Tanzeggit
Tanzeggit d tigezmi tameqrant ger igzumen n tusnakt d win i izerwen tiwellatin neɣ Udmawen deg tallunin s myal tasekta, deg tsekta Tis kraḍ (3) nezmar ad nemmeslay ɣef tenzeggit tuklidt. Ass-a ur nezmir ad d nesbin tilisa yellan ger tenzeggit akk d wesebrurez neɣ d uljiber, acku ur nezmir ad nemmeslay ɣef tenzeggit taljibirt ma yella ur d-neddim ara kra din kra da.
Tanzeggit diɣen d tafurk n tusnakt i izerrwen talɣiwin, tallunin, imeccaqen, tiɣmar d wassaɣen gar tneqqiḍin d yiḥricen n talɣiwin. Tesɛa azal deg usket n wakal d lebni, maca tuɣal daɣen d taɣult tamatut n tusnakt i yezmer ad tzerrew tallunin ur nezmir ara ad nwali s wallen, neɣ tallunin i yesɛan ugar n kraḍ n tsektiwin.
1. Amezruy n tenzeggit
[ẓreg | ẓreg aɣbalu]Awal agraɣlan géométrie yekka-d seg yigrigiyen i d-yemmalen asket n wakal. Isketen d yisiḍen inzeggane ttwasqedcen deg Maṣer, Mezupuṭamya d tɣermiwin nniḍen uqbel tanzeggit n Yigrigiyen. Ur ilaq ara ad nerṛ tanzeggit meṛṛa d asnulfu n yiwen ugdud kan.
| Tallit | Amhaz agenjdan |
|---|---|
| Taglest | Asket n wakal, lebni, asiḍen n teɣzi, n wanrar d ubleɣ. |
| Tasut tis 3 uqbel
tallit-nneɣ |
Aklid deg Iskandariya yesdukkel tizrawin deg "Iferdisen". |
| Tallit talemmast | Tigawt n yimusnawen deg tmura n Tineslemt, Lehend d Cinwa: asektiɣmer(trigonométrie), asali d
tsenallunt. |
| Tasut tis 17 | Descartes d Fermat qqnen talɣiwin ɣer imsidag d tegdiwin. |
| Tasut tis 19 | Lobachevsky d Bolyai snernan tanzeggit taknudemt; Gauss d Riemann snernan tazrawt n
tiknit d tallunin. |
| 1872 d mbeɛd | Klein yeskan azal n yibeddilen d yigrawen; Hilbert yessegzi lsas s yinagrawen n yigejdan. |
| Tasut tis 20–21 | Tanzeggit taljibṛant, taneẓlayt, tukrist, asiḍen s uselkim d uẓeṭṭa n twuriwin gar tfurka. |
"Iferdisen" n Aklid yesseqdac isbaduten d yigejdan iwakken ad yebnu isekkuden. Abrid-a yesɛa tazrirt tameqqrant deg tusnakt. Maca tanzeggit tamirant ur teqqim ara deg yigejdan n Aklid: tazrawt n umṣadaɣ tewwi-d timusniwin ɣef tallunin nniḍen. Tanzeggit tamirant tettawi ula d tallunin yellan deg tfizikt, isefka, tigniwin d yisiḍen. Amezruy-is d anerni n yisental d terrayin, mačči d asemselsi n yiwet n tanzeggit s tayeḍ kan.
2. Isbaduten, igejdan d usfukel
[ẓreg | ẓreg aɣbalu]Asbadu yessegza tamiḍrant s wawalen yettwasnen. Agejdi d tafyirt i neqqbel deg yiwen unagraw war asfukel deg unagraw-nni. Asekkud(tiẓri) d ayen ara d-nsuffeɣ seg yigejdan d yisekkuden yettwasfuklen yakan.
Deg umedya n Aklid, yezmer ad yettwabnu yiɣil n teɣda gar snat n tneqqiḍin, ad yettwasɣezef yiɣil, ad tettwabnu twinest s wammas d waqqar yettwabedren. Yella daɣen agejdi ɣef tɣmar timaɣudin d wayen yeqqnen ɣer umṣadaɣ.
Amedya n useɣẓen
[ẓreg | ẓreg aɣbalu]Deg uɣawas n Aklid, i teneqqiṭ P i ur nelli ara ɣef taɣda d, tella yiwet kan n teɣda iɛeddan seg P ur nettmagar ara d. Talɣa-a n yigejdi n umṣadaɣ tettwassen s yisem n Playfair; d talɣa tagdazalt i yigejdi wis semmus n Aklid deg lsas n tanzeggit tamagnut.
Tugna 2. Tiɣmert d wassaɣ n umṣadaɣ. Tigniwin ttɛawanent deg tɣuri n yisekkuden; isfukal ilaq ad d-begnen ayɣer iiṣeḥḥa wassaɣ.
Deg tugna, tineqqiḍin ttbanent sɛant tehri, izirigen sɛan tuzert, imeccaqen zemren ur ttwaḥerzen ara s tiseddi. Deg tusnakt, taneqqiṭ ur tesɛi ara tehri. Ihi ma tugna teskan-d belli sin n yimeccaqen d igdazalen, ayen-a ur d-yettawi ara asfukel.
