Ad argvmentvm transi
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Coordinatae polares et cvrvae polares

Svmmarivm

Haec lectio altervm modvm pvncta plani nominandi exponit: per eorvm distantiam a pvncto fixo, polo, et per angvlvm qvem cvm semirecta fixa, axe polari, efficivnt. Conventionem adhibet qvae radios negativos admittit, omnes repraesentationes vnivs eivsdemqve pvncti describit, formvlas transitvs inter coordinatas polares et Cartesianas dedvcit, cvm correctione arcvs tangentis secvndvm qvadrantem, et distantiam inter dvo pvncta ope theorematis cosinvvm obtinet. Deinde aeqvationes polares rectarvm et circvmferentiarvm, criteria symmetriae et delineationem per tabvlas valorvm cardioidis, rosarvm, spirae Archimedeae et cochleae tractat, et in aeqvatione polari sectionvm conicarvm qvarvm focvs in polo est finit, ex qva genvs earvm, vertices et latvs rectvm legvntvr.

Proposita discendi

  1. Pvncta in coordinatis polaribvs repraesentare, omnes repraesentationes vnivs eivsdemqve pvncti agnoscere et a coordinatis polaribvs ad Cartesianas transire et vicissim.
  2. Distantias in coordinatis polaribvs compvtare et aeqvationes rectarvm et circvmferentiarvm ex vno systemate in altervm transferre, polo nec amisso nec addito.
  3. Criteria symmetriae et tabvlas valorvm ad delineationem cvrvarvm polarivm adhibere: cardioidvm, rosarvm, spirarvm et cochlearvm.
  4. Aeqvationem polarem sectionvm conicarvm qvarvm focvs in polo est dedvcere et in ea earvm eccentricitatem, vertices et latvs rectvm legere.

Coordinatae polares

Coordinatae Cartesianae, in lectione cvrsvs I de aeqvatione rectae et systematibvs Cartesianis tractatae, pvnctvm per eivs distantias cvm signo a dvabvs rectis perpendicvlaribvs nominant. Atqvi mvltae cvrvae natvralivs describvntvr per distantiam a pvncto fixo et per directionem qva ex eo conspicivntvr: circvmferentia circa svvm centrvm, spira qvae aeqvabili modo se aperit, avt orbita planetae circa Solem. Ad eas apta est ratio coordinatarvm polarivm, qvae constat pvncto fixo OO, polo, et semirecta fixa ex eo orta, axe polari. Vt dvo systemata simvl consistant, polvs in origine ponitvr et axis polaris in semiaxe positivo abscissarvm.

Sint rr et θ\theta dvo nvmeri reales. Pvnctvm coordinatarvm polarivm (r,θ)(r, \theta) est pvnctvm plani coordinatarvm Cartesianarvm

x=rcos⁡θ,y=rsin⁡θx = r \cos \theta, \qquad y = r \sin \theta

Si r>0r > 0, est pvnctvm qvod distantia rr a polo abest in latere terminali angvli θ\theta in positione normali, in lectione de angvlis eorvmqve mensvra tractati: est pvnctvm (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta) circvmferentiae vnitariae, in lectione de rationibvs trigonometricis definitvm, coordinatis eivs per rr mvltiplicatis. Nvmervs rr radivs vocatvr, et angvlvs θ\theta, in radianis mensvs, angvlvs polaris.

Definitio etiam r=0r = 0 et r<0r < 0 admittit, et in hoc consistit conventio hvivs lectionis. Cvm r=0r = 0 polvs obtinetvr, qvicvmqve est θ\theta. Cvm r<0r < 0 obtinetvr pvnctvm qvod distantia −r-r a polo abest in directione contraria directioni θ\theta: per symmetrias cos⁡(θ+π)=−cos⁡θ\cos(\theta + \pi) = -\cos \theta et sin⁡(θ+π)=−sin⁡θ\sin(\theta + \pi) = -\sin \theta lectionis de rationibvs trigonometricis, paria (−r,θ)(-r, \theta) et (r,θ+π)(r, \theta + \pi) idem pvnctvm dant.

(−rcos⁡θ,−rsin⁡θ)=(rcos⁡(θ+π),rsin⁡(θ+π))\left( -r \cos \theta, -r \sin \theta \right) = \left( r \cos(\theta + \pi), r \sin(\theta + \pi) \right)

Mvlti libri r≥0r \geq 0 postvlant; radios negativos admittere dvo commoda habet qvibvs haec lectio vtetvr: recta per polvm dvcta cvm directione α\alpha tota aeqvatione θ=α\theta = \alpha describitvr, et cvrvae vt rosae continvo percvrrvntvr. Pretivm est qvod vnvm idemqve pvnctvm plvres repraesentationes admittit. Ex qvo seqvitvr regvla qvam iam nvnc statvere expedit: pvnctvm ad cvrvam aeqvationis polaris pertinet si aliqva eivs repraesentatio aeqvationi satisfacit, etiamsi aliae non satisfacivnt.

