Inputs and coordinate system
Canvas points use x coordinates increasing to the right and y coordinates increasing downward. The tool converts point differences to vectors, so translation does not change an angle. Image zoom changes the view transform; point coordinates remain in the workspace coordinate system used by the calculation.
The methods calculate an angle from user-selected geometry; they do not run edge detection or infer the real-world plane. Results are rounded for display while exports retain additional decimal precision. Degenerate inputs, such as a zero-length ray or line, do not produce a valid angle.
Formulas used by each mode
- Three-point: for endpoints A and B with vertex V, θ = arccos(((A−V)·(B−V))/(|A−V||B−V|)).
- Multiple rays: sort all endpoints by atan2 direction around the shared vertex, then apply the three-point formula to every adjacent pair; the semicircle set does not add a closing angle between the two outermost rays.
- Two-line: compute atan2 for direction vectors P2−P1 and Q2−Q1, take their absolute directional difference, then reduce it to the smaller intersection angle.
- Overlay: normalize the manually aligned baseline rotation into the 0°–180° range; tick selection remains a visual user decision.
- Complement = 90°−θ when 0°≤θ≤90°. Supplement = 180°−θ. Reflex = 360°−θ for a nonzero interior angle.
- Radians = degrees×π/180. For a slope angle θ measured from horizontal: percent grade = 100×tan(θ), run-per-unit-rise = 1/tan(θ), and roof rise per 12 = 12×tan(θ).
Validation and repeatability
- Unit tests cover 0°, acute, right, obtuse, straight, and reflex classifications plus degree/radian and slope conversions.
- Known coordinate fixtures verify three-point and two-direction calculations.
- Metadata tests require unique titles, descriptions and H1 values, self-referencing canonicals, sitemap coverage and parseable visible-content schema.
- A repeatability check should move endpoints farther from the vertex, repeat placement, and compare the spread rather than reporting false instrument precision.
Error model
- Point-placement error grows when endpoints are close to the vertex or edges are thick and blurred.
- Perspective error occurs when a photographed plane is tilted relative to the sensor; this can change the visible angle even with perfect point placement.
- Lens distortion is often strongest near frame edges. Scan skew and unequal x/y scaling can change document geometry.
- Rounding affects the displayed last digits, while source uncertainty usually dominates browser arithmetic.
- Sensor mode adds device calibration, case alignment, orientation handling, filtering, and drift errors.
Version and update record
Methodology version 1.1 — 2026-08-10: added the homepage multiple-ray set, direction sorting, adjacent-angle labels, complete-set saving, and automated interaction and geometry tests. Version 1.0 — 2026-08-03: documented the three-point vector, two-line direction, overlay rotation, derived angle, and slope formulas; added contextual outputs and automated SEO/content checks. Future behavior changes must update this record and the corresponding tests.