Sectiones Conicae: Characterizatio et Graphice Parabolarum, Ellipseon et Hyperbolarum
Summarium:
In hac lectione sectiones conicas (parabolas, ellipses et hyperbolas) recognoscemus, ab eorum aequationibus canonicis et generalibus incipientes. Explicatur quomodo unaquaeque curva cognosci et characterizari possit, cum attentione speciali ad elementa clavis sicut vertex, focus et axis symmetriae in parabolis, atque ad distinctionem inter ellipses et hyperbolas secundum signa coefficientium suorum.
Proposita Discendi:
Hoc capite perfecto discipulus poterit
- Agoscere aequationes canonicas sectionum conicarum (parabolarum, ellipseon, hyperbolarum)
- Computare unamquamque proprietatem sectionum conicarum: longitudinem semiaxium, distantiam focalem, directricem, etc.
INDEX CONTENTORUM
Sectiones conicae
Recognitio Parabolarum
Recognitio Ellipseon et Hyperbolarum
Characterizatio Ellipseos
Characterizatio Hyperbolae
Exercitia Soluta
Sectiones conicae
Sectiones conicae appellantur omnes curvae quae oriuntur ex sectione superficiei coni plano facto. Familia sectionum conicarum constat ex circulis et ellipsibus, et hyperbolis: omnes sunt curvae quas iam investigavimus.
Nunc recensionem faciemus de technicis quibus unaquaeque harum curvarum agnosci et characterizari potest. In formis canonicis specialiter incumbemus, quia hae formae saepissime occurrunt atque minime informationis explicite exhibent. Aequationes generales, contra, fere totam characterisationem geometricam manifestant.
Recognitio Parabolarum
Omnis parabola repraesentatur per aequationem huius formae
y=ax^2 + bx + c, cum a\neq 0
Ex hac forma obtinuimus
- Coordinatae Verticis: \displaystyle (x_0, y_0)=\left( -\dfrac{b}{2a}, c - \dfrac{b^2}{4a} \right)
- Positio focalis: \displaystyle f=\dfrac{1}{4a}
- Coordinatae foci: \displaystyle focus=\left( -\dfrac{b}{2a}, c - \dfrac{b^2}{4a} + f \right) =\left( -\dfrac{b}{2a}, c + \dfrac{1- b^2}{4a} \right)
- Aequatio directricis: \displaystyle y= c - \dfrac{b^2}{4a} - f = c - \dfrac{1+b^2}{4a}
- Aequatio axis symmetriae: \displaystyle x= -\dfrac{b}{2a}
- Intersections cum axe x (si existunt): \displaystyle x_{1,2}= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}
Et his iam habemus omnem informationem necessariam ad graphice repraesentandam quamlibet parabolam.
Recognitio Ellipseon et Hyperbolarum
Ellipses et hyperbolae, ut vidimus, habent expressionem canonicam huiusmodi formae.
Ax^2 + Bx + Cy^2 + Dy + E = 0
Ubi A et C sunt constantes a zero distinctae, et ex iis quae studuimus, habetur quod:
- Si A et C idem signum habent, tunc est ellipsis.
- Si A et C signa contraria habent, tunc est hyperbola.
Ad casus utrumque clare separandos, scribemus:
- \alpha x^2+ \beta x + \gamma y^2 + \delta y + \epsilon = 0 est ellipsis.
- \alpha x^2+ \beta x - \gamma y^2 + \delta y + \epsilon = 0 est hyperbola.
Ubi \alpha, \beta, \gamma, \delta et \epsilon sunt numeri reales quicumque, et \alpha et \gamma semper positi sunt. Sic scribere nobis permittit casus ambos clare distinguere. Ex hoc possumus sequentia concludere:
Characterizatio Ellipseos
Partiendo ab aequatione canonica habemus deductionem sequentem:
| (1) | \alpha x^2+ \beta x + \gamma y^2 + \delta y + \epsilon = 0 | ; aequatio canonica ellipsium. |
| (2) | \displaystyle \alpha \left( x^2+ \dfrac{\beta}{\alpha }x\right) + \gamma \left(y^2 + \dfrac{\delta}{\gamma }y\right) =- \epsilon | ; factorando et terminos regruppando |
| (3) | \displaystyle \alpha \left( x + \dfrac{\beta}{2 \alpha }\right)^2 + \gamma \left(y + \dfrac{\delta}{2 \gamma } \right)^2 =\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon | ; quadrata complendo et terminos regruppando |
| (4) | \displaystyle \alpha \dfrac{\left( x + \dfrac{\beta}{2 \alpha }\right)^2}{\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)} + \gamma \dfrac{\left(y + \dfrac{\delta}{2 \gamma } \right)^2}{\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)} = 1 | ; dividendo omnia per \displaystyle \dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon |
| (5) | \displaystyle \dfrac{\left( x + \dfrac{\beta}{2 \alpha }\right)^2}{\dfrac{1}{\alpha }\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)} + \dfrac{\left(y + \dfrac{\delta}{2 \gamma } \right)^2}{\dfrac{1}{ \gamma}\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)} = 1 | ; reordinando \alpha et \gamma |
| (6) | \displaystyle \left( \dfrac{ x - \left(-\dfrac{\beta}{2 \alpha }\right)}{\sqrt{\dfrac{1}{\alpha}\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)}}\right)^2 + \left( \dfrac{y - \left(-\dfrac{\delta}{2 \gamma } \right)}{\sqrt{\dfrac{1}{\gamma}\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)}}\right)^2 = 1 | ; restruendo per radices |
In hac deductione gradus (3) est praesertim delicatus, quia si coefficientis \displaystyle \dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon valor est negativus, tunc ellipsis exsistere non potest.
