The geometry model is part of the answer
This compound miter calculator models two equal-thickness panels symmetrically around one finished seam. Included angle C is the angle between finished faces. Slope S tilts the workpiece reference edge from a horizontal reference plane. M is horizontal rotation from a square-cut saw-table zero; B is blade tilt from vertical zero. The default is an inside joint, left piece, face up.
The controls map an inside or outside joint, left or right piece, and face-up or face-down orientation to signed directions. Magnitudes describe the corresponding absolute scale readings. These definitions do not include a crown spring angle, unequal thickness, a profiled back, or a manufacturer-specific alternate zero. Confirm the named zero conventions on the actual equipment.
From two face normals to a cut-plane setting
- Enter the finished included angle C strictly between 0.1° and 179.9°, then enter slope S between −89° and 89°.
- Choose inside or outside joint, the left or right symmetric piece, and whether the referenced face is up or down. Each toggle changes a documented direction sign.
- Read signed M and B to retain orientation, or use the adjacent absolute scale values only with the displayed direction labels.
- Compare the top and end views and the reconstructed included-angle check. A flat S = 0° case must return B = 0°.
- Print or copy the complete reference, including model and toggles, so a bare angle is never detached from its convention.
Right-handed vector construction
- The local axes are x along the reference edge toward the cut, y left across the face, and z out of a face-up workpiece. This is a right-handed coordinate system.
- For a flat symmetric joint, the half-angle plane normal is q0 = (cos(C/2), sin(C/2), 0). It gives ordinary symmetric miter M = C/2 and B = 0.
- A right-hand rotation about x by S creates q = (cos(C/2), cos S sin(C/2), sin S sin(C/2)). The two face planes and their cross product establish the seam direction.
- M = atan2(qy, qx), and B = asin(clamp(qz)). Piece, joint, and face selections map signs after this geometric solve rather than changing the absolute construction.
- The implementation normalizes the plane normal, reconstructs C with 2 atan2(hypot(qy,qz),qx), checks symmetry, and never returns NaN or Infinity for an accepted input.
Worked compound and flat references
- Flat square corner: C = 90° and S = 0° gives |M| = 45° and |B| = 0°, the ordinary symmetric miter limit.
- Sloped square corner: C = 90° and S = 30° gives |M| ≈ 40.893° and |B| ≈ 20.705° under this model. The reconstructed included angle remains 90°.
- Obtuse panel joint: C = 120° and S = 25° gives |M| ≈ 57.501° and |B| ≈ 21.469°. Swapping left and right reverses direction signs, not magnitudes.
- Changing face up to face down reverses bevel direction while preserving its absolute setting. Outside-joint mapping likewise changes the named orientation, not the underlying symmetric plane.
Why compound calculators disagree
A displayed 35° may be measured from square, from the board edge, from vertical blade zero, or from the blade plane. Face-down stock and opposite pieces can reverse signs. A formula can be correct under one convention and wrong when its labels are reused under another. This page keeps input and output definitions next to the results.
Simple miter as a verification boundary
At S = 0° the normal has no z component, so bevel is exactly zero and miter reduces to C/2. That reduction, left/right sign symmetry, unit-normal length, and reconstructed C are used as numerical property tests. Full derivation and known limits are recorded in the methodology document.
Model limits, calibration, and material behavior
This is a geometry reference, not a universal saw setting or operating procedure. It does not model crown molding spring angle, asymmetric pieces, unequal thickness, blade kerf, fence error, zero calibration, stock cupping, material movement, or machine limits. Calculator differences often come from different definitions. Consult the equipment manual and use suitable professional judgment and a reference fit for real material.
The calculation is browser-local. Share URLs retain only C, S, joint, piece, and face selections.