Ad argvmentvm transi
Topos Uranos
Latine

Geometria per nvmeros complexos

Svmmarivm

Haec lectio nvmeris complexis vt coordinatis plani vtitvr et notiones geometriae elementaris in operationes algebraicas transfert: distantia est modvlvs differentiae, pvnctvm medivm et centrvm gravitatis svnt mediae arithmeticae, et motvs plani svnt formvlae breves. Ostendit R(z)=c+eiφ(z−c)R(z) = c + e^{i\varphi}(z - c) esse conversionem circa cc angvlo φ\varphi, reflexionem in recta qvae per originem transit cvm angvlo α\alpha esse S(z)=e2iαz‾S(z) = e^{2i\alpha}\conj{z}, et compositionem dvarvm reflexionvm esse conversionem. Collinearitatem et perpendicvlaritatem per qvotientem describit, orientationem triangvli per signvm partis imaginariae eivs qvotientis definit, ostendit tria pvncta triangvlvm aeqvilatervm formare si et solvm si z1+ωz2+ω2z3=0z_1 + \omega z_2 + \omega^2 z_3 = 0, cvm ω=e2πi3\omega = e^{\frac{2\pi i}{3}}, et demonstrat pvncta qvorvm distantiae a dvobvs pvnctis fixis rationem constantem servant circvmferentiam formare, eam qvae Apollonii dicitvr. Problemata theoremate Napoleonis conclvdvntvr.

Proposita discendi

  1. Pvncta, distantias, pvncta media et centra gravitatis per nvmeros complexos repraesentare, et translationes, conversiones, homothetias et reflexiones vt fvnctiones variabilis complexae exprimere.
  2. Formvlas conversionis circa pvnctvm et reflexionis in recta demonstrare, et compositiones reflexionvm et conversionvm investigare.
  3. Collinearitatem, perpendicvlaritatem et angvlvm dvorvm segmentorvm per qvotientem dvarvm differentiarvm diivdicare, orientationem triangvli per signvm partis imaginariae eivs qvotientis determinare et triangvlvm aeqvilatervm per radicem cvbicam vnitatis ω\omega describere.
  4. Circvmferentias per modvlvm describere, theorema de circvmferentia Apollonii demonstrare et haec instrvmenta in problematis classicis adhibere, vt in eis Napoleonis et Van Avbel.

Pvncta, distantiae et centra

Inde a lectione cvrsvs Algebra et Geometria I de operationibvs cvm nvmeris complexis, qvisqve complexvs z=a+biz = a + bi cvm pvncto (a,b)(a, b) plani identificatvr, et differentia z−wz - w est vector qvi a pvncto ww ad pvnctvm zz dvcit; per theorema Pythagorae distantia inter ambo pvncta est ∣z−w∣\abs{z - w}. In hac lectione ratio inspiciendi invertitvr: non nvmeri per planvm investigantvr, sed planvm per nvmeros. Figvra per complexos pvnctorvm svorvm describitvr, et proprietas geometrica per aeqvalitatem inter eos.

Instrvmentvm praecipvvm est exponentialis complexa lectionis de exponentiali complexa et formvla Evleri. Pro omni reali φ\varphi, nvmervs eiφe^{i\varphi} est pvnctvm circvmferentiae radii 11 cvivs argvmentvm est φ\varphi, et proprietates infra scriptas habet; vltima inde provenit qvod cos⁡(−φ)=cos⁡φ\cos(-\varphi) = \cos \varphi et sin⁡(−φ)=−sin⁡φ\sin(-\varphi) = -\sin \varphi:

eiφ=cos⁡φ+isin⁡φ,∣eiφ∣=1,eiαeiβ=ei(α+β),eiφ‾=e−iφe^{i\varphi} = \cos \varphi + i \sin \varphi, \qquad \abs{e^{i\varphi}} = 1, \qquad e^{i\alpha} e^{i\beta} = e^{i(\alpha + \beta)}, \qquad \conj{e^{i\varphi}} = e^{-i\varphi}

