What is ellipses: Definition and 40 Discussions

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity



e


{\displaystyle e}
, a number ranging from



e
=
0


{\displaystyle e=0}
(the limiting case of a circle) to



e
=
1


{\displaystyle e=1}
(the limiting case of infinite elongation, no longer an ellipse but a parabola).
An ellipse has a simple algebraic solution for its area, but only approximations for its perimeter (also known as circumference), for which integration is required to obtain an exact solution.
Analytically, the equation of a standard ellipse centered at the origin with width



2
a


{\displaystyle 2a}
and height



2
b


{\displaystyle 2b}
is:







x

2



a

2




+



y

2



b

2




=
1.


{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1.}
Assuming



a

b


{\displaystyle a\geq b}
, the foci are



(
±
c
,
0
)


{\displaystyle (\pm c,0)}
for



c
=



a

2




b

2






{\textstyle c={\sqrt {a^{2}-b^{2}}}}
. The standard parametric equation is:




(
x
,
y
)
=
(
a
cos

(
t
)
,
b
sin

(
t
)
)


for


0

t

2
π
.


{\displaystyle (x,y)=(a\cos(t),b\sin(t))\quad {\text{for}}\quad 0\leq t\leq 2\pi .}
Ellipses are the closed type of conic section: a plane curve tracing the intersection of a cone with a plane (see figure). Ellipses have many similarities with the other two forms of conic sections, parabolas and hyperbolas, both of which are open and unbounded. An angled cross section of a cylinder is also an ellipse.
An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:




e
=


c
a


=


1




b

2



a

2






.


{\displaystyle e={\frac {c}{a}}={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}.}
Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the Solar System is approximately an ellipse with the Sun at one focus point (more precisely, the focus is the barycenter of the Sun–planet pair). The same is true for moons orbiting planets and all other systems of two astronomical bodies. The shapes of planets and stars are often well described by ellipsoids. A circle viewed from a side angle looks like an ellipse: that is, the ellipse is the image of a circle under parallel or perspective projection. The ellipse is also the simplest Lissajous figure formed when the horizontal and vertical motions are sinusoids with the same frequency: a similar effect leads to elliptical polarization of light in optics.
The name, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his Conics.

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  1. opus

    Solving an algebraic identity with ellipses

    Homework Statement Prove the following relation. It is assumed that all values of x and y which occur are such that the denominators in the indicated fractions are not equal to 0. $$\frac{x^n-1}{x-1}=x^{n-1}+x^{n-2}+...+x+1$$ Homework EquationsThe Attempt at a Solution Please see attached...
  2. S

    I Identifying Identical Points Using Overlapping Error Ellipses

    I'm not very familiar with statistics, so that's my main problem. I use a least squares software to generate the coordinates of points and their associated 95% error ellipses. If the error ellipses meet a certain pre determined tolerance then I'm done. However, I have a situation where I have...
  3. Tris Fray Potter

    B Can Parabolas Transform into Ellipses?

    I know the difference between the two, but I was wondering if parabolas ever became so steep that they turned back into an elliptical shape.
  4. C

    Calculating Equations of Ellipses Within a Cone

    Hello. So, I'm designing an equatorial platform mount for my telescope at the moment. I'm also going to use it for another telescope that I'm in the process of building. I know that for both of the bearings, I can use small sections of two circles cut from a cone with an angle between the axis...
  5. K

    MHB Conics- Word problem with ellipses.

