Ad argvmentvm transi
Topos Uranos
Latine

Rotatio axivm et aeqvatio generalis secvndi gradvs

Svmmarivm

Haec lectio stvdivm conicarvm absolvit casv qvi in svspenso manserat: aeqvatione secvndi gradvs qvae terminvm in xyxy continet. Per coordinatas polares et formvlas differentiae dedvcit qvomodo coordinatae pvncti mvtentvr cvm axes angvlo θ\theta convertvntvr, et demonstrat rotationem distantias conservare, ita vt nvllivs figvrae formam alteret. Coefficientes aeqvationis generalis post rotationem compvtat, et demonstrat angvlvm qvi cot⁡2θ=A−CB\cot 2\theta = \frac{A - C}{B} satisfacit terminvm in xyxy eliminare, svmmamqve A+CA + C et discriminans B2−4ACB^2 - 4AC rotatione non mvtari. Inde seqvitvr classificatio omnivm aeqvationvm secvndi gradvs per signvm discriminantis: ellipses, parabolae et hyperbolae, cvm casibvs earvm degeneratis. Finitvr agnoscendo vt conicas rotatas hyperbolam xy=1xy = 1, ellipsim et parabolam qvarvm axes obliqvi svnt.

Proposita discendi

  1. Formvlas dedvcere qvae coordinatas pvncti in dvobvs systematibvs Cartesianis rotatione differentibvs inter se conectvnt, et demonstrare rotationem distantias conservare.
  2. Coefficientes aeqvationis secvndi gradvs post rotationem compvtare et angvlvm eligere qvi terminvm in xyxy eliminat.
  3. Demonstrare A+CA + C et B2−4ACB^2 - 4AC rotatione invariantia esse, eisqve vti ad formam redactam sine svbstitvtione obtinendam.
  4. Aeqvationem generalem secvndi gradvs per signvm discriminantis eivs classificare, elementa eivs in coordinatis rotatis invenire eaqve in coordinatis originalibvs exprimere.

Rotatio axivm

In lectione cvrsvs I de sectionibvs conicis demonstratvm est omnem aeqvationem secvndi gradvs sine termino in xyxy parabolam, ellipsim, hyperbolam avt vnvm e casibvs earvm degeneratis repraesentare, qvarvm axes symmetriae axibvs coordinatarvm paralleli svnt. Iam vero ellipsis cvivs axis maior in recta y=xy = x iacet etiam aeqvationem secvndi gradvs habet, et in ea terminvs in xyxy apparet. Vt agnoscatvr, expedit alivd systema axivm eligere, respectv primi rotatvm, in qvo cvrva formam svam canonicam recvperet. Nvnc examinatvr qvomodo coordinatae pvncti mvtentvr cvm axes convertvntvr.

Sit systema Cartesianvm axivm xx et yy, qvale in lectione cvrsvs I de aeqvatione rectae et systematibvs Cartesianis, et sit θ\theta nvmervs realis. Systema angvlo θ\theta rotatvm est systema Cartesianvm qvod eandem originem et eandem vnitatem longitvdinis habet, cvivs semiaxis positivvs abscissarvm est semirecta ab origine proficiscens cvm angvlo θ\theta in positione normali, et cvivs semiaxis positivvs ordinatarvm est semirecta angvli θ+π2\theta + \frac{\pi}{2}. Novae coordinatae litteris maivscvlis scribvntvr: idem pvnctvm PP coordinatas (x,y)(x, y) in systemate originali habet et (X,Y)(X, Y) in rotato. Cvm dvo systemata eandem orientationem habeant, in rotato qvoqve coordinatae polares definivntvr, eodem polo et semiaxe positivo ipsivs XX pro axe polari.

TheoremaFormvlae rotationis

Si pvnctvm coordinatas (x,y)(x, y) in systemate Cartesiano habet et coordinatas (X,Y)(X, Y) in systemate angvlo θ\theta rotato, tvm coordinatae eivs novae ex originalibvs, et originales ex novis, per formvlas infra scriptas obtinentvr.

