Circles¶
Recall (doc/postdoc). Not a first course.
Job: metric spheres \(S^1(c,r)\) in Euclidean plane; group of rotations; power of a point.
Core¶
\[
(x-h)^2+(y-k)^2=r^2,\qquad
C=2\pi r,\quad A=\pi r^2
\]
| Object | Definition |
|---|---|
| Chord | segment joining two points on circle |
| Diameter | chord through center; length \(2r\) |
| Tangent | line meeting circle in exactly one point; \(\perp\) radius |
| Arc / sector / segment | portion of circle / pie / chord-cap |
Central angle \(=\) arc measure; inscribed angle \(=\) half the subtended arc (see theorems).
Used later for¶
- Circle theorems: angle/power facts
- Complex analysis: \(|z-a|=r\); Möbius send circles/lines to circles/lines
- Fourier: \(S^1\) analysis
- Differential geometry: curvature of plane curves
Check¶
Is every circle in \(\mathbb{R}^2\) the image of \(S^1\) under an affine map?
Under similarities yes (isometry + scale + translate). General affine maps send circles to ellipses — not circles unless conformal linear part (scalar \(\times\) orthogonal).