    Hi. I am given the following problem. A small bridge is shaped like a semi-ellipse. Given that its maximum height is 3m and that its foci are located 4m from the centre find the height of the bridge at a distance of 2m from its edge. So the problem give me the values b= 3 and c=4. With this we...
  6. A

    MHB Equation of the Normal and Ellipses Questions

    Hey guys, I have a couple of questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For 1a, I used implicit differentiation and isolated dy/dx. This gave me the following answer: dy/dx = (-3x^2 -4y)/(4x+12y) Which I...
  7. J

    Sketching Ellipses: Comparing 0 and 8

    Hi guys, What is the best way to sketch 0 = x2 - 2x + 4y2 And 8 = x2 - 2x + 4y2 ? How do I sketch these two and how do I know they're both ellipses? Thank you.
  8. M

    Method for unique collisions between 2 subdivided ellipses

    Hello, I would like find a way to figure out how many unique collision there are between 2 equally subdivided ellipsoids (velocity=1). When you have 2 ellipsoids and you let them collide than you have an infinite amount of possible outcomes. The goal is to reduce this infinite number to...
  9. E

    Kepler Orbits and ellipses

    Homework Statement I am trying to see if I am on the right track with this. The problem: A kepler orbit (an ellipse) in Cartesian coordinates is: $$(1−\epsilon^2)x^2 + 2\alpha \epsilon x + y^2 = \alpha^2$$. The task is to show that the major and minor axes are: $$a =...
  10. N

    Generalizing symmetry axis of constant-contour ellipses

    Hi I am looking at the contours of the following function f, which trace out an ellipse: f(x, y, z) = \exp(-x^2a)\exp(-y^2b) Here a\neq b are both positive, real constants. The axis of these ellipses is along z. Now, I am wondering how to generalize the function f such that the symmetry...
  11. P

    Finding the points of intersection of two ellipses

    Does anyone know where I can find an algorithm for the points of intersection of two ellipses existing with arbitrary center points and rotations and having 0, 1, 2, 3 or 4 points of intersection?
  12. P

    Question Regarding Ellipses

    I'm trying to write an algorithm that will create the smallest possible ellipse to encompass any number of points on 2D euclidean space. I've gotten it to the point where I can attain the major axis A by taking the furthest two points in the set and likewise the centerpoint C as the average of...
  13. veronica1999

    MHB Graphing Ellipses: How to Change Formats

    How do I graph this ellipse? It doesn't seem to be in the right form. (x+2)^2 /5 + 2 (y-1)^2 = 1 I don't know what to do with the 2 in front of the (y-1)^2 Doesn't an ellipse have to be x^2/a^2 + y^2/b^2 = 1
  14. DaveC426913

    Tech drawing - ellipses in perspective

    I'm doing some 2D pencil diagrams (CUD - computer-unaided design). So I've got a square in perspective (2 point perspective but 1 point perspective is good enough). Now I want to draw an ellipse within it, so that the ellipse is properly tangential to all four midpoints of the square. I have...
  15. P

    Double integral over region surrounded by two ellipses

    Homework Statement A thin plate has the form of the intersection of the regions limited by \frac{x^2}{9} + \frac{y^2}{4} = 1 and \frac{x^2}{4} + \frac{y^2}{9} = 1 Which is the plate's mass if his density is δ(x, y) = |x| 2. The attempt at a solution I've tried using u, v...
  16. D

    Finding Apogee and Perigee of Moon's Elliptical Orbit

    Homework Statement The Moon orbits the Earth in an elliptical path with the center of Earth at one focus. The major and minor axes of the orbit have lengths of 768,800 kilometers and 767,640 kilometers, respectively. a) Find the greatest distance, called the apogee, and the lest distance...
  17. J

    Troubles with Ellipses (Cartesian -> Polar)

    Okay, so I have just broken into the polar coordinate system, and I like to derive things on my own to strengthen my intuition. I decided to try and derive the equation of an ellipse swiftly on my own, and had the high ambitions of eventually deriving the area of an ellipse with polar...
  18. S

    Confusion about ellipses

    This is a really dumb question, but could someone quickly explain why it is that if b > a in the equation of an ellipse then y is the major axis. Just intuitively I want to think that y^2/5 as opposed to y^2/3 is going to be smaller for a given value of y since each value is being limited by...
  19. S

