Triangles on a ball
Ukurasa huu bado haujatafsiriwa kwa Kiswahili, kwa hiyo unaonyeshwa kwa English.
Saidia kutafsiri →Swali
On flat paper the angles of every triangle add up to 180°, and a² + b² = c² in every right triangle. Draw a triangle on a ball instead. Is either rule still true?
Fanya jaribio kabla ya kuendelea kusoma. Hakuna njia mbaya ya kuchunguza.
Jaribio
Pembetatu kwenye nyuso zilizopinda
Buruta vipeo. Kwenye ndege bapa jumla ya pembe ni 180°; kwenye tufe ni zaidi, na kwenye ndege hipaboliki ni chini.
Buruta kipeo ili kukisogeza; buruta mahali pengine popote ili kugeuza tufe. Pande ni matao ya duara kuu — njia fupi kabisa juu ya tufe.
- ∠A80.6°
- ∠B66.6°
- ∠C64.7°
∠A + ∠B + ∠C = 212.0°
- Ziada ya pembe (jumla − 180°)
- 32.0° = 0.558 rad
- Eneo, limepimwa kwa hesabu za namba
- 0.558 R²
- Sehemu ya tufe zima (eneo ÷ 4πR²)
- 4.44 %
∠A + ∠B + ∠C − π = Eneo / R²Kwenye tufe lenye nusukipenyo R, ziada kwa radiani ni sawa na eneo likigawanywa kwa R². Eneo linakokotolewa kando, kwa kuunganisha kwa namba (integration), kwa hiyo kulingana kwao ni ukaguzi, si dhana.
∠A + ∠B + ∠C − π = K · Eneo
| Ndege bapa | K = 0 | ∠A+∠B+∠C = 180° |
|---|---|---|
| Tufe | K = 1/R² | ∠A+∠B+∠C > 180° |
| Ndege hipaboliki | K = −1 | ∠A+∠B+∠C < 180° |
Huu ni mtazamo wa kwanza wa nadharia ya Gauss–Bonnet: kwa pembetatu ambayo pande zake ni jiodesiki (njia fupi kabisa), jumla ya pembe toa π ni sawa na mpindo wote ulio ndani yake. Hapa mpindo K ni uleule kila mahali; kwenye nyuso nyingine K hujumlishwa (huunganishwa) juu ya eneo.
Vidokezo, ukivitaka
Kinachoendelea
Kinachoendelea
On a sphere the "straight lines" are great circles — circles whose centre is the centre of the sphere, like the equator. The walk in the first hint makes a triangle with three right angles: its angles add up to .
Albert Girard found the rule in 1629. On a sphere of radius , a triangle with angles (in radians) has
Our triangle covers one eighth of the sphere, so its area is , and the excess is — exactly the extra .
Pythagoras fails too. All three sides of that triangle are quarter-circles of the same length , so . The spherical version of the theorem is
and for small triangles it is very close to — which is why flat geometry works so well for drawing a field or a town, even though the Earth is round.
The difference between the angle sum and measures how curved a surface is. On a saddle-shaped surface — the hyperbolic plane — triangles have angle sums less than . Gauss, Bolyai, Lobachevsky and Riemann turned these observations into non-Euclidean geometry, and Einstein used curved geometry to describe gravity.
Hii inaelekea wapi
- Shahada ya kwanza (mwanzo) · Somo linapatikanaNon-Euclidean geometry: spherical and hyperbolicWhat happens when the parallel postulate fails: geometry on the sphere, the Poincaré disc and half-plane models of the hyperbolic plane, angle sums and area, curvature as the measure of the difference, and the independence of the parallel postulate.
- Shahada ya kwanza (juu) · Somo linapatikanaDifferential geometry of curves and surfacesCurvature and torsion of space curves, the first and second fundamental forms of surfaces, Gaussian and mean curvature, Gauss's Theorema Egregium, geodesics and the Gauss–Bonnet theorem.
- Mwaka wa kwanza wa uzamili · Somo linapatikanaRiemannian geometryCurved spaces studied from the inside: Riemannian metrics, the Levi-Civita connection, geodesics and the exponential map, curvature tensors, Jacobi fields, the global Gauss–Bonnet theorem, Hopf–Rinow, and comparison theorems such as Bonnet–Myers.