|
|
| Line 1: |
Line 1: |
| [[Image:An image describing the semi-major and semi-minor axis of eclipse.png|thumb|upright=1.8|The semi-major (in red*) and semi-minor axis (in blue*) of an ellipse.<br />
| | <br><br>Another day I woke up and realised - At the moment I have also been single for a little while and after much bullying from pals I today find luke bryan presale tickets ([http://www.senatorwonderling.com senatorwonderling.com]) myself registered for web dating. They assured me that there are a lot of interesting, sweet and ordinary individuals to fulfill, therefore the pitch is gone by here!<br>[http://Browse.Deviantart.com/?qh=§ion=&global=1&q=I+strive I strive] to keep as toned as possible being at the gymnasium several-times per week. I love my sports and make an effort to perform or see because many a potential. I am going to often at Hawthorn matches being winter. Note: I've seen the carnage of fumbling matches at stocktake sales, Supposing that you would contemplated purchasing a sport I don't mind.<br>My pals and household [http://lukebryantickets.pyhgy.com luke bryan tickets boston] are awesome and spending time together at pub gigabytes or dishes is obviously a necessity. I haven't ever been in to night clubs as I come to [http://www.google.com/search?q=realize&btnI=lucky realize] that you could never have a significant conversation against the noise. Additionally, I got two really adorable and definitely cheeky canines that are consistently excited to meet up new people.<br><br>Feel free to visit my homepage [http://minioasis.com luke bryan past tour dates] |
|
| |
| (* on some browsers)]]
| |
| | |
| In [[geometry]], the '''semi-minor axis''' (also '''semiminor axis''') is a [[line segment]] associated with most [[conic section]]s (that is, with [[ellipse]]s and [[hyperbola]]s) that is at [[right angle]]s with the [[semi-major axis]] and has one end at the center of the conic section. It is one of the [[axis of symmetry|axes of symmetry]] for the curve: in an ellipse, the shorter one; in a hyperbola, the one that does not intersect the hyperbola.
| |
| | |
| ==Ellipse==
| |
| | |
| The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the [[Focus (geometry)|foci]]) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.
| |
| | |
| The semi-minor axis ''b'' is related to the [[semi-major axis]] <math>a</math> through the [[eccentricity (mathematics)|eccentricity]] <math>e</math> and the [[semi-latus rectum]] <math>l</math>, as follows:
| |
| | |
| :<math>b = a \sqrt{1-e^2}\,\!</math>
| |
| | |
| :<math>al=b^2\,\!</math>.
| |
| | |
| The semi-minor axis of an ellipse is the [[geometric mean]] of the maximum and minimum distances <math>r_{max}</math> and <math>r_{min}</math> of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis:
| |
| | |
| :<math> b = \sqrt{r_{max}r_{min}}.</math>
| |
| | |
| A [[parabola]] can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping ''l'' fixed. Thus ''a'' and ''b'' tend to infinity, ''a'' faster than ''b''.
| |
| | |
| The length of the semi-minor axis could also be found using the following formula,<ref>http://www.mathopenref.com/ellipseaxes.html,"Major / Minor axis of an ellipse",Math Open Reference, 12 May 2013</ref>
| |
| :<math> 2b = \sqrt{(p+q)^2 -f^2} </math> where f is the distance between the foci, p and q are the distances from each focus to any point in the ellipse.
| |
| | |
| ==Hyperbola== | |
| | |
| In a hyperbola, a '''conjugate axis''' or '''minor axis''' of length 2''b'', corresponding to the minor axis of an ellipse, can be drawn perpendicular to the '''transverse axis''' or '''major axis''', the latter connecting the two [[Vertex (curve)|vertices]] (turning points) of the hyperbola, with the two axes intersecting at the center of the hyperbola. The endpoints (0, ±''b'') of the minor axis lie at the height of the asymptotes over/under the hyperbola's vertices. Either half of the minor axis is called the semi-minor axis, of length ''b''. Denoting the semi-major axis length (distance from the center to a vertex) as ''a'', the semi-minor and semi-major axes' lengths appear in the equation of the hyperbola relative to these axes as follows:
| |
| | |
| :<math>\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.</math>
| |
| | |
| The semi-minor axis and the semi-major axis are related through the eccentricity, as follows:
| |
| | |
| :<math>b = a \sqrt{e^2-1}.</math>
| |
| | |
| Note that in a hyperbola ''b'' can be larger than ''a''.
| |
| [http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node27.html] | |
| | |
| ==References== | |
| {{reflist}}
| |
| | |
| == External links ==
| |
| *[http://www.mathopenref.com/ellipsesemiaxes.html Semi-minor and semi-major axes of an ellipse] With interactive animation
| |
| | |
| {{orbits}}
| |
| | |
| [[Category:Conic sections]]
| |
| [[Category:Orbits]]
| |
Another day I woke up and realised - At the moment I have also been single for a little while and after much bullying from pals I today find luke bryan presale tickets (senatorwonderling.com) myself registered for web dating. They assured me that there are a lot of interesting, sweet and ordinary individuals to fulfill, therefore the pitch is gone by here!
I strive to keep as toned as possible being at the gymnasium several-times per week. I love my sports and make an effort to perform or see because many a potential. I am going to often at Hawthorn matches being winter. Note: I've seen the carnage of fumbling matches at stocktake sales, Supposing that you would contemplated purchasing a sport I don't mind.
My pals and household luke bryan tickets boston are awesome and spending time together at pub gigabytes or dishes is obviously a necessity. I haven't ever been in to night clubs as I come to realize that you could never have a significant conversation against the noise. Additionally, I got two really adorable and definitely cheeky canines that are consistently excited to meet up new people.
Feel free to visit my homepage luke bryan past tour dates