X=xcos⁡θ+ysin⁡θ,Y=−xsin⁡θ+ycos⁡θX = x \cos \theta + y \sin \theta, \qquad Y = -x \sin \theta + y \cos \theta
x=Xcos⁡θ−Ysin⁡θ,y=Xsin⁡θ+Ycos⁡θx = X \cos \theta - Y \sin \theta, \qquad y = X \sin \theta + Y \cos \theta

Idea est in coordinatis polaribvs axes convertere distantiam a polo non mvtare et tantvm θ\theta ab angvlo polari svbtrahere; formvlae additionis et svbtractionis, qvae in lectione de formvlis additionis et svbtractionis demonstrantvr, reliqva efficivnt.

Demonstratio

  1. x=rcos⁡φ,y=rsin⁡φ\dato{xy}{x = r \cos \varphi, \quad y = r \sin \varphi}

    Si PP est origo, coordinatae eivs in vtroqve systemate nvllae svnt et formvlae valent. Si non est, repraesentationem polarem (r,φ)(r, \varphi) cvm r>0r > 0 respectv axis polaris originalis svmimvs, vt in lectione de coordinatis polaribvs.

  2. X=rcos⁡(φ−θ),Y=rsin⁡(φ−θ)\dato{XY}{X = r \cos(\varphi - \theta), \quad Y = r \sin(\varphi - \theta)}

    In systemate rotato polvs idem est, ita vt distantia ipsivs PP a polo adhvc rr sit. Sit ψ\psi angvlvs polaris ipsivs PP in systemate rotato. In circvmferentia vnitaria novvs axis polaris per P(θ)P(\theta) transit; inde arcvm ψ\psi percvrrere dvcit, per proprietatem conivnctionis axiomatis percvrsvs ex lectione de angvlis eorvmqve mensvra, ad P(θ+ψ)P(\theta + \psi), qvod esse debet pvnctvm P(φ)P(\varphi) semirectae qvae a polo ad PP tendit. Per lemma angvlorvm coterminalivm, θ+ψ\theta + \psi et φ\varphi mvltiplo ipsivs 2π2\pi differvnt, ita vt ψ=φ−θ\psi = \varphi - \theta svmere possimvs.

  3. X=rcos⁡(φ−θ)=rcos⁡φcos⁡θ+rsin⁡φsin⁡θ=xcos⁡θ+ysin⁡θX = \resaltar{r \cos(\varphi - \theta)} = r \cos \varphi \cos \theta + r \sin \varphi \sin \theta = \resaltar{x \cos \theta + y \sin \theta}

    Cosinvm differentiae evolvimvs et in qvoqve termino coordinatas originales agnoscimvs.

  4. Y=rsin⁡(φ−θ)=rsin⁡φcos⁡θ−rcos⁡φsin⁡θ=−xsin⁡θ+ycos⁡θY = \resaltar{r \sin(\varphi - \theta)} = r \sin \varphi \cos \theta - r \cos \varphi \sin \theta = \resaltar{-x \sin \theta + y \cos \theta}

    Eodem modo, cvm sinv differentiae.

  5. x=rcos⁡((φ−θ)+θ)=Xcos⁡θ−Ysin⁡θ,y=rsin⁡((φ−θ)+θ)=Xsin⁡θ+Ycos⁡θx = \resaltar{r \cos((\varphi - \theta) + \theta)} = \resaltar{X \cos \theta - Y \sin \theta}, \qquad y = \resaltar{r \sin((\varphi - \theta) + \theta)} = \resaltar{X \sin \theta + Y \cos \theta}

    Vicissim, angvlvs originalis est svmma φ=(φ−θ)+θ\varphi = (\varphi - \theta) + \theta; formvlae svmmae coordinatas originales per novas dant.

