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. As such, 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
{\displaystyle 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.
The equation of ellipse reduces to :
$$(2x+3)^2+(3y+2)^2=8$$
$$\frac{(x+3/2)^2}{8/4}+\frac{(y+2/3)^2}{8/9}=1$$
Center of ellipse =##\left(\frac{-3}{2},\frac{-2}{3}\right)##
##b^2=a^2(1-e^2)=8/9## and ##a^2=8/4##
Therefore ##e=\frac{\sqrt{5}}{3}##
Distance between foci=##\frac{2\sqrt{10}}{3}##...
I think I have completed the exercise but since I have seldom used polar coordinates I would be grateful if someone would check out my work and tell me if I have done everything correctly. Thanks.
My solution follows.
Since ##\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1## it follows...
My initial idea was to first parametrize the ellipse as ##(a\sin(\theta')\cos(\phi'),b\sin(\theta')\sin(\phi'),c\cos(\theta'))## and then calculate ##\theta,\phi## in terms of these coordinates. I then did the coordinate transform ##x\to\frac{x}{a},y\to\frac{y}{b},z\to\frac{z}{c}## to convert it...
I want to calculate eccentric anomaly of all points of ellipse-circle intersection.
Ellipse is not rotated and its center is in origin.
Circle can be translated to (Cx, Cy) coordinates.
I am using python for calculations.
Only solution I found, is this...
Can time on elliptical orbit be expressed analytically? Which relations are capable of analytic expression?
The distance from focus can be expressed as a function of position angle θ:
r=a(1-e2)/(1+e cos θ)
The length linearly along the ellipse famously cannot be expressed analytically.
The total...
When the rocket accelerates in space does its trajectory which is an ellipse change in size and not the focal points because the Earth is still in one of two and also the current height doesn't increase, right?
Listen to the following arguments:
Earth's orbit isn't perfect ellipse because classically there is the gravitational field of moon and possibly of Mars and Venus which affect it
According to general relativity isn't perfect ellipse because there is the curvature of space time which doesn't...
Hi everyone
The solution for this question has b^2 as 4, but I don't see why it has to be 4. I've tried using different values of b for the ellipse on Desmos, and it is possible to make ellipses with smaller values of b that pass through (6,0).
Have I missed something in the question? Or has...
While fighting a CAD program, today, I might have stumbled on a potential way to easily calculate the circumference of an ellipse. I checked my method against a half a dozen online ellipse calculators and while my formula gives different results, I can't see where I'm making any logical errors...
Given:
x^2+xy+y^2=18
x^2+y^2=12
Attempt:
(x^2+y^2)+xy=18
12+xy=18
xy=6
y^2=12-x^2
(12)+xy=18
xy=6
Attempt 2:
xy=6
x=y/6
y^2/36+(y/6)y+y^2=18
43/36y^2=18
y ≠ root(6) <- should be the answer
Edit:
Just realized you can't plug the modified equation back into its original self
I plugged y=6/x...
While reading the Strang textbook on tilted ellipses in the form of ax^2+2bxy +cy^2=1, I got to thinking about ellipses of the form ax^2 + 2bx + 2cxy + 2dy + ey^2=1 and wondered if I could model them through 3x3 symmetric matrices. I think I figured out something that worked for xT A x, where x...
My physics textbook does the approximation that $$r=\frac{r_0}{1-\frac{A}{r_0}\sin\theta}\approx r_0\left( 1+\frac A r_0\sin\theta\right)$$ when ##A/r_0 \ll 1##. Can someone please explain how it is done?
An ellipse is a conic section - a slice through a cone. The axis of the cone coincides with the focus of the ellipse.(Correct?)
But ellipses can have two foci, not just one, as shown by this method of construction:
Questions:
1. Is F1 in the diagram the same as the axis of the conic section...
Summary:: Questions about finding largest ellipse in polygon
Was wondering if someone could help me understand two concepts involved with finding largest ellipse in a polygon? Some background:
First, I set up a set of half-plane representations for the example polygon below and as you can...
Hi all:)
In my recent exploration of Elliptic Function, Curves and Motion I have come upon a handy equation for creating orbital motion.
Essentially I construct a trigonometric function and use the max distance to foci as the boundary for my motion on the x-plane.
When I plot a point rotating...
As part of the final stage of a problem, there is some algebraic manipulation to be done (from the solution manual):
But I'm getting lost somewhere:
Also a bit of general advice needed: This is part of a self-study Calculus course, and I often have difficulty with bigger algebraic...
My Attempt :We need to maximize
## D=\sqrt{x^2+(y+2)^2} ##
subject to the constraint
##4x^2 + 5y^2 = 20##.
From the constraint equation, we can write
##x^2=\frac{20-5y^2}{4}##
Using this in the formula for distance,
##D=\sqrt{\frac{20-5y^2}{4}+(y+2)^2}##
Differentiating this wrt y, and...
Substituting :
a = (9.15x10^7 mi)+(9.45x10^7mi) = 1.86x10^8 mi
b = ( a/2 ) = 9.3x10^7 mi
For this, I used six terms and got :
1.075x10^9 miles
Is my math wrong?;
The question asks, "A one-way road has an overpass in the form of a semi-ellipse, 15 ft high at the center, and 40 ft wide. Assuming a truck is 12 ft wide, what is the tallest truck that can pass under the overpass?"
