on each axis. Its reconstruction residual is printed to make the inverse auditable.
The angular answer does not identify hardware or prove that a camera's corrected, illuminated and stackable field reaches that dimension.
Maximum overlap is a coverage boundary
omax = (nF − R)/[(n − 1)F]
When a required span exceeds one panel field but fits in n fields at zero overlap, this inverse finds the largest overlap that leaves exactly enough ideal coverage.
It is an upper geometric limit rather than a recommended setting. Margins must represent any desired allowance, and real observations may need further reduction after footprint verification.
Panel centres are local coordinates
xi = (i − (columns − 1)/2)sx; yj = ((rows − 1)/2 − j)sy
The centred grid gives each panel an X/Y offset in arcminutes relative to the mosaic centre. Row one is drawn at positive Y and column one at negative X.
These offsets do not contain celestial coordinates, projection, parity, rotator angle, declination scaling, guide-star feasibility, slew timing or a mount command.
Follow the numbers
Build a rotated-target mosaic with independent overlaps
- Rotate the entered 180 × 90 arcminute target box by 25° and calculate its detector-axis bounding width and height.
- Add 6 arcminutes to each horizontal side and 4 arcminutes to each vertical side to obtain the required rectangular coverage.
- Convert 15% horizontal overlap on a 90-arcminute field to a 76.5-arcminute column step.
- Convert 20% vertical overlap on a 60-arcminute field to a 48-arcminute row step.
- Round each continuous panel requirement upward only after preserving the exact one-panel boundary.
- Centre the resulting grid, number it by the selected serpentine path and retain every panel centre and rectangular bound in the ledger.
The result is a complete ideal detector-frame layout with explicit excess coverage, ready for a separate real-sky footprint and equipment check.
Quick guide
How to use this calculator
- Choose minimum grid to calculate rows and columns from a usable field, required field to size a panel footprint for a fixed grid, or maximum overlap to find the geometric coverage limit of a fixed grid.
- Enter a rectangular target or framing bound in arcminutes and its rotation relative to the planned detector axes. Add explicit per-side margins for any allowance you want included in the arithmetic.
- Use a field already reduced to usable rectangular coverage when known. The optics option derives an ideal sensor field and cannot subtract vignetting, stacking crop or distortion for you.
- Read the numbered panel-centre ledger in its declared detector frame. Convert it to a real sky, mount or acquisition plan only with the appropriate coordinate, orientation and equipment tools.
Calculation method
Calculation and interpretation
Turn entered detector-frame target bounds and camera coverage into an auditable rows-by-columns layout with exact panel centres, coverage margins and inverse requirements.
Rotated bounds: Wb = |W cos φ| + |H sin φ| and Hb = |W sin φ| + |H cos φ|; step = field × (1 − overlap); coverage = field + (panels − 1) × step; minimum panels = 1 + ceil((required span − field)/step); required field = required span/[1 + (panels − 1)(1 − overlap)]; maximum overlap = (panels × field − required span)/[(panels − 1) × field].
Worked example
Build a rotated-target mosaic with independent overlaps
The result is a complete ideal detector-frame layout with explicit excess coverage, ready for a separate real-sky footprint and equipment check.
Rotated bounds: Wb = |W cos φ| + |H sin φ| and Hb = |W sin φ| + |H cos φ|; step = field × (1 − overlap); coverage = field + (panels − 1) × step; minimum panels = 1 + ceil((required span − field)/step); required field = required span/[1 + (panels − 1)(1 − overlap)]; maximum overlap = (panels × field − required span)/[(panels − 1) × field].
Supported inputs
Precision and limits
Ideal rectangular usable fields
Every panel is treated as the same axis-aligned usable rectangle. Chip gaps, obstruction, vignetting, field curvature, distortion, sensor tilt and stack rejection are not modelled.
Entered target box
The target is only the entered rotated rectangle. The page does not fetch an object catalogue, trace an irregular outline, transform a sky coordinate or determine a position angle.
Overlap is geometric
Entered and solved overlaps describe ideal footprint duplication. They do not guarantee common stars, successful registration, equal exposure depth, flat-field quality or a seamless stitched image.
No acquisition commands
Panel centres are local detector-frame offsets. Numbering is a planning ledger and does not prescribe mount order, RA/Dec moves, guide-star acquisition, meridian handling or rotator control.
Verify the real footprint
Dithers, mosaic rotation, pointing uncertainty, stacking crop and usable edge quality can all change coverage. Check the final footprint in the appropriate sky and equipment planning software.
No exposure or storage plan
Panel geometry does not calculate per-panel integration, filter allocation, overhead, data volume, signal-to-noise ratio or storage. Those remain separate workflows.
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