3. Taneqqiṭ, taɣda d tallunt
[ẓreg | ẓreg aɣbalu]Taneqqiṭ d amḍiq war tehri; taɣda tettwasɣezef deg sin yinumak war tilisa; iɣil n teɣda yesɛa sin n yixfawen; tazinaɣda tesɛa yiwen n yixef, tettwasɣezef deg yiwen unamek. Aɣawas d tajumma ilweɣ i deg yezmer ad yili wudlif s sin n yimsidag.
Tugna 3. Tisektiwin 0, 1, 2 d 3. Aɣawas n tugna d amur ifuken kan; aɣawas n tusnakt ur yesɛi ara tilisa.
Tasekta n tallunt tettawi amḍan n yimsidag ilelliyen i ilaqen iwakken ad nesbadu amḍiq. Deg taɣda, yiwen x ikfa; deg uɣawas, nesra (x,y); deg tallunt s kraḍ n tsektiwin, nesra (x,y,z). Tasekta ur tɛelliq ara ɣef wamek ara d-nsuneɣ talɣa: amaḍal s tajumma kan yesɛa snat n tsektiwin, ɣas ma yettwaseknen deg tallunt s kraḍ.
ℝⁿ = {(x₁, x₂, …, xₙ) : xᵢ d imḍanen ilawen}
Tallunin s n > 3 ttwasqedcent deg tusnakt d yisefka. Ma yal amedya n yisefka yesɛa 10 n yimuttiyen ilelliyen, yezmer ad yettwagenses s tneqqiṭ deg ℝ¹⁰. Ayen-a ur d-yemmal ara belli nezmer ad t-nwali s wallen; imsidag d yilugan fkan-d allalen n tazrawt.
Tajumma d wanrar
[ẓreg | ẓreg aɣbalu]Tajumma d talɣa n snat n tsektiwin; anrar d asket-is. S yiwet n tɣawsa, takurt tesɛa daxel-is d ubleɣ, ma amaḍal d tajumma n rrif-is. Asbadu n wayen nzerrew yesɛa azal uqbel yal asiḍen.
4. Ikerdisen d usekkud n Pythagore
[ẓreg | ẓreg aɣbalu]Akerdis yesɛa kraḍ n yidis, kraḍ n tqacucin d kraḍ n tiɣmar. Deg uɣawas n Aklid, timernit n tɣmar-is d 180°. Akerdis agdudisan yesɛa kraḍ n yidis igdazalen; akerdis akendis yesɛa ma drus sin n yidis igdazalen; akerdis amaɣud yesɛa yiwet n tiɣmert n 90°.
Tugna 4. Tama taẓelmaḍt: anrar n umkuẓ ɣef yidis 5 d 25 = 9 + 16. Tama tayeffust: kuẓ n yikerdisen s yidis a d b, d umkuẓ alemmas s yidis c.
a² + b² = c², ma c d alubdis n ukerdis amaɣud
Asfukel n tugna tayeffust: anrar n umkuẓ ameqqran d (a+b)². Kuẓ n yikerdisen sɛan anrar 4 × ab/2 = 2ab. Ayen i d-yeqqimen d c²; ihi c² = (a+b)² − 2ab = a²+b². Ayen-a d aseɣẓen s wanrar, mačči d asket n teɣzi ɣef tugna. Asekkud-a yesɛa daɣen talɣa tamsuɣalt: ma a²+b²=c² deg ukerdis, tiɣmert i d-yezzin i c d 90°. Deg tanzeggit taknudemt neɣ ɣef umaḍal, tanfalit-a ur tettwasemres ara akka war abeddel.
5. Tales(Thales) d tarwast
[ẓreg | ẓreg aɣbalu]Sin n yikerdisen imarwasen sɛan tiɣmar igdazalen d yidis akniwen s yiwen wassaɣ. Ma tasɣent tessenɣas neɣ tessemɣer akk imeccaqen s yiwen umuskir k > 0, tḥerrez tiɣmar d talɣa. D tarwast.
Tugna 5. DE d BC msadaɣen, ihi ikerdisen ADE d ABC d imarwasen. Deg tugna tayeffust, amuskir n teɣzi d 2; win n wanrar d 4.
AD/AB = AE/AC = DE/BC
Deg umedya, A d yiwet n tqacuct n sin yikerdisen; D yella ɣef [AB], E ɣef [AC], yerna DE ∥ BC. Ihi assaɣ n Thales yessegza ayɣer akk ibḍan-a d igdazalen. Ma nessen kra n teɣziwin, nezmer ad nsiḍen teɣziwin ur nessin ara s useqdec n wassaɣen-a.
Teɣzi: × k; anrar: × k²; ableɣ: × k³
Ma nsemɣer asrem s umuskir 3, tehri d teɣzi-nni ad nernunt kraḍ n tikal; anrar ad yili 9 n tikal. Ma nsemɣer agasas s 3, ableɣ ad yili 27 n tikal. Ihi amḍan n tsektiwin yesɛa azal deg wamek i yettbeddil usket.
Tagdukta d abeddel iḥerzen imeccaqen; d amedya uslig n tarwast s k = 1. Ur ilaq ara ad nsemmi snat n talɣiwin d igdazalen s wesket ma ttemcabint kan deg talɣa.
