Arc, chord, sagitta — interactively.
For readers who learn by seeing.
This page is the gentle on-ramp to the rest of the site. The three concepts you will meet here — the arc, the chord, and the sagitta — are the vocabulary every other page assumes you know. If you have not seen these shapes before, this is the place to start. Move the sliders. Watch the shapes change. You do not need to read the equations; the visuals carry the story.
When you reach the bottom of the page, the same geometry will be ready to translate from circles into relationships between minds — the subject of every other page on this site.
Start with the simplest shape. A circle has a center, a radius, and a rim. The rim is the curved boundary. A quadrant is one quarter of the way around — a 90° slice of pie. Two quadrants is half, three is three-quarters, four is the whole circle.
Pick a radius. Pick a quadrant. Watch the arc length and the sector area change.
Two things to notice. First, when you make the radius larger, both the arc length and the area grow — bigger circle, more rim to walk along and more pie to eat. Second, when you sweep a wider slice, again both grow — same circle, more of it covered.
This is orientation. Next: what happens when you connect two points across the curve instead of along it.
Now we have two points on the rim of a circle, separated by some angle. There are two ways to measure the distance between them. Both are real distances; neither is wrong; they are not the same.
The arc (shown in ochre) is the path that follows the curve. It is what you walk if you stay on the rim.
The chord (shown in red) is the straight line across. It is what you walk if you cut through the interior.
Watch what happens as you widen the angle:
At small angles, the arc and chord are nearly the same length. Look at the numbers when θ is 10° or 20° — the difference is tiny. The curve barely matters at small scales; a short walk along the rim is almost the same as a short walk straight across.
But as you widen the angle, they pull apart. The arc keeps growing; the chord can only grow so far. At θ = 180° (a full half-circle), the chord is a straight diameter while the arc is the entire top half — the arc is more than 50% longer than the chord. By the time you sweep the whole circle (360°), the arc is the full circumference and the chord has collapsed back to nothing.
This is the central distinction the rest of this site rests on. The chord is what you get when you ignore the curvature. The arc is what you get when you honor it. Both are real measurements. They are not the same measurement. And the gap between them grows as the relationship between the two points becomes wider, deeper, more curved.
So the arc and the chord differ. But how do we measure how they differ? Not just in length — in shape. In what's between them.
Drop a perpendicular line from the midpoint of the chord up to the arc. That perpendicular distance is called the sagitta — Latin for arrow. It is the height of the arc above the chord at its highest point.
Watch the cyan line as you change the angle:
At narrow angles, the sagitta is tiny. The chord is a good approximation of the arc — the gap between them is small. At wide angles, the sagitta grows large. The chord is missing more and more of what the arc actually traces.
The sagitta is what we mean by “the perpendicular dimension the chord can't see.” It is not just length; it is height. It is the part of the geometry that a straight line, by construction, cannot represent. To draw the chord is to write down a measurement between two points. To know the sagitta is to know how far the actual curve rises above that straight-line measurement.
If you take only one thing from this primer, take this: the sagitta is the part of the relationship that survives only when the curvature is honored.
That is the whole geometric vocabulary you need: arc, chord, and the sagitta between them.
On the rest of this site, we use the same vocabulary to describe something different — not lines on a circle, but relationships between minds. The chord becomes the explicit artifact that connects two parties: the manuscript, the transcript, the recorded exchange. The arc becomes the trajectory the relationship is actually traversing through the curved surface of meaning. The sagitta becomes the perpendicular dimension that the explicit artifact, by construction, cannot capture — what we call the subtended heart.
The four geometric properties on the rest of the site — angular fidelity, the implied center, torque at orthogonality, and scale invariance — each name a different consequence of the chord-vs-arc distinction you just watched on the sliders. And the article that ties them together — The Subtended Heart — is about what living intelligences must do to keep the chord touching the arc, instead of letting the curvature collapse.