A circle is not drawn: it is generated by a condition — all points that are at the same distance from a center. And the distance of a point to the origin is the hypotenuse of a right triangle whose legs are the x and y themselves. The circle is born from the triangle, and here you can see it happening.
Do: drag x and y and watch the hypotenuse form. Then fix the required distance r and press sweep all points at this distance: the circle appears as result of the condition, not as a pencil stroke.
1. The distance is the hypotenuse. Move the point and observe the triangle: the legs are exactly the x and the y of the point, and the distance to the origin is the hypotenuse. It's not an analogy — it's the same triangle from the previous slice, with the legs laid over the axes. That's why the triangle comes before the circle.
2. The circle is the set that satisfies a condition. Choose a r and sweep: the points that fulfill distance = r are marked, and their trail is the circle. No one drew it — it is what remains when one demands one thing only from every point of the plane.
3. And the condition, written, is the equation. Squaring removes the root and leaves x² + y² = r². Watch the reading line as you move: when the sum x² + y² touches r², the point is on the circle. The equation is not a formula to memorize — it's the condition, expressed in symbols.
A set of points defined by a condition is called a geometric locus. The circle of radius 1 is the geometric locus of the points of the plane whose distance to the origin is 1 — and the condition, written, is
x² + y² = 1.
This set has a proper name in mathematics: S¹. And here lies a trap that is worth disarming before it catches you, because on this same page appear three superscripts with three meanings:
x² | the 2 is power — x multiplied by itself |
ℝ² | The 2 is the dimension of the space where the thing lives — the plane has two axes |
S¹ | the 1 is the dimension of the thing itself — the circle is a curve, and a curve is one-dimensional |
A circle drawn on the plane is a one-dimensional object within a two-dimensional space: moving along it gives one freedom only — forward or backward along the curve. It's the same idea from the first slice, one level up.
Why sin² + cos² = 1. Put r = 1. A point of the unit circle has coordinates (cos θ, sin θ) — it is the slice of the circle that shows this, with the cosine along the floor and the sine perpendicular to it. Substituting in the condition: cos²θ + sin²θ = 1. The fundamental identity is Pythagoras in
the circle of radius 1, and not a new formula.
Why a circle needs two accounts to be written. Test the rule of
the function: for x = 0 and r = 2 there are two points,
y = +2 and y = −2. One input, two outputs — the circle
is not a function, and breaks the requirement that the first slice establishes. It is exactly
for this reason that it is written as (r·cos θ, r·sin θ): an angle enters, and the
two numbers come out together.
Local, self-contained instrument: no network, no external library. Between
o-triangulo and o-circulo in the order of the family.