Circle - from Eratosthenes to GPS

240 BC, Alexandria. Eratosthenes reads that in Syene at noon the sun reflects vertically in wells - no shadow. In Alexandria there is a shadow. Two angles, one circle, the distance between cities - and Earth's radius more accurate than many medieval scholars. Error: 2%. Today, GPS determines coordinates through the intersection of four spheres - circles in 3D. Each satellite contributes one. Four satellites give a point. Circle geometry literally lives in every pocket on the planet.

• **GPS triangulation:** position = intersection of 4 spheres, each defined by the circle equation in 3D - (x-xi)^2 + (y-yi)^2 + (z-zi)^2 = di^2
• **Game engines:** circle-circle collision via d^2 < (r1+r2)^2 - O(1) without sqrt, scales to 60+ fps
• **Monte Carlo pi:** random points in a square, fraction inside the circle times 4 converges to pi - foundation of Monte Carlo methods
• **Computer graphics:** Bresenham circle algorithm draws circles using only integer arithmetic, exploiting the inscribed angle invariant

Предварительные знания

• Polygons

Circle Equation and GPS Interpretation

240 BC, Alexandria. Eratosthenes reads that in Syene at noon the sun reflects vertically in wells - no shadow. In Alexandria there is a shadow. Two angles, one circle, the distance between cities - and Earth's radius is more accurate than many medieval scholars achieved. Error: 2%. The tool: the definition of a circle as the set of all points equidistant from one center.

**Standard form:** (x - a)² + (y - b)² = r² **Expanded form:** x² + y² + Dx + Ey + F = 0 Center: (-D/2, -E/2), radius: r² = D²/4 + E²/4 - F **Point test:** < r² inside, = r² on circle, > r² outside

GPS determines coordinates through the intersection of circles - spheres in 3D space. Each satellite contributes one sphere: the set of points at distance d from the satellite. Four satellites give four spheres. Their intersection is a point. Circle geometry literally lives in every pocket.

In game engines: compare d² vs (r1+r2)² for circle-circle collision - no sqrt, roughly 2x faster in tight 60 fps loops.

Chord, Arc, and Inscribed Angle

A chord connects two points on a circle. An arc is the portion of the circle between them. An inscribed angle has its vertex on the circle and subtends an arc. A central angle subtends the same arc from the center. The relationship is non-obvious: the inscribed angle is exactly half the central angle, regardless of where on the circle the vertex sits.

**Inscribed angle theorem:** inscribed angle = 1/2 * central angle on the same arc ∠BAC (inscribed) = 1/2 * ∠BOC (central) **Thales' theorem:** an angle inscribed in a semicircle (subtending a diameter) equals 90°.

Thales' theorem is foundational for computer graphics. The Bresenham circle algorithm draws circles using only integer operations, exploiting the invariance of the inscribed angle: no matter which point on the semicircle one chooses, the angle remains 90°. GPU rasterization exploits this same symmetry at hardware level.

All inscribed angles subtending the same arc are equal - a direct consequence of the inscribed angle theorem. This is an invariant, not a coincidence.

Tangent Line - Perpendicular to Radius

A tangent to a circle touches it at exactly one point. The key property: the tangent is perpendicular to the radius at the point of tangency. This is not an axiom but a theorem - and it holds at every scale, from compass and straightedge to orbital mechanics.

**Tangent to x² + y² = r² at point (x₀, y₀):** x*x₀ + y*y₀ = r² **Tangent length from external point P(px, py) to circle (a, b, r):** t = sqrt((px-a)² + (py-b)² - r²)

Monte Carlo estimation of pi: random points are cast into a unit square, those landing inside the unit circle are counted. The fraction multiplied by 4 converges to pi. GPS inverts the same idea - instead of random points, a system of equations; instead of a square, spheres. The tangent plane to each sphere defines the uncertainty surface.

Power of a Point - Circle Invariant

The power of a point P with respect to a circle is d² - r², where d is the distance from P to the center. Positive outside, zero on the circle, negative inside. This is not just a number - it is an invariant: draw any secant through P, and the product of the two signed distances to the circle is always the same.

**Power of a point theorem:** if two secants through P intersect the circle at A, B and C, D: PA * PB = PC * PD = |power of P| For a tangent PT: PT² = PA * PB

UMAP and other manifold learning algorithms build low-dimensional representations through local distances. At their core lies an idea analogous to the power of a point: a distance invariant that is preserved when projecting onto a manifold. Circle geometry is not a school subject - it is the language modern ML speaks.

Power of a point as a signed distance function: negative inside, positive outside. Level-set methods in computer vision use exactly this structure - just for complex contours rather than circles.

Key Ideas

• **Equation:** (x-a)²+(y-b)²=r²; point test: compare d² vs r², no sqrt needed
• **Inscribed angle** = half the central angle; angle in semicircle = 90° (Thales) - invariant regardless of vertex position
• **Tangent** is perpendicular to radius; length from external point t = sqrt(d²-r²); GPS uses this in 3D
• **Power of a point:** PA*PB = PC*PD = PT² - invariant across all secants through the same point

Related Topics

Circles are the foundation for solids of revolution, trigonometry, and projective geometry:

• Areas and Perimeters — Circle area pi*r² and circumference 2*pi*r - the next step
• Solids of Revolution — Rotating a circle generates a sphere or torus
• Projective Geometry — Circle inversion and cross-ratio in projective geometry

Вопросы для размышления

• Eratosthenes measured Earth's radius through shadow angles and city distance. Which exact property of circles did he use - and could the experiment be replicated today?
• GPS needs 4 satellites, not 3. Why? What does the fourth sphere provide that three cannot?
• The power of a point is negative inside a circle. How does this relate to signed distance functions used in computer vision level-set methods?

Связанные уроки

• geo-05 — Circle area and circumference follow directly
• geo-03 — Polygons and proportions as foundation
• trig-01 — Inscribed angles and arcs connect through sine and cosine
• calc-03-limits-intro — Circumference as limit of inscribed polygon perimeters
• prob-04-bayes — Monte Carlo pi estimation uses circle membership
• trig-03