LemmaRepraesentationes pvncti

Sit (r1,θ1)(r_1, \theta_1) repraesentatio pvncti PP a polo diversi. Par (r2,θ2)(r_2, \theta_2) etiam PP repraesentat si et solvm si r2=r1r_2 = r_1 et θ2=θ1+2kπ\theta_2 = \theta_1 + 2k\pi pro aliqvo nvmero integro kk, avt r2=−r1r_2 = -r_1 et θ2=θ1+(2k+1)π\theta_2 = \theta_1 + (2k + 1)\pi pro aliqvo nvmero integro kk. Polvs est pvnctvm qvod paribvs (0,θ)(0, \theta) repraesentatvr, qvolibet θ\theta.

Adhibentvr identitas Pythagorica et characterizatio angvlorvm coterminalivm ex lectione de angvlis eorvmqve mensvra: dvo angvli idem pvnctvm circvmferentiae vnitariae dant exacte cvm mvltiplo integro nvmeri 2π2\pi differvnt.

Demonstratio

  1. x2+y2=r2cos⁡2θ+r2sin⁡2θ=r2→r2=r1∨r2=−r1x^2 + y^2 = \resaltar{r^2 \cos^2 \theta + r^2 \sin^2 \theta} = \resaltar{r^2} \rightarrow \dato{rr}{r_2 = r_1 \vee r_2 = -r_1}

    Distantia pvncti a polo non pendet ex repraesentatione electa: per identitatem Pythagoricam, x2+y2x^2 + y^2 aeqvat r2r^2. Ergo dvae repraesentationes eivsdem pvncti habent r22=r12r_2^2 = r_1^2, et polvs est vnicvm pvnctvm cvm r=0r = 0.

  2. cos⁡θ2=cos⁡θ1∧sin⁡θ2=sin⁡θ1↔θ2=θ1+2kπ\resaltar{\cos \theta_2 = \cos \theta_1 \wedge \sin \theta_2 = \sin \theta_1} \leftrightarrow \dato{co}{\theta_2 = \theta_1 + 2k\pi}

    Si r2=r1≠0r_2 = r_1 \neq 0, coordinatas vtrivsqve pvncti per r1r_1 dividimvs: pvncta (cos⁡θ1,sin⁡θ1)(\cos \theta_1, \sin \theta_1) et (cos⁡θ2,sin⁡θ2)(\cos \theta_2, \sin \theta_2) circvmferentiae vnitariae congrvvnt, qvod exacte fit cvm angvli coterminales svnt.

  3. θ2+π=θ1+2kπ↔θ2=θ1+(2k−1)π\resaltar{\theta_2 + \pi} = \theta_1 + 2k\pi \leftrightarrow \resaltar{\theta_2 = \theta_1 + (2k - 1)\pi}

    Si r2=−r1r_2 = -r_1, par (−r1,θ2)(-r_1, \theta_2) idem pvnctvm dat ac (r1,θ2+π)(r_1, \theta_2 + \pi), et per gradvm svperiorem θ2+π\theta_2 + \pi coterminalis est cvm θ1\theta_1. Cvm 2k−12k - 1 omnes nvmeros impares percvrrat dvm kk nvmeros integros percvrrit, hoc est qvod lemma affirmat.

Sic pvnctvm coordinatarvm polarivm (2,π3)\left( 2, \frac{\pi}{3} \right) etiam repraesentationes (2,7π3)\left( 2, \frac{7\pi}{3} \right) et (2,−5π3)\left( 2, -\frac{5\pi}{3} \right) admittit, qvae vna conversione plvs avt vna minvs facivnt, et (−2,4π3)\left( -2, \frac{4\pi}{3} \right) vel (−2,−2π3)\left( -2, -\frac{2\pi}{3} \right), qvae in directionem contrariam spectant et retro cedvnt.

A coordinatis Cartesianis ad polares

Formvlae x=rcos⁡θx = r \cos \theta et y=rsin⁡θy = r \sin \theta transitvm a coordinatis polaribvs ad Cartesianas dant. Transitvs contrarivs postvlat vt repraesentatio inter omnes possibiles eligatvr; qvae vvlgo svmitvr ea est qvae habet r>0r > 0 et θ\theta in intervallo (−π,π]\ioc{-\pi}{\pi}. Ad angvlvm inveniendvm adhibetvr fvnctio arcvs tangentis ex lectione de fvnctionibvs trigonometricis inversis, qvae valores in (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} capit ideoqve solos angvlos qvadrantvm primi et qvarti videt: in ceteris dvobvs corrigenda est addendo vel svbtrahendo π\pi.

TheoremaA coordinatis Cartesianis ad polares

Omne pvnctvm (x,y)(x, y) a polo diversvm vnicam repraesentationem polarem (r,θ)(r, \theta) habet cvm r>0r > 0 et θ∈(−π,π]\theta \in \ioc{-\pi}{\pi}. Radivs eivs est r=x2+y2r = \sqrt{x^2 + y^2}, et angvlvs eivs polaris qvinqve casibvs formvlae infra scriptae datvr.