6. Igeddisen, tiwinas d yiḍebsiyen
[ẓreg | ẓreg aɣbalu]Ageddis d talɣa n uɣawas iḥerzen yiwen rrif yettwabnan s yiɣallen. I ugeddis afraray, war amager n yidis gar-asen, timernit n tiɣmar n daxel d (n−2) × 180°, anda n d amḍan n yidis. Abḍu n ugeddis affensu s yiwet n tqacuct yessegza tanfalit-a: ad d-naf n−2 n yikerdisen.
Tugna 6. Tawinest d rrif; aḍebsi yesɛa daxel. Ageddis s semmus n yidisan yettwabḍa ɣef kraḍ n yikerdisen.
Tawinest s wammas O d waqqar r tesdukkel tineqqiḍin P i yesɛan d(O,P)=r. Aḍebsi yeɣleq yesdukkel tineqqiḍin i yesɛan d(O,P) ≤ r. Akdu d 2r. Taslayt i twinest deg P tebdd ɣef waqqar OP.
Azzikit n twinest: P = 2πr; anrar n uḍebsi: A = πr²
Ma r yettwaskat s yimitren, P ad yili s yimitren, A s yimitren imkuẓen. Pi d amḍan agraɣlan π ≈ 3.14159. Ur ilaq ara ad nsexleḍ amḍan n yisekkilen n tamawaẓt d wazal amatu n π.
7. Asektiɣmer d radyan
[ẓreg | ẓreg aɣbalu]Asektiɣmer (trigonométrie) izerrew assaɣ gar tɣmar d teɣziwin. Deg ukerdis amaɣud, sin θ d assaɣ n yidis i d-yezzin i θ ɣer ulubdis; cos θ d assaɣ n yidis n tama-s ɣer ulubdis; tan θ = sin θ / cos θ mi cos θ ≠ 0.
Tugna 7. Tawinest s waqqar 1 tefk amur ilaw cos θ d amur yebdden sin θ. Udlifen n sin d cos sseknen abeddel-nsen d wallus-nsen.
sin² θ + cos² θ = 1; 180° = π rad
Radyan d asket n tɣmert s wassaɣ θ = s/r, anda s d teɣzi n uḥric n twinest. Ma θ = 1 rad, s = r. Tuzzya tummidt d 2π rad, neɣ 360°. Aseqdec n radyan yeshel tigawt n usiḍen aneẓlay d tarrayt n yizellumen.
c² = a² + b² − 2ab cos γ
Asaḍuf n cosine yessegza assaɣ gar kraḍ n yidis n ukerdis d tɣmert γ i d-yezzin i c. Ma γ = 90°, cos γ = 0, ad d-naf asekkud n Pythagore. Assaɣen-a ttɛawanen deg usket n tiddi n wayen ur nezmir ara ad naweḍ srid d usiḍen n yiberdan.
8. Teɣzi, anrar d ubleɣ
[ẓreg | ẓreg aɣbalu]Teɣzi tettwaskat s tiget n teɣzi, anrar s tiget n teɣzi ɣer sin, ableɣ s tiget n teɣzi ɣer kraḍ. Azzikit d teɣzi n rrif. Tajumma tella d talɣa; anrar d azal n wesket-is. Deg tallunt, ilaq ad nessemgired anrar n rrif d ubleɣ n daxel.
Tugna 8. Tiddi h tebdd ɣef uzadur b. Awlellu d takurt sɛan isketen n ubleɣ; amaḍal d rrif n takurt.
| Talɣa | Asket |
|---|---|
| Asrem s yidis a d b | A = ab; P = 2(a+b) |
| Akerdis s uzadur b d tiddi h | A = bh/2 |
| Agasas s yidis a | V = a³; anrar n rrif = 6a² |
| Awlellu s waqqar r d tiddi h | V = πr²h |
| Akaswar s waqqar r d tiddi h | V = πr²h/3 |
| Takurt / amaḍal s waqqar r | V = 4πr³/3; anrar n umaḍal = 4πr² |
I talɣiwin ur nesɛi ara tanfalit tafessast, nezmer ad nebḍu talɣa ɣef yiḥricen, ad nessemres aɣred neɣ asiḍen n temawaẓt. Ma yella kra n uḥric i neḥseb snat n tikal, ilaq ad t-nekkes yiwet n tikkelt iwakken ur t-nettḥasab ara snat n tikal.
9. Asali s tesɣunt d ukumpas
[ẓreg | ẓreg aɣbalu]Deg tanzeggit tamensayt, tesɣunt war yizamulen n wesket tettawi-d taɣda iɛeddan seg snat n tneqqiḍin. Akumpas yettawi-d tawinest s wammas d waqqar. Asali d asartu n tigawin s yilugan, mačči d asefru n talɣa s tseddi n tugna kan.
Tugna 9. Sin n yisugnen n twinest sɛan yiwen waqqar, i yugar AB/2. Amager-nsen d P d Q. Taɣda PQ tebdd ɣef AB deg tlemmast M
I usali n tabeddayt i yiɣil [AB] deg tlemmast-is, nbennu tawinest s wammas A d tayeḍ s wammas B, s yiwen waqqar i yugar AB/2. Ad mlilent deg P d Q. Acku PA=PB d QA=QB, P d Q bɛaden s yiwen umeccaq seg A d B; taɣda-nsen d tabeddayt n tlemmast n [AB].