Secvndvm lectionem de forma polari nvmerorvm complexorvm, omnis complexvs non nvllvs scribitvr z=reiθz = re^{i\theta}, cvm r=∣z∣r = \abs{z} et θ\theta aliqvo argvmento ipsivs zz; dvobvs complexis mvltiplicatis modvli mvltiplicantvr et argvmenta addvntvr, et eis divisis modvli dividvntvr et argvmenta svbtrahvntvr. Praesertim, mvltiplicare zz per eiφe^{i\varphi} modvlvm eivs servat et argvmentvm eivs qvantitate φ\varphi avget: pvnctvm zz angvlo φ\varphi circa originem convertit, contra motvm horologii si φ>0\varphi > 0. Prodvctvm per i=eiπ2i = e^{\frac{i\pi}{2}}, in lectione cvrsvs I de nvmeris complexis vt conversio qvadrantis investigatvm, est casvs φ=π2\varphi = \frac{\pi}{2}.

Pvncta segmenti qvoqve per complexos describvntvr. Secvndvm lectionem cvrsvs I de vectoribvs et aeqvationibvs parametricis rectae, pvncta rectae qvae per z1z_1 et z2z_2 transit svnt formae z1+t(z2−z1)z_1 + t(z_2 - z_1), cvm tt reali, et pvncta segmenti qvod ea ivngit valoribvs 0≤t≤10 \leq t \leq 1 respondent.

LemmaPvnctvm medivm et centrvm gravitatis in complexis

Pvnctvm medivm segmenti cvivs extrema svnt z1z_1 et z2z_2 est semisvmma m=z1+z22m = \frac{z_1 + z_2}{2}. Tres medianae triangvli cvivs vertices svnt z1z_1, z2z_2 et z3z_3 per pvnctvm g=z1+z2+z33g = \frac{z_1 + z_2 + z_3}{3} transevnt, qvod centrvm gravitatis appellatvr et in qvaqve mediana dvabvs tertiis itineris iacet qvod a vertice ad pvnctvm medivm lateris oppositi dvcit. Lemma per vectores demonstratvm est in problemate solvto de medianis lectionis cvrsvs I de vectoribvs et aeqvationibvs parametricis rectae; demonstratio qvae seqvitvr id ad complexos transfert. Figvra eam seqvitvr cvm verticibvs z1=−2−iz_1 = -2 - i, z2=3−iz_2 = 3 - i et z3=2iz_3 = 2i.

Demonstratio

  1. m=z1+12(z2−z1)=z1+z22\dato{m}{m = z_1 + \resaltar{\frac{1}{2}}(z_2 - z_1) = \resaltar{\frac{z_1 + z_2}{2}}}

    Pvnctvm segmenti qvod valori t=12t = \frac{1}{2} respondet est semisvmma extremorvm.

  2. m−z1=z2−z12=z2−m→∣m−z1∣=∣z2−m∣m - z_1 = \resaltar{\frac{z_2 - z_1}{2}} = z_2 - m \rightarrow \resaltar{\abs{m - z_1} = \abs{z_2 - m}}

    Ab vtroqve extremo aeqve distat, qvia dvo vectores qvi id cvm eis ivngvnt aeqvales svnt: est pvnctvm medivm.

  3. z3+23(z1+z22−z3)=z1+z2+z33z_3 + \resaltar{\frac{2}{3}} \left( \frac{z_1 + z_2}{2} - z_3 \right) = \dato{g}{\resaltar{\frac{z_1 + z_2 + z_3}{3}}}

    In mediana qvae z3z_3 cvm pvncto medio lateris oppositi ivngit, pvnctvm qvod valori t=23t = \frac{2}{3} respondet hoc est.

  4. z1+23(z2+z32−z1)=g=z2+23(z1+z32−z2)z_1 + \frac{2}{3} \left( \frac{z_2 + z_3}{2} - z_1 \right) = \resaltar{g} = z_2 + \frac{2}{3} \left( \frac{z_1 + z_3}{2} - z_2 \right)

    Expressio obtenta in tribvs verticibvs symmetrica est; qvod cvm ita sit, idem calcvlvs cvm vtralibet ex dvabvs aliis medianis idem pvnctvm dat, et omnes tres per gg transevnt. Cvm verticibvs figvrae, g=13g = \frac{1}{3}.

Itaqve pvnctvm medivm est media arithmetica dvorvm complexorvm, et centrvm gravitatis trivm.