    Solving Ellipses on Spheres: Finding Point Sources in Galaxies

    I have galaxies in the shapes of ellipses, and I have point sources around these galaxies. I need to find a formula to determine whether or not the point sources are within a specific galaxy or outside. It would be extremely simple with circles, but I can't figure it out with ellipses...
  20. R

    A simple inequality with ellipses

    Assume: p>1, x>0, y>0 a \geq 1 \geq b > 0 \frac{a^2}{p^2}+(1-\frac{1}{p^2})b^2 \leq 1 \frac{x^2}{a^2}+\frac{y^2}{b^2} \leq 1 Prove: \frac{x}{p}+y\sqrt{1-\frac{1}{p^2}} \leq 1 I've been trying for 3 days and it's driving me crazy. Any ideas?
  21. P

    Question about ellipses described in the two-body problem

    Good afternoon, Suppose a two-body problem, where both bodies, of masses m1 and m2, are under influence of their mutual attractions. I know that, for the motion of one body relative to the other, this body describes a conic section whose eccentricity is given by the magnitude of the following...
  22. B

    Calculating Lamp Location Above x-Axis: Elliptical Shadow Problem

    Homework Statement A lamp is located three units to the right of the y-axis and a shadow is created by the elliptical region x^2 + 4y^2 = 5. If the point (-5,0) is on the edge of the shadow, how far above the x-axis is the lamp located? 2. The attempt at a solution Ive calculated the...
  23. Z

    Points on two ellipses with identical tangent lines

    Hi, I'm trying to get this working for a program I'm making. I've been working on this for a while, but I can't seem to figure it out. I have multiple rotated ellipses. Imagine you took a rubber band and stretched it around the ellipses. The rubber band would follow the curve of the outside...
  24. G

    Intersection of ellipses and equivalent problems

    Does anyone know how to determine whether two ellipses intersect? I don't need the precise points but rather only the answer whether there are points. All my attempts led to 4th order polynomials, which are heavy to solve, but considering that I don't need the actual points I assume there must...
  25. E

    Solving KE for a Bead Rolling Along an Ellipse Using Lagrange's Method

    I'm trying to find the equation of motion of a bead, which is constrained to roll along the bottom half of this frictionless ellipse, by using langrange's method - L(\phi,\dot{\phi}). Here's the setup: given the bottom half of an ellipse: \mathbf{r}(\phi) =...
  26. M

    Newtons theory of gravity : satelite orbits and ellipses

    The figure shows two planets of mass m orbiting a star of mass M. The planets are in the same orbit, with radius r, but are always at opposite ends of a diameter. This is the equation eq. I used: mv^2/r=GMm/r^2+Gmm/(2r)^2 This is what I came up with but it is not the right answer. Where...
  27. jacksonpeeble

    Ellipses - Basic Concept Question

    In class today, my instructor went over conic sections and ellipses (and hyperbolas, although that's irrelevant). We pretty much learned the basics - foci, semi-major and semi-minor axes, etc. However, the equation c²= a² − b² where c is the distance from the focus to vertex and b is the...
  28. I

    Also help with conicsmainly ellipses?

    Homework Statement x^2 + 4y^2 - 2x - 32y = 0 Get into standard form Homework Equations The Attempt at a Solution x^2 + 4y^2 - 2x - 32y = 0 Complete the square two times. (x^2 - 2x + 1) +4(y^2 - 8y +16 )=0 + 4(16) + 1 Simplfy.. (x-1)^2 + 4(y-4)^2 =65 Try to get the equation...
  29. X

    Find Height of Elliptical Arch Spanning 118ft & 8ft High

    Homework Statement A bridge is built in the shape of a semielliptical arch. It has a span of 118 feet. The height of the arch 25 feet from the center is to be 8 feet. Find the height of the arch at its center? Homework Equations not sure if the 25 feet from the center is the focal axis or...
  30. S