Nezmer ad nebnu akerdis agdudisan, talemmast n yiɣil neɣ abḍu n tiɣmert ɣef sin. Maca mačči yal asali yezmer ad yettwaxdem s sin n wallalen-a: amedya, abḍu n yal tiɣmert ɣef kraḍ n yiḥricen igdazalen ur yezmir ara s wudem amatu. Abḍu n kra n tiɣmar usligen yezmer.
Tanzeggit tagellamt tessemres tigriyin iwakken ad d-tessekna talɣiwin n kraḍ n tsektiwin deg uɣawas. D lsas i tesneḍrit n lebni d kra n terrayin n usunneɣ aṭekniki.
10. Imsidag d yimawayen
[ẓreg | ẓreg aɣbalu]Tanzeggit tusliṭ teqqen talɣa ɣer tegda. S yimsidag n Descartes, taneqqiṭ A tettwasbadu s (x,y); taɣda s ax+by+c=0, anda a d b ur llin ara i sin d 0. Tarrayt-a tettawi talɣiwin ɣer yisiḍen(leḥsab) n uljibṛ.
Tugna 10. Imeccaqen Δx = 4 d Δy = 3 fkan d = 5. Amaway u+v yettwabna s unadi n v seg yixef n u; amḍiq n tazwara n umaway ur ibeddel ara talɣa-s.
d(A,B) = √[(x₂−x₁)² + (y₂−y₁)²]
Talemmast n [AB] tesɛa imsidag ((x₁+x₂)/2, (y₁+y₂)/2). Deg umedya A(1,1), B(5,4), talemmast d (3,2.5). Tanfalit n umeccaq tekka-d seg Pythagore; tanfalit n tlemmast tekka-d seg wassaɣ n yimawayen.
u · v = u₁v₁ + u₂v₂ = ‖u‖ ‖v‖ cos θ
Amaway yesɛa teɣzi d tanila. Afaris afesnan n sin yimawayen yezmer ad yessegza tiɣmert gar-asent. Ma u·v = 0 d sin yimawayen ur nelli ara d 0, imawayen-a bedden gar-asen. Tanzeggit tamawayt teshel anerni n yisiḍen-a ar tallunin s waṭas n tsektiwin.
11. Ibeddilen d tujjuṛt
[ẓreg | ẓreg aɣbalu]Tagdukta d abeddel iḥerzen imeccaqen. Tasikelt, tuzzya d tujjuṛt s ugellus d imedyaten deg uɣawas n Aklid. Abeddel yezmer ad yerr talɣa deg umḍiq nniḍen, maca teɣzi, tiɣmar d wanrar ad qqimen d yiwen.
Tugna 11. Talɣa tazegzawt d tin n tazwara. Tasikelt d tuzzya ḥerzent aniri; tujjuṛt s ugellus tbeddel aniri.
Tasikelt: x′ = x + a; y′ = y + b
Tuzzya: x′ = x cos θ − y sin θ; y′ = x sin θ + y cos θ
Tujjuṛt s ugellus y tettawi (x,y) ɣer (−x,y). Tasikelt s umaway v tettawi P ɣer P+v. Taludegt s wammas O d umuskir k tettawi OP ɣer kOP; tessenɣas neɣ tessemɣer imeccaqen s |k|. Ma |k| ≠ 1, mačči d tagdukta.
Timhelt n uḍfar n yibeddilen tesɛa azal: tuzzya deffir tasikelt ur tefk ara dima ayen i d-tefk tasikelt deffir tuzzya. Ibeddilen i yeṭṭafen talɣa meṛṛa ur tettbeddil ara sdukklen agraw n tujjuṛt-is. D allal i tazrawt n yizeṭṭa, lebni d talɣiwin n ugama.
12. Igezzumen n ukaswar
[ẓreg | ẓreg aɣbalu]Igezzumen n ukaswar (coniques) d tiwinas, tiglayin, tikenzirarin d yiknudem. Zemren ad d-ilin s umager n uɣawas d ukaswar asinan; tili n ugzam tettawi talɣa yemgaraden. Deg taenzeggit tusliṭ, d imedyaten n tegdiwin s tfesna 2.
Taglayt tesɛa sin n yidiken. Takenzirart tesɛa adik d teɣda tanilayt. Aknudem yesɛa sin n yifurkan. Ilugan n tigniwin llan deg yizwal-nsent.
Taglayt tesdukkel tineqqiḍin i yesɛan timernit n yimeccaqen seg sin n yidiken d yiwen wazal. Takenzirart tesdukkel tineqqiḍin i bɛaden s yiwen umeccaq seg udik d teɣda tanilayt. Aknudem yesɛa tameẓla tamagdazt n yimeccaqen seg sin n yidiken d yiwen wazal.
Ax² + Bxy + Cy² + Dx + Ey + F = 0
I tegda n tfesna 2 war talɣiwin yettwasenɣasen, B²−4AC < 0 yettawi talɣa n teglayt; = 0 talɣa n tkenzirart; > 0 talɣa n uknudem, s tawtilin n tili n talɣa tilawt. Kra n tegdiwin zemrent ad fkent taɣda, snat teɣdiwin, taneqqiṭ neɣ tagrumma tilemt. Ihi ur ilaq ara ad nsemmu talɣa s yiwen umḍan kan.