I don't think this is a super complicated question yet it proves to be too...
I was able to find the equation of an ellipse where its major axis is shifted and rotated off of the x,y, or z axis. However, I could not find anywhere an equation for a spheroid that does not have its axis or revolution along the x,y, or z axis. How might I go about deriving such an...
I'm tasked with drawing the trajectory of the Moon around the Earth (in 2D), taking into account the fact that the trajectory also undergoes precession, so the elliptical orbit rotates about it's center (I think it should rotate around the Earth-Moon barycenter, but for the first step I...
Homework Statement
We want to send a satellite from a low Earth orbit of 320 km to mars. Calculate the change in velocity required to join the transfer ellipse.
Homework Equations
Earth velocity: (μS/REarthRev)1/2
Transfer velocity at perihelion: (2μSRMarsRev/(REarthRev(REarthRev+RMarsRev))1/2...
I'm trying to build an oval bicycle wheel and need a formula to calculate the radii so that I can use the results to calculate the length for each spoke. See diagram for details. Hope someone can help.
Homework Statement
Graph the ellipse 4x² + 2y² = 1
Homework Equations
4x² + 2y² = 1
The Attempt at a Solution
2x² + y²/2 = 1/2
I searched for exercises on Google, and i didn't find an equation like that. I watched videoleassons too but it didn't teach this type of equation.
There seem to be many kinds of examples where the behavior of a quantum particle having been constrained to move on a curve or surface is investigated. The simplest is the case of a particle on a circular path or a spherical surface, where the energy eigenstates are equal to the angular momentum...
Hello everyone :)
Not too long ago, I was thinking about planetary motion around a sun, both with circular orbits and elliptic orbits. However, when thinking a little longer about these two cases in a broader sense, I spotted a big difference which I found quite odd (assume purely classical...
Hi :)
The question is in dutch so i'l translate it.
on an ellipse E with vertex P and P' on the major axis and vertex Q and Q' on the minor axis. chose R(x1,y1), the projection of R on the major axis is R' and on the minor axis is R''. Define the perpendicular projection of the intrersection...
Homework Statement
13.71 The path of the 3.6-kg particle P is an ellipse given by ##R = \frac {R_0} {(1+ecosθ)}##.
where R_0 = 0.5m and e=2/3. Assuming that the angular speed of line OP is constant
at 20 rad/s, calculate the polar components of the force that acts on the particle when it is at...
Homework Statement
The problem comes from S. Lang's "Basic mathematics", chapter 7, §1:
"Consider the following generalization of a dilation. Let ##a > 0, b > 0##. To each point ##(x, y)## of the plane, associate the point ##(ax, by)##. Thus we stretch the x-coordinate by ##a## and the...
1. The problem statement, aall variables, and given/known data
I need to find the angle between a point on the ellipse and the ellipse's center shown in the figure below:
The known variables are 'd', the distance between point O (the center of the ellipse) and point A, and 'a', the angle...
What I would like to be able to calculate is the following:
Suppose a hoop or ring is held up perpendicular to the ground and you stood in front of it, it would look perfectly circular and knowing the radius you could calculate the area of this circle. Now if this hoop was tilted over backwards...
Dear all,
I have a question regarding the computation of the area of an ellipse. The parametric form of the ellipse with axes a and b is
$$x(t) = a\cos{(t)}, \ \ \ y(t) = b\sin{(t)} $$
Using this to evaluate the area of the ellipse, usually one takes one halve or one quarter of the ellipse...
I am working on planetary orbits in an ellipse where the sun is at the foci, not the centre of the ellipse.
I need a formula that describes the 'Change of Distance by Change of Theta angle' where Distance represents the distance between a planet to the sun (Planet is on parameter, Sun is at...
Homework Statement
Let |z+1| + |z-1| = 7 where z are complex numbers. Show that the solutions to this equation form an ellipse with foci at (+/-)1
Homework Equations
(x^2 / a) + (y^2 / b) = 1 equation for an ellipse
The Attempt at a Solution
I set z = a + bi and so |z-1| = ((a-1)^2 + b^2)^1/2...
Homework Statement
Given an ellipse ##\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1## , where ##a \ne b##, find the equation of the set of all points from which there are two tangents to the curve whose slopes are (a) reciprocals and (b) negative reciprocals
Homework Equations
Equation of a tangent...
I did and exam a while ago and one of the questions asked to find a formula for an "equilateral ellipse". In my mind an equilateral ellipse should be a circle, but it seems I am wrong.
I searched online for the term, but didn't find anything useful. Does anyone know what it is, can find a...
We are told that planets and comets orbit the sun in an ellipse (Kepler's 3 laws) as shown below:
We are also told that according to Einstein's theory of gravity, there is no force applied. Implied is that the planets move in straight lines through curved space. We know that the effect of...
My textbook (on celestial mechanics) makes a passing reference to position on an ellipse being expressed as:
##r = a(1 - e \cos E)## before moving on to the substance of the chapter. E is the eccentric anomaly, and r is the distance from the