Dvo paria formvlarvm eandem strvctvram habent: secvndvm ex primo obtinetvr θ\theta in −θ-\theta mvtando, qvia systema originale est id qvod oritvr novvm angvlo −θ-\theta convertendo. Exempli gratia, cvm θ=π4\theta = \frac{\pi}{4}, pvnctvm (1,1)(1, 1) coordinatas novas X=2X = \sqrt{2} et Y=0Y = 0 habet: in novo axe abscissarvm iacet, distantia 2\sqrt{2} ab origine. Expedit rotationem axivm cvm rotatione pvnctorvm non confvndere: axes angvlo θ\theta convertere pvncta immota relinqvit eorvmqve nomina mvtat, et idem efficit ac axes immotos relinqvere et totvm planvm angvlo −θ-\theta convertere.

LemmaRotatio distantias conservat

Distantia inter dvo pvncta, per coordinatas eorvm in systemate rotato compvtata, cvm ea congrvit qvae per coordinatas eorvm originales compvtatvr.

Res iam in lectione de coordinatis polaribvs animadversa est: distantia inter dvo pvncta ab angvlis eorvm polaribvs tantvm per differentiam eorvm pendet, et planvm convertere est constantem eis angvlis addere. Hic directe in coordinatis Cartesianis confirmatvr, per formvlas rotationis et identitatem Pythagoricam ex lectione de rationibvs trigonometricis.

Demonstratio

  1. X1−X2=ucos⁡θ+vsin⁡θ,Y1−Y2=−usin⁡θ+vcos⁡θ\dato{dif}{X_1 - X_2 = u \cos \theta + v \sin \theta, \quad Y_1 - Y_2 = -u \sin \theta + v \cos \theta}

    Sint P1P_1 et P2P_2 dvo pvncta, coordinatarvm (x1,y1)(x_1, y_1) et (x2,y2)(x_2, y_2) in systemate originali et (X1,Y1)(X_1, Y_1) et (X2,Y2)(X_2, Y_2) in rotato. Formvlas rotationis pro vtroqve pvncto scriptas svbtrahimvs, cvm u=x1−x2u = x_1 - x_2 et v=y1−y2v = y_1 - y_2.

  2. (X1−X2)2+(Y1−Y2)2=(ucos⁡θ+vsin⁡θ)2+(−usin⁡θ+vcos⁡θ)2=u2(cos⁡2θ+sin⁡2θ)+v2(sin⁡2θ+cos⁡2θ)(X_1 - X_2)^2 + (Y_1 - Y_2)^2 = \resaltar{(u \cos \theta + v \sin \theta)^2 + (-u \sin \theta + v \cos \theta)^2} = \resaltar{u^2\left(\cos^2 \theta + \sin^2 \theta\right) + v^2\left(\sin^2 \theta + \cos^2 \theta\right)}

    Ad qvadratvm elevamvs et addimvs. Prodvcta dvpla, 2uvsin⁡θcos⁡θ2uv \sin \theta \cos \theta et −2uvsin⁡θcos⁡θ-2uv \sin \theta \cos \theta, se invicem tollvnt.

  3. (X1−X2)2+(Y1−Y2)2=u2+v2=(x1−x2)2+(y1−y2)2(X_1 - X_2)^2 + (Y_1 - Y_2)^2 = \resaltar{u^2 + v^2} = (x_1 - x_2)^2 + (y_1 - y_2)^2

    Per identitatem Pythagoricam qvaeqve parenthesis 11 valet, et restat qvadratvm distantiae originalis.

Lemma conseqventiam habet qvae omnia deinceps tractata ivstificat. Parabola, ellipsis et hyperbola in lectionibvs antecedentibvs per proprietates distantiarvm a focis et directricibvs definitae svnt; cvm rotatio distantias conservet, cvrva qvae in systemate rotato est ellipsis semiaxivm aa et bb est, in systemate originali conspecta, ellipsis eorvndem semiaxivm, qvamvis axes eivs symmetriae obliqvi sint. Idem valet de ceteris conicis, de focis earvm, verticibvs et asymptotis, qvae ex vno systemate in altervm per formvlas rotationis transfervntvr.