Talɣiwin-a sɛant aseqdec deg tsenallunt, imeskaren n tafat d usali n kra n yimeslaḍen.
13. Tanzeggit talawsant d tseggart
[ẓreg | ẓreg aɣbalu]Tanzeggit talawsant tḥerrez imeẓla n umsadaɣ d wassaɣen n tneqqiḍin ɣef yiwet n teɣda. Abeddel alawsan yesɛa talɣa x′=Ax+b, anda A d isirew imutti. Yezmer ad yessenɣes yiwen yidis ugar n wayeḍ, ihi tiɣmar d teɣziwin ur ilaq ara ad qqiment d yiwen.
Tugna 13. Tama taẓelmaḍt: amkuẓ yuɣal d talɣa s yidis imsadaɣen. Tama tayeffust: tigri n snat teɣdiwin timsadaɣin tettban tettawi-ten ɣer yiwet n tneqqiṭ.
Tanzeggit taseggart tzerrew ayen i yeqqimen d yiwen s tigri. Deg uɣawas aseggar, snat teɣdiwin yemgaraden mlilent dima deg yiwet n tneqqiṭ. I taɣdiwin timsadaɣin n uɣawas alawsan, amager-a yettwagenses s tneqqiṭ deg war taggara.
[x : y : z] = [λx : λy : λz], ma λ ≠ 0
Deg yimsidag imsariyen, z ≠ 0 yezmer ad yettwarr ɣer z = 1 s ubḍu. Tineqqiḍin s z = 0 d taɣda n war taggara. Snat teɣdiwin timsadaɣin sɛant taneqqiṭ n tanila i ten-yesdukklen. «War taggara» dagi d amur n unagraw n yimsidag, mačči d amḍiq i nettawi s useɣzef n ugdil.
Assaɣ n umager d talɣa n teɣda ttwaḥerzen s tigri, maca umsadaɣ neɣ talemmast ur ilaq ara ad qqimen. Tanzeggit-a tettawi-d allalen i tigniwin n ukamera d tmuɣli n lebni.
14. Tanzeggit ur nelli ara n Aklid
[ẓreg | ẓreg aɣbalu]Tanzeggit taknudemt d tanzeggit n umaḍal sseknayent belli agejdi n umṣsadaɣ ur yelli ara d ayen iwatan i yal tallunt. Deg uɣawas aknudeman, seg P i ur nelli ara ɣef d, llant aṭas n teɣdiwin i ur nettmagar ara d. Ɣef umaḍal, iberdan i yettaken adeg n teɣdiwin d tiwinas timeqqranin.
Tugna 14. Akerdis ɣef umaḍal: seg Ugafa ɣer sin n tneqqiḍin deg usebges i yemgaraden s 90°, timernit d 270°. Deg uḍebsi n Poincaré, yidis n ukerdis d iḥricen n twinas i yebdden ɣef rrif.
Deg tanzeggit n umaḍal s waqqar R, akerdis s yidis n twinas timeqqranin, deg yiwen umur iwatan, yesɛa timernit n tɣmar tugar π. Anrar-is d R²(α+β+γ−π), anda tiɣmar ttwasqedcent s radyan. Deg uɣawas aknudeman s tiknit −1, anrar n ukerdis d π−(α+β+γ).
Deg umedya n Poincaré, rrif n uḍebsi ur yelli ara deg tallunt taknudemt: tineqqiḍin ttwaliɣent qqrebent ɣer rrif, maca ameccaq aknudeman yezmer ad yettnerni war tilisa. Ur ilaq ara ad nsket imeccaqen n tallunt-a s tesɣunt ɣef tugna amagnut.
15. Tanzeggit taneẓlayt d tiknit
[ẓreg | ẓreg aɣbalu]Tanzeggit taneẓlayt tessemres izellumen d yiɣraden iwakken ad tzerrew udmawen d tijummiwin. I udem ilweɣ s yiwen uɣewwaṛ t, r(t) = (x(t),y(t)), taslayt teqqen ɣer r′(t). Tiknit tettawi-d asket n ubeddel n tanila mi nettnerni deg teɣzi n ubrid.
κ = |x′y″ − y′x″| / (x′² + y′²)^(3/2), ma r′(t) ≠ 0
Tugna 15. Amedya n tiknit n Gauss: tajumma n umaḍal s K > 0, aɣawas s K = 0, d talɣa n ukersi n userdun s K < 0.
Tawinest s waqqar R tesɛa tiknit 1/R. Taɣda tesɛa tiknit 0. Ɣef tajumma deg tallunt s kraḍ n tsektiwin, llant snat n tikniyin tigejdanin k₁ d k₂; tiknit n Gauss d K = k₁k₂. K tezmer ad tili tugar 0, d 0 neɣ ddaw 0.
Tiknit n Gauss d taɣara n daxel: yezmer ad tettwasekza seg yisketen deg tajumma, war asket n wamek i tettwabna deg tallunt n berra. D ayen i d-yeskan usekkud n Gauss yettwasnen s Theorema Egregium. Ihi tajumma tettban teɛwej seg berra ur ilaq ara adtesɛu K ≠ 0. Awlellu, amedya, yesɛa K = 0.
