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CHECK YOUR CONVENTIONS / THREE WORKED EXAMPLES
From numbers to orientation
Verified against the converter’s passive matrices and its visible output formatter. Expected values below use MATLAB format, six decimal places, degrees, and output sequence 321 (ZYX).
01
Single-axis rotation
A quarter-turn about Z
With Passive (C), degrees, and ZYX selected, enter Z = 90, Y = 0, X = 0. The product reduces to C_Z(90°). Multiplying C by [1, 0, 0]ᵀ gives [0, −1, 0]ᵀ: the same vector has new coordinates. Transposing C gives the paired active matrix. The equivalent negative quaternion is [0, 0, −0.707107, −0.707107].
Determinant = 1; orthogonality residual = 0 apart from floating-point roundoff. The ideal zeros in this matrix may appear as very small numbers internally.
Use Passive (C), degrees, and ZYX with Z = 10, Y = 20, X = 30. The actual product is C_X(30°) C_Y(20°) C_Z(10°). To reproduce it in Sequence mode, create Step 1: Z = 10; Step 2: Y = 20; Step 3: X = 30, with every other angle zero. Step 1 acts first. Reversing those steps generally produces a different result.
The output Euler vector is [Z, Y, X], not [X, Y, Z]. Rebuilding the passive matrix from these values returns the same matrix to floating-point precision; the output axis-angle expresses the same combined orientation as one rotation.
Use Passive (C), degrees, and ZYX with Z = 0, Y = 89.95, X = 0. The middle angle is close enough to the singularity to trigger the solver’s |sin(β)| ≥ 1 − 10⁻⁶ branch. It sets the displayed middle Euler angle to 90°. The matrix, quaternion, and quaternion-derived axis-angle still encode 89.95°. This is intentional documentation of the existing tolerance, not a change to the solver.
A near-singularity warning appears. Reconstructing from [0, 90, 0] differs from the original matrix by about 0.000873 in the largest entry. Use the original matrix or quaternion when that difference matters. At exact lock the first and third angles are not uniquely recoverable.
Different numbers can describe the same orientation
A unit quaternion q and its opposite −q produce the same rotation matrix. Euler angles can differ by full turns or by an equivalent branch; at exact gimbal lock, individual first and third angles are not unique. Compare matrices under the same convention rather than insisting that parameter strings match.
Expected outputs here are rounded to six decimal places. Tiny roundoff and the explicitly documented singular-branch tolerance should be distinguished from a frame or component-order mismatch. See the numerical checks guide.