θ={arctan⁡yxx>0arctan⁡yx+πx<0∧y≥0arctan⁡yx−πx<0∧y<0π2x=0∧y>0−π2x=0∧y<0\theta = \begin{cases} \arctan \frac{y}{x} & x > 0 \\ \arctan \frac{y}{x} + \pi & x < 0 \wedge y \geq 0 \\ \arctan \frac{y}{x} - \pi & x < 0 \wedge y < 0 \\ \frac{\pi}{2} & x = 0 \wedge y > 0 \\ -\frac{\pi}{2} & x = 0 \wedge y < 0 \end{cases}

Notio est problema ad circvmferentiam vnitariam redvcere, vbi declivitas rectae per polvm dvctae dvo pvncta opposita determinat, et signvm abscissae inter ea decernit.

Demonstratio

  1. r=x2+y2,(cos⁡θ,sin⁡θ)=(xr,yr)r = \sqrt{x^2 + y^2}, \qquad \dato{q}{(\cos \theta, \sin \theta) = \left( \frac{x}{r}, \frac{y}{r} \right)}

    Per lemma svpra positvm, r2=x2+y2r^2 = x^2 + y^2, et cvm r>0r > 0 radivs est radix qvadrata. Divisione per rr facta, angvlvs qvaesitvs est angvlvs pvncti QQ circvmferentiae vnitariae, qvod exstat per proprietatem operimenti axiomatis percvrsvs, in lectione de angvlis eorvmqve mensvra admissi, et qvod in (−π,π]\ioc{-\pi}{\pi} vnicvm est, qvia dvo angvli hvivs intervalli, cvivs longitvdo est 2π2\pi, coterminales esse non possvnt.

  2. x=0→Q=(0,1)∨Q=(0,−1)x = 0 \rightarrow Q = \resaltar{(0, 1)} \vee Q = \resaltar{(0, -1)}

    Si x=0x = 0, pvnctvm QQ est (0,1)(0, 1) vel (0,−1)(0, -1) secvndvm signvm nvmeri yy, et per valores insignes sinvs et cosinvs angvlvs est π2\frac{\pi}{2} vel −π2-\frac{\pi}{2}.

  3. tan⁡θ=sin⁡θcos⁡θ=yx,α=arctan⁡yx\tan \theta = \frac{\resaltar{\sin \theta}}{\resaltar{\cos \theta}} = \resaltar{\frac{y}{x}}, \qquad \dato{al}{\alpha = \arctan \frac{y}{x}}

    Si x≠0x \neq 0, tangens angvli θ\theta est declivitas m=yxm = \frac{y}{x} rectae qvae polvm cvm pvncto ivngit. Angvlvs α=arctan⁡m\alpha = \arctan m hanc tangentem habet et in (−π2,π2)\ioo{-\frac{\pi}{2}}{\frac{\pi}{2}} iacet, vbi cosinvs positivvs est.

  4. u2+(mu)2=1↔u=11+m2∨u=−11+m2u^2 + \resaltar{(mu)^2} = 1 \leftrightarrow \resaltar{u = \frac{1}{\sqrt{1 + m^2}} \vee u = -\frac{1}{\sqrt{1 + m^2}}}

    In circvmferentia vnitaria exacte dvo pvncta svnt cvm declivitate mm a polo: si (u,v)(u, v) vnvm eorvm est, v=muv = mu et u2+v2=1u^2 + v^2 = 1. Opposita svnt, ita vt altervm abscissam positivam habeat, altervm negativam; per symmetriam angvli α+π\alpha + \pi, svnt pvncta angvlorvm α\alpha et α+π\alpha + \pi.

  5. x<0∧y<0→θ=α+π−2π=α−π∈(−π,−π2)x < 0 \wedge y < 0 \rightarrow \theta = \resaltar{\alpha + \pi - 2\pi} = \resaltar{\alpha - \pi} \in \ioo{-\pi}{-\frac{\pi}{2}}

    Si x>0x > 0, pvnctvm QQ abscissam positivam habet et θ=α\theta = \alpha. Si x<0x < 0, est pvnctvm angvli α+π\alpha + \pi. Cvm insvper y≥0y \geq 0, declivitas negativa vel nvlla est, α∈(−π2,0]\alpha \in \ioc{-\frac{\pi}{2}}{0} et α+π\alpha + \pi iam in (π2,π]\ioc{\frac{\pi}{2}}{\pi} iacet; cvm y<0y < 0, declivitas positiva est et vna conversio svbtrahenda est, α+π−2π=α−π\alpha + \pi - 2\pi = \alpha - \pi.

Exempli gratia, pvnctvm (−1,1)(-1, 1) in secvndo qvadrante iacet: radivs eivs est 2\sqrt{2}, et qvia arctan⁡(−1)=−π4\arctan(-1) = -\frac{\pi}{4}, angvlvs eivs polaris est −π4+π=3π4-\frac{\pi}{4} + \pi = \frac{3\pi}{4}. Machina calcvlatoria qvae arcvm tangentis nvmeri yx\frac{y}{x} dat, signis nvmerorvm xx et yy separatim non consideratis, redderet −π4-\frac{\pi}{4}, qvi est angvlvs pvncti oppositi (1,−1)(1, -1).