16. Tanzeggit n Riemann d yiberdan
[ẓreg | ẓreg aɣbalu]Tanḍa d tallunt i yettemcabin, deg yal amḍiq ameẓẓyan, ɣer ℝⁿ, maca talɣa-is meṛṛa tezmer ad temgarad. Amaḍal d akruri zemren ad ttwabḍun s yikarṭiyen adiganen s sin n yimsidag. Ur ilaq ara yiwen ukarṭi ad yeṭṭef akk tallunt.
Tanzeggit n Riemann ternu asbadu n wesket i yal taneqqiṭ s isirew gᵢⱼ(x) iḍemnen belli teɣzi n yal amaway ur nelli ara d 0 tugar 0. Teɣzi n uḥric ameẓẓyan tettwaru s talɣa-a:
ds² = Σᵢⱼ gᵢⱼ(x) dxᵢ dxⱼ
Tugna 16. Awlellu yezmer ad yeldi ɣer uɣawas war abeddel n yimeccaqen adiganen. Taɣda deg tugna tayeffust tettawi abrid yezzin ɣef uwlellu.
Abrid agejdis (géodésique, asegzi dagi) d abrid iḥerzen tanila s wamek i t-id-tesbadu tallunt. Deg tallunt n Riemann, iberdan-a sɛan taɣara n ubrid ameẓẓyan s teɣzi deg yiḥricen meẓẓiyen iwulmen. Ur ilaq ara yal abrid agejdis ad yili d ameẓẓyan akk gar sin n yixfawen s wudem amatu.
Deg tfizikt n Einstein, tallunt d wakud ttwazerwen s tanzeggit n Lorentz: asket-is ur yesɛi ara yiwen unagraw n yizamulen am Riemann. Ihi «tajumma yeɛwejjen» d asegzi i d-iqerben kan, ur d-yessegzi ara iman-is akk tiẓri n wassaɣ n tallunt d wakud (relativité).
17. Tasnidegt d talɣa meṛṛa
[ẓreg | ẓreg aɣbalu]Tasnidegt tzerrew timitar i ur nettbeddil ara s ubeddel amaɣlal i yesɛan amutti amaɣlal. Zemrent ad ttwazerwent tuqqna, ixjaḍ d wamek i tettwasdukkel tallunt. Teɣzi d tɣmar ur ilaq ara ad ttwaḥerzent. Ihi tasnidegt teqqen ɣer tanzeggit, maca ur telli ara d yiwet n taɣult s sin n yismawen.
Tugna 17. Amaḍal d ukruri mgaraden deg umḍan n yixjaḍ. amaḍal ur yesɛi ara axjiḍ, a akruri yesɛa yiwen uxjiḍ iɛeddan deg tlemmast-is, Tigniwin-a sseknayent tijummiwin n rrif kan, mačči d talɣiwin yeččuren d axel.
χ = N₀ − N₁ + N₂; i tajumma iɣelqen tusɣilt: χ = 2 − 2g
I tajumma yettwabḍan ɣef talɣiwin, N₀ d amḍan n tqacucin, N₁ n yidis d N₂ n wudmawen. χ d taskant n Euler. Ɣef rrif n ugasas, 8−12+6 = 2. Ɣef akruri, χ = 0. Deg tnfalit tis snat, g d amḍan n yixjaḍ s unamek n wanaw(genre), tajumma ilaq ad tili teqqen, teɣleq, tusɣilt d war rrif.
Tazrawt n talɣa meṛṛa tezmer ad d-tefk tilisa i tiknit. Asekkud n Gauss-Bonnet yeqqen aɣred n tiknit n Gauss ɣer χ n tajumma. D amedya ameqqran n tuqqna gar usket adigan d talɣa n tallunt meṛṛa.
18. Tanzeggit taljibṛant
[ẓreg | ẓreg aɣbalu]Tanzeggit taljibṛant tzerrew tegrummiwin n tifrat n tegdiwin n yigetfulen. Ma P(x,y)=0, tineqqiḍin i yettḥeqqiqen tagda-a zemrent ad d-fkent azlig. Tarrayt-a tesdukkel aljibṛ d talɣiwin; tazrawt tamatut tezmer ad tili ɣef yimḍanen ilawen, isemlalen neɣ taɣulin nniḍen.
Tugna 18. Tama taẓelmaḍt: azlig y² = x³ − x + 1 war taneqqiṭ tusligt deg uɣawas ilaw. Tama tayeffust: y² = x³ yesɛa taneqqiṭ tasuft deg (0,0).
I uzlig P=0, taneqqiṭ tasuft tḥeqqiq P = 0 d akk izellumen imezwura ∂P/∂x = ∂P/∂y = 0. Deg y²−x³=0, ayen-a yettili deg (0,0). Deg y²−x³+x−1=0, ulac taneqqiṭ i yeṭṭfen akk tawtilin-a. Amedya yessekza ayɣer aljibṛ yezmer ad yessegzi tikniyin d yimeẓla n talɣa.