    Eye of Ellipses: Photographing Magnetic Lines of Constant Scalar Potential

    Photographing Magnetic Lines of Constant Scalar Potential http://www.sendspace.com/pro/dl/3hchd6 This post is intended as an introduction to my paper, Photonic Dipole Contours. I have been asked to address some deficiencies in my paper such as background information and prior work...
  31. T

    Conic Sections - Ellipses

    Homework Statement Center is at (4, -1) Vertex is at (4, -5) Focus is at (4, -3.5) Find the equation of the ellipse. Homework Equations horizontal ellipse: ((x-h)^2)/(a^2)) + ((y-k)^2)/(b^2)) = 1 Vertical ellipse: ((y-k)^2)/(a^2)) + ((x-h)^2)/(b^2)) = 1 c^2 = a^2 - b^2 The...
  32. C

    Level Curves of T(x, y) and V(x, y) - Revisiting Ellipses

    Homework Statement I need to sketch level curves of T(x, y) = 50(1 + x^2 + 3y^2)^{-1} and V(x, y) = \sqrt{1 - 9x^2 -4y^2} The Attempt at a Solution Is it correct that they are ellipses? ie [tex] 1 = \frac{9}{1 - c^2} x^2 + \frac{4}{1 - c^2}y^2[/itex] for V(x, y) = c = constant I feel so...
  33. P

    Is This Ellipse Equation Conversion Correct?

    Homework Statement an ellipse is represented by the equation: x|^2 + 4y^2 - 4 x + 8y - 60 = 0 express the equation in standard form: ((x-2)^2 / 68) + ((y-4)^2/17) = 1 can anyone tell me if this is accurate? thanks ~Amy
  34. R

    Factoring and Ellipses: How to Solve for Width?

    I'm trying to put this into factored form help me out and help me with ellipses I don't know what to do... Would i have to place the C that i solved into a^2 = 10^2 + 25.4^2 http://s65.photobucket.com/albums/h237/runicrice/?action=view&current=scan.jpg
  35. siddharth

    Exploring Precessing Ellipses: Showing How F Changes Orbit Shape

    Suppose that the force of attraction between the sun and the Earth is F = GMm(\frac{1}{r^2} + \frac{\alpha}{r^3}) Where \alpha is a constant. Show that the orbit does not close on itself but can be described as a precessing ellipse. Find an expression for the rate of precession of the...
  36. P

    Confused on Orbits: Ellipses vs Circles

    As someone who only studied first year physics and maths and have taken no interest in it since than I was rather surprised to wake up one morning and realize along with 99.5% of the population that I really had no idea how the planets orbits worked beyond the vague word ellipse which I didn't...
  37. T

    How do you solve problems involving ellipses centered at (h,k)?

    My math teacher has been out for 2 days and the subs have just been putting notes up and I'm totally not getting this. So maybe someone here can explain everything in other ways. Note that my book does a crappy job of explaining this subject. But it does break it down into two distince areas -...
  38. V

    Solving Precal Problems: Ellipses & Hyperbolas

    Here are two problems that stumped our entire precal class. And we have a test soon, so I would like to be able to know how to work these type of problems. 1. Write the equation of the hyperbola, x^2 + 4xy + y^2 - 12 = 0, in standard form. Okay, I know the formula needs to be x^2/a^2 -...
  39. A

    Finding the Area Within Two Ellipses Using Integration?

    i am given the equations for two ellipses: (x^2)/3 + y^2 =1 and x^2 + (y^2)/3 =1 and am told to find the area within both. i can just find the volume of both ellipses by taking the integral of the equations and then add them right? That's what first comes to mind when i saw this problem...
  40. Coelum

    Minimum distance between ellipses

    Forum, I'm addressing the problem of computing the minimum possible distance between two non-interacting bodies on elliptical orbits. From a general point of view, it looks like a minimization problem of a function of two variables, e.g. in the domain [0,2*pi)*[0,2*pi). This problem can be...
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