Radius and center—your two anchors
Every point on a circle is the same distance from the center. Freehand, the easiest way to respect that rule is to keep a steady mental picture of the radius—how far the golden dot appears from your stroke. If the gap grows or shrinks, your brain is telling you the radius changed. Correct gently while maintaining pace.
Chords, arcs, and why roundness matters
A chord is a straight line between two points on the circle; an arc is the curved path between them. When a stroke momentarily behaves like a chord (tiny straight segment), we see a flat spot and roundness drops. Drills that lengthen continuous curvature—like Clock Drill and Size Ladder—reduce these chord‑like moments and lift roundness.
Tangent and normal—seamless closures
The tangent at a point is the line that just touches the circle and points in the direction of travel; the normal points toward the center. Closure seams happen when you approach the start point with the wrong tangent. The fix is to align your tangent early and “aim through” the point; the normal stays the same length if your radius is steady.
Angles, quadrants, and pacing
Think in quadrants (0°–90°–180°–270°). Many people tense near 90° (top) and 270° (bottom). A steady 2.5–3.0 second pace smooths these transitions. Calling out the quadrants in your head—“twelve, three, six, nine”—is a surprisingly effective way to keep arc length consistent and prevent speed spikes.
Applying geometry to practice
- Roundness focus: Whole‑arm motion and even arcs (Clock Drill, Tempo).
- Centering focus: Keep radius visually constant (Center Lock, Crosshair).
- Closure focus: Match tangent before the join (Portal, Blend).
These simple ideas—radius, tangent, consistent arc length—are all you need to bring “math‑level clarity” to your freehand practice. For the math used under the hood, see The Kasa Method.