Deg tanzeggit taljibṛant, imḍanen isemlalen zemren ad fken talɣa tummidt n tifrat: ayen yettban d sin n yifurkan deg tugna tilawt yezmer ad yesɛu tuqqniwin nniḍen deg tallunt tasemlalt. Ur ilaq ara tugna tilawt ad tettwaḥsab d tamusni tummidt n talɣa. Tanzeggit taljibṛant tesɛa iseqdacen deg tiẓri n yimḍanen, ladɣa deg tazrawt n tifrat n tegdiwin s yimḍanen ummiden neɣ imeɣẓanen, izligen ilipsiyen d teɣri n tegdiwin. Iwakken ad tili tɣuri n lsas, ilaq aljibṛ imzireg d kra n tiẓri n tegrummiwin.
19. Tanzeggit d usiḍen s uselkim
[ẓreg | ẓreg aɣbalu]Tanzeggit n usiḍen tzerrew ilugan i asiḍen n wassaɣen anzeggane: amager n yiɣallen, taɣlaft taffensut, abḍu n tajumma ɣef ikerdisen d tazrawt n yiberdan. Tanzeggit tafrarayt tzerrew daɣen tegrummiwin n tneqqiḍin d talɣiwin tinzegganin, d wassaɣen gar-asent.
Tugna 19. Yiwen wamud n 15 n tneqqiḍin: taɣlaft taffensut, abḍu ɣef ikerdisen n Delaunay d taɣulin n Voronoi. Tilisa i yettbanen deg ugdil d tamuɣli n uḥric n uɣawas.
Tagrumma taffensut teṭṭef iɣil meṛṛa i yeqqnen yal snat n tneqqiḍin-is. Taɣlaft taffensut n wamud d tagrumma taffensut tameẓẓyant i yeṭṭfen akk tineqqiḍin n wamud-nni. Deg wudlif n Voronoi, yal taneqqiṭ n wamud tesɛa taɣult i yesdukklen akk tineqqiḍin i iqerben ɣur-s ugar n tneqqiḍin nniḍen n wamud, neɣ i yebɛaden s yiwen umeccaq ɣur-s d kra seg-sent.
Abḍu n Delaunay d abḍu ɣef ikerdisen, s tawtilt-a: i yal akerdis, ulac taneqqiṭ nniḍen n wamud deg daxel n twinest iɛeddan ɣef kraḍ n tqacucin-is. Deg waddad amatu, abḍu n Delaunay d wudlif n Voronoi qqnen gar-asen akka: snat n tneqqiḍin n wamud ttwaqqnent s yiɣil deg ubḍu n Delaunay ma taɣulin-nsent n Voronoi ccerkent yiwen n yidis.
Ilaq ad ttwasxedmen yisiḍen s tiseddi: tuccḍa tameẓẓyant tezmer ad d-tefk agmuḍ ur neṣḥiḥ ara mi ara nsenqed ma sin n yiɣallen ttemlilin neɣ ala. Tarrayin-a sɛant azal deg usnulfu n tigniwin s kraḍ n tsektiwin (3D), deg tarubut, deg usnulfu n yikarṭiyen d lebni n yimedyaten n tfizikt.
20. Tarrayt n wesket n umeccaq
[ẓreg | ẓreg aɣbalu]Tallunt timizrirt d tagrumma X akked tesɣent d i d-yettaken ameccaq gar yal snat n tneqqiḍin. Ilaq d(x,y) ≥ 0; d(x,y)=0 ma x=y kan; d(x,y)=d(y,x); d(x,z) ≤ d(x,y)+d(y,z).
Tawtilt taneggarut d targadda n ukerdis: abrid usrid ur yugar ara abrid iɛeddan seg yiwet n tneqqiṭ nniḍen.
Tugna 20. Kraḍ n terrayin n umeccaq. Yal talɣa tesdukkel tineqqiḍin i bɛaden s umeccaq ≤ 1 seg (0,0); rrif-is yesɛa ameccaq = 1.
d₂ = √(Δx² + Δy²); d₁ = |Δx| + |Δy|
d∞ = max(|Δx|, |Δy|)
Deg umedya A(0,0), B(3,4), ameccaq n Aklid d 5, win n Manhattan d 7, win n Chebyshev d 4. Manhattan yettemcabi ɣer ubrid deg uẓeṭṭa n yiberdan ibedden: nezmer ad neddu kan s sin n yidis n uẓeṭṭa. Ihi «aqrab» yeqqen ɣer tarrayt n wesket, mačči ɣer tugna kan.
Tarrayt n umeccaq ur teqqin ara kan ɣer tallunt i nettwali. Nezmer ad nesbadu ameccaq gar tesɣunin, imḍan n yibeddilen gar sin n yiḍrisen, neɣ amgired gar talɣiwin. Maca yal tanfalit i nsemmi «ameccaq» ilaq ad tettasenqed ma tḥerrez akk tawtilin n tallunt timizrirt.
Asekkud n Pythagore yesɛa azal deg umeccaq n Aklid. Ma nbeddel tarrayt n wesket, tanfalit n twinest d ubrid ameẓẓyan ad tbeddel. Tigniwin-a sseknayent umgired gar uẓeṭṭa n yimsidag d wesket i yettwafernen deg-s.