Recordemur aequationem generalem ellipsium huius formae esse:
\displaystyle \left( \dfrac{x-h}{a} \right)^2 + \left(\dfrac{y-k}{b} \right)^2 = 1
Hoc ultimo resultato habemus nunc relationem directam inter parametra formulae generalis quae permittit nobis totam informationem contentam in expressione canonica manifestare:
- Coordinatae centri: \displaystyle (h,k) = \left( -\dfrac{\beta}{2\alpha}, -\dfrac{\delta}{2\gamma}\right)
- Longitudo semiaxis horizontalis: \displaystyle a = \sqrt{\dfrac{1}{\alpha}\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)}
- Longitudo semiaxis verticalis: \displaystyle b = \sqrt{\dfrac{1}{\gamma}\left(\dfrac{\beta^2}{4\alpha } + \dfrac{\delta^2}{4\gamma } - \epsilon\right)}
His iam possibile est ellipsin agnoscere et graphice repraesentare directe ex forma canonica. Figura eius talis videbitur:
Characterizatio Hyperbolae
Ratione omnino simili utens potes, ab aequatione canonica, characterisationem completam hyperbolarum efficere. Re vera, analysis tam similis est ut exemplum ellipseos hic applicetur, cum paucis mutationibus.
| (1) | \alpha x^2+ \beta x - \gamma y^2 + \delta y + \epsilon = 0 | ; aequatio canonica hyperbolarum. |
| (2) | \displaystyle \alpha \left( x^2+ \dfrac{\beta}{\alpha }x\right) - \gamma \left(y^2 - \dfrac{\delta}{\gamma }y\right) =- \epsilon | ; factorando et terminos regruppando |
| (3) | \displaystyle \alpha \left( x + \dfrac{\beta}{2 \alpha }\right)^2 - \gamma \left(y - \dfrac{\delta}{2 \gamma } \right)^2 =\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon | ; quadrata complendo et terminos regruppando |
| (4) | \displaystyle \alpha \dfrac{\left( x + \dfrac{\beta}{2 \alpha }\right)^2}{\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)} - \gamma \dfrac{\left(y - \dfrac{\delta}{2 \gamma } \right)^2}{\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)} = 1 | ; dividendo omnia per \displaystyle \dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon |
| (5) | \displaystyle \dfrac{\left( x + \dfrac{\beta}{2 \alpha }\right)^2}{\dfrac{1}{\alpha}\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)} - \dfrac{\left(y - \dfrac{\delta}{2 \gamma } \right)^2}{\dfrac{1}{\gamma}\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)} = 1 | ; reordinando terminos \alpha et \gamma |
| (6) | \displaystyle \left( \dfrac{ x - \left(-\dfrac{\beta}{2 \alpha }\right)}{\sqrt{\dfrac{1}{\alpha}\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)}}\right)^2 - \left( \dfrac{y - \left(\dfrac{\delta}{2 \gamma } \right)}{\sqrt{\dfrac{1}{\gamma}\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)}}\right)^2 = 1 | ; restruendo per radices |
Ex hoc nunc habemus relationem directam inter aequationem canonicam et aequationem hyperbolarum, quae nobis permittit celeriter figuram earum depingere.
\displaystyle \left(\dfrac{x-h}{a} \right)^2 - \left(\dfrac{y-k}{b} \right)^2 =1
Nunc autem, dissimiliter ab ellipsibus, hic rectius est loqui de “arca generatrice”, ut in sequenti figura demonstrabitur:
- Coordinatae centri: \displaystyle (h,k) = \left( -\dfrac{\beta}{2\alpha}, \dfrac{\delta}{2\gamma}\right)
- Longitudo semiaxis horizontalis: \displaystyle a = \sqrt{\dfrac{1}{\alpha}\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)}
- Longitudo semiaxis verticalis: \displaystyle b = \sqrt{\dfrac{1}{\gamma}\left(\dfrac{\beta^2}{4\alpha } - \dfrac{\delta^2}{4\gamma } - \epsilon\right)}
Ex his analysibus iam possumus quemlibet membrum familiae sectionum conicarum sine ulla difficultate graphice repraesentare.