Intuition: best‑fit circles in plain English
When you draw, the app collects many points along your stroke. A best‑fit circle is the single circle—defined by a center and a radius—that comes closest on average to those points. If your stroke is smooth and well centered, most points sit at almost the same distance from that center. If the distance wiggles, roundness drops. If the center drifts from the target, centering drops. If the start and end don’t meet, closure drops. The Kasa method is a fast way to compute that best‑fit circle.
The core formula (no heavy math)
The Kasa method uses algebra to find a center (a, b) and radius r that minimize squared distance errors. You don’t need the derivation to benefit from it, but here’s the gist:
- We compute simple sums of your points: the sums of
x,y,x²,y², andxy. - From those, we solve a tiny linear system that yields the best‑fit center.
- The average distance from the points to that center is the best‑fit radius.
This approach is numerically stable and blazing fast in the browser, perfect for instant feedback after each stroke.
From fit to score: roundness, centering, closure
- Roundness. We measure how tightly the distances from your points to the fitted center cluster around the radius (low variance = high roundness).
- Centering. We measure how far the fitted center is from the golden dot. Less distance = better centering.
- Closure. We measure the distance between the start and end points of your path, and whether the end’s tangent matches the start.
The overall score blends these elements (see Roundness Explained and Centering Techniques for practical tips).
Why this method is robust for drawing
- Noise tolerant. It averages over all points, so tiny jitters don’t dominate.
- Direction agnostic. Whether you draw clockwise or counter‑clockwise, the fit is the same.
- Scale friendly. Works for small to large radii; we adjust grid and tempo so small circles aren’t unfairly penalized.
How to use the insights to improve
- If roundness is low, reduce micro‑corrections: lead with shoulder/elbow and keep a 2.5–3.0s tempo.
- If centering is low, track the dot–stroke distance; use grid rails.
- If closure is low, aim through the start point and match tangent earlier (see Closure Techniques).
FAQ
Is Kasa the only way to fit a circle?
No—there are geometric and algebraic alternatives. Kasa is a great trade‑off between speed, stability, and accuracy for real‑time feedback.
Does drawing faster trick the fit?
No. Speed changes the pattern of your errors but the fit still reflects actual distances. If wobble increases, roundness drops accordingly.