Enter Path Samples
Provide a sequence of points (x, y). Use commas, spaces, or semicolons. One point per line. The calculator estimates how many times the path winds around the origin.
Formula Used
The winding number counts how many times a closed curve wraps around the origin. For a smooth closed curve (x(t), y(t)), it can be written as:
For sampled points, this calculator estimates the total angle change by summing the turning of each segment. The robust method uses:
This avoids manual unwrapping and is stable for most well-sampled loops that do not pass through the origin.
How to Use This Calculator
- Enter points describing your path, one per line.
- Keep the path centered around the origin if needed.
- Enable auto-close unless your last point equals the first.
- Choose the robust method for reliable results.
- Click compute, then export CSV or print to PDF.
Example Data Table
A simple unit circle sampled at four quadrants (counterclockwise) should give a winding number near +1 when closed.
| # | x | y | Comment |
|---|---|---|---|
| 1 | 1 | 0 | Start |
| 2 | 0 | 1 | Quarter turn |
| 3 | -1 | 0 | Half turn |
| 4 | 0 | -1 | Three quarters |
| 5 | 1 | 0 | Closed |
Technical Article
1) What the winding number measures
The winding number is a dimensionless count of how many full turns a closed planar path makes around a reference point (the origin here). It is topological: as long as the loop stays closed and does not cross the reference point, the count does not change under smooth deformations.
2) Connection to phase, vortices, and defects
Many wave and field problems use a phase angle that winds around singular cores. If a complex field is A = |A| e^{i theta}, then theta can change by an integer multiple of 2*pi around a vortex. That integer is the winding number, with sign showing counterclockwise or clockwise circulation.
3) Discrete sampling and resolution effects
Real data arrive as sampled points. Reliable counting needs enough samples so each segment represents a small rotation about the origin. A practical check is closeness to an integer: refine sampling until the deviation is small (for example, below 0.02 for clean traces), especially for tight curves or multi-turn loops.
4) Robust segment angle summation
The recommended method sums incremental angle changes between consecutive radius vectors. For segment i it uses atan2(det, dot), where det = x_i*y_{i+1} - y_i*x_{i+1} and dot = x_i*x_{i+1} + y_i*y_{i+1}. Summing increments gives total rotation; dividing by 2*pi yields the winding estimate.
5) Angle unwrapping as a diagnostic
Unwrapping computes theta = atan2(y, x) at each point and wraps each difference into (-pi, pi]. It is intuitive for debugging and comparisons, but it can amplify noise when points pass near the origin. For most datasets, edge-sum is more stable because it aggregates segment geometry directly.
6) Origin tolerance and stability checks
Near the origin, direction is ill-defined and small numerical noise can cause large angle jumps. The tool counts points within a user tolerance (epsilon). If you see hits, increase clearance from the origin, shift the reference point (by subtracting x0,y0 from samples), or sample more densely away from the singular region.
7) Interpreting non-integer outputs
Ideal winding is an integer for a closed loop that avoids the reference point. Non-integers usually mean the loop is not closed, the path grazes the origin, or sampling is too coarse. Use the nearest-integer and deviation values as quick quality control, then improve closure or sampling as needed.
8) Practical applications and reporting
Typical uses include verifying vortex charge in phase maps, checking contour-orientation in simulations, counting turns in tracked planar motion, and validating loop-like Lissajous trajectories. Export CSV for lab logs and print to PDF for reports. Recording method, tolerance, and closure distance helps keep results reproducible.
FAQs
1) What does the sign of the winding number mean?
Positive values typically mean the path rotates counterclockwise around the origin. Negative values mean clockwise rotation. The sign follows the standard orientation of the x-y plane used by atan2.
2) Do I need to repeat the first point at the end?
No. If Auto-close is enabled, the tool connects the last point to the first. If you disable Auto-close, you should provide a closed loop explicitly to obtain a standard winding number.
3) My result is 0.73 instead of an integer. Why?
Non-integers usually indicate an open curve, insufficient sampling, or points too near the origin. Try enabling Auto-close, adding more points along the curve, and increasing the origin clearance using a better path or reference point.
4) Can I compute winding around a point that is not the origin?
Yes. Subtract the reference point from every sample: use x' = x - x0 and y' = y - y0, then paste the shifted points. The computed winding then applies around (x0, y0).
5) Which method should I choose?
Use the robust edge-sum method for most cases. Use angle unwrapping when you want a diagnostic view of stepwise angle changes or when you are comparing with a separate unwrapping pipeline.
6) What is a good epsilon value?
Epsilon should reflect your coordinate scale. For unit-scale data, values like 1e-9 to 1e-6 are common. If your coordinates are large or noisy, increase epsilon so near-origin warnings remain meaningful.
7) Can this handle multiple turns like w = 3 or w = -2?
Yes. If the curve loops around the origin multiple times without crossing it, the total angle change scales accordingly. Increase sampling density so each loop is resolved and the output stays close to an integer.