How sharply does a road bend?
Ukurasa huu bado haujatafsiriwa kwa Kiswahili, kwa hiyo unaonyeshwa kwa English.
Saidia kutafsiri →Swali
Driving along a winding road, how sharply does it bend at each point? Is there a circle that fits the road best at each point — and what is its radius?
Fanya jaribio kabla ya kuendelea kusoma. Hakuna njia mbaya ya kuchunguza.
Jaribio
Mpindo wa mkunjo
Nukta husafiri kando ya mkunjo. Duara busu ni duara linalolingana vizuri zaidi na mkunjo kwenye nukta hiyo; nusukipenyo chake ni 1/κ.
- Mpindo wenye ishara
- κ = 0.447
- Nusukipenyo cha mpindo
- 1/|κ| = 2.239
κ ni chanya mahali mkunjo unapopinda kushoto unaposafiri juu yake, na hasi unapopinda kulia. N ni tanjenti T iliyogeuzwa 90° kushoto; kitovu cha duara busu kiko umbali wa 1/|κ|, upande ambao mkunjo unapindia.
∫ κ ds = 6.283 = 1.000 × 2π
Namba ya mgeuko 1: ukizunguka mara moja, tanjenti hugeuka jumla ya 1 × 360°.
Umlaufsatz ya Hopf: kwa mkunjo laini uliofungwa, ∫ κ ds = 2π × (namba ya mgeuko). Mkunjo sahili uliofungwa — usiojikata wenyewe — ukipitiwa kinyume cha saa daima una namba ya mgeuko 1. Limasoni yenye kitanzi cha ndani ina 2; umbo la nane lina 0.
Urefu wa tao na ∫ κ ds hukokotolewa kwa namba kutoka vinyambuo kamili kwenye maelfu ya nukta, kwa hiyo tarakimu ya mwisho inaweza kuyumba.
Mpindo wenye ishara κ dhidi ya urefu wa tao s
Eneo lenye ishara chini ya grafu hii ni ∫ κ ds (kijani-samawati juu ya mhimili huhesabiwa chanya, udongo chini yake hasi). Thamani kubwa sana karibu na ncha kali hukatwa.
Vidokezo, ukivitaka
Kinachoendelea
Kinachoendelea
The circle that fits a curve best at a point — the limit of circles through three nearby points — is the osculating circle ("kissing circle"). If its radius is , the curvature of the curve there is : a gentle bend has a large circle and small curvature, a hairpin has a small circle and large curvature, and a straight road has curvature .
For the graph of ,
For the parabola at its lowest point, and , so and the osculating circle has radius .
Curvature has a striking global property. Walk once around a simple closed curve, adding up the curvature as you go (counting bends to the left as positive and to the right as negative): the total is always . Heinz Hopf proved this turning-tangent theorem in 1935. Road and railway engineers use curves whose curvature grows steadily — clothoids, or Euler spirals — so that the steering wheel turns smoothly instead of suddenly.
Gauss extended curvature to surfaces, and found that the curvature of a surface can be measured by creatures living inside it, without leaving it. Riemann extended it to spaces of any dimension, and Einstein used that curvature to describe gravity.
Hii inaelekea wapi
- 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.
- Mwaka wa kwanza wa uzamili · Somo linapatikanaThe geometry of spacetime: special and general relativityPhysics as geometry: Minkowski spacetime and the Lorentz group, proper time and causal structure, Lorentzian manifolds, the equivalence principle, Einstein's field equations, the Schwarzschild solution, geodesics of light and matter, and cosmological models.