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2009 年 11 月 30 日星期一

Wikipedia 點人 ge

分類: Uncategorised

Parameters

Various parameters are associated with a conic section.

conic section equation eccentricity (e) linear eccentricity (c) semi-latus rectum (ℓ) focal parameter (p)

circle $x^2+y^2=r^2 \,$ $0$ $0$ $r \,$ $\infty$

ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $\frac{\sqrt{a^2-b^2}}{a}$ $\sqrt{a^2-b^2}$ $\frac{b^2}{a}$ $\frac{b^2}{\sqrt{a^2-b^2}}$

parabola $y 2 = 4 a x$ $1$ $a\,$ $2a \,$ $2a\,$

hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $\frac{\sqrt{a^2+b^2}}{a}$ $\sqrt{a^2+b^2}$ $\frac{b^2}{a}$ $\frac{b^2}{\sqrt{a^2+b^2}}$

Conic parameters in the case of an ellipse

Conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negative number e, are the locus of points whose distance to F equals e times their distance to L. F is called the focus, L the directrix, and e the eccentricity.

The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci).

The latus rectum (2ℓ) is the chord parallel to the directrix and passing through the focus (or one of the two foci).

The semi-latus rectum (ℓ) is half the latus rectum.

The focal parameter (p) is the distance from the focus (or one of the two foci) to the directrix.

The following relations hold:

$p e = \ell \,$
$a e = c \,$

source: http://en.wikipedia.org/wiki/Semi-latus_rectum#Parameters

Mathematics of the ellipse

Main article: Ellipse

Figure 4: Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axis b and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = r_min and for θ = 180°, r = r_max.

In the ellipse equation

$r=\frac{p}{1+\varepsilon\cdot\cos\theta}$