21. Tanzeggit n fraktal
[ẓreg | ẓreg aɣbalu]Tanzeggit n fraktal tzerrew kra n talɣiwin i yesɛan telqey i yettbanen tikelt deffir tayeḍ deg tfesniwin timeẓyanein. Deg yimedyaten i yettemcabin s wudem aḥeqqani, yal aḥric d anɣel imeẓẓi n talɣa meṛṛa. Tanzeggit-a tessemgired teɣzi tamagnut d terrayin nniḍen n wesket.
Tugna 21. Agliz(Udem) n Koch s 5 n wallusen; akerdis n Sierpiński s 6. Deg tusnakt, fraktal amatu d talast n wallus war taggara; tugna tesɛa telqey ifuken.
D = ln(N) / ln(1/r), deg umedya yettemcabin s tawtilin iwulmen
Deg Koch, yal aḥric yettwabeddel s 4 n yiḥricen, yal yiwen s teɣzi 1/3 n tin tamezwarut. Tasekta n Hausdorff d ln(4)/ln(3) ≈ 1.262. Deg Sierpiński, llant 3 n talɣiwin s umuskir 1/2; tasekta d ln(3)/ln(2) ≈ 1.585. Iwakken formula n tarwast ad tefk tasekta n Hausdorff, ilaqt awtilin n ferru gar yiḥricen.
Ur ilaq ara tasekta n fraktal ad tili d amḍan arummid dima; ur ilaq ara yal talɣa i yettemcabin ɣer yiman-is ad tili d fraktal. Deg ugama, amcabi yezmer ad yili kan gar kra n tfesniwin. Rrif n yilel neɣ ifurkan n useklu ur sɛin ara telqey war tilisa.
22. Ifurkan nniḍen n tanzeggit
[ẓreg | ẓreg aɣbalu]Tanzeggit tamirant tesɛa aṭas n ifurkan i yesɛan allalen d yisental yemgaraden. Kra seg-sent ttwasuddusent s wamek i tettwabna tallunt; kra s yibeddilen i yettwaḥerzen; kra s yilugan n tfizikt neɣ s umḍan n yiferdisen.
Tugna 22. Tama taẓelmaḍt: udmawen n yiwen wazal n Hamiltonien H. Tama tayeffust: aɣawas n Fano; tawinest n tlemmast d yiwet seg «taɣdiwin» n unagraw, mačči d tawinest s unamek n Aklid.
Tanzeggit tukris (symplectique) tzerrew tanḍiwin yesɛan talɣa taneẓlayt ω n tfesna 2, teɣleq yerna ur tefsiy ara. Deg imsidag adiganen iwulmen, ω = Σ dqᵢ ∧ dpᵢ. Tasekta n tallunt ilaq ad tili tayugant. Taɣult-a teqqen ɣer tiẓri n umussu n Hamilton d tallunt n waddaden n tfizikt.
Tanzeggit ifuken tesɛa amḍan ifuken n tneqqiḍin d «teɣdiwin». Aɣawas n Fano d amedya s 7 n tneqqiḍin d 7 n teɣdiwin; yal taɣda tesɛa 3 n tneqqiḍin. Zemren ad ttwabnun s yimawayen n ℱ₂³ war 0 d yimḍiqen n tsekta 2. D amedya n tanzeggit taseggart ɣef taɣult ssin n yiferdisen.
Tanzeggit timizrirt tzerrew tallunin s umeccaq amatu; tanzeggit n Lorentz teqqen ɣer tallunt d wakud; tanzeggit tasemlalt tessemres imsidag isemlalen d tesɣunin iwulmen. Tfuriwin-a zemrent ad mlilent deg yiwen unadi.
23. Taɣulin n useqdec
[ẓreg | ẓreg aɣbalu]Tanzeggit d allal n tusnakt d unadi usnan. Tettawi-d ilugan i usket, usali n yimedyaten, asenqed n yiberdan d usmekti n talɣiwin. Aseqdec iwatan ilaq ad yessegzi amedya n tallunt, tiget n wesket d tilisa n tamawaẓt.
Tugna 23. Akal yettwagenses s yikerdisen; abrid ilipsi n umtiweg s Yiṭij deg yiwen udik. Sin n yimedyaten-a d asali n usegzi, mačči d ikarṭiyen neɣ isefka yettwaskaten.
| Taɣult | Aseqdec |
|---|---|
| Lebni d tesneḍrit | Anrar, ableɣ, tasaksart, asali d tigriyin. |
| Tasenallunt d wakal | Abeddel n yimsidag, ikarṭiyen d yiberdan. |
| Tafizikt | Tiknit, amussu, tallunt n waddaden d tallunt-wakud. |
| Tigzi n tmacint d tigniwin | Asmekti n talɣiwin, tigriyin n ukamera d tallunin n yisefka. |
| Tarubut d usiḍen | Abrid war amager, tanila d usali n tallunt. |
| Tasnuddirt d tmeddurt | Talɣiwin n yiferdisen, tijummiwin d yizḍwa. |
Deg lebni, tugna tasemlalt neɣ tigniwin n uselkim ur suffɣent ara tamusni n usket: amedya 3D yezmer ad yili yecbeḥ maca imeccaqen-is ur wulmen ara. Deg yal taɣult, asenqed n wassaɣ gar amedya d wayen yettwaskaten d aḥric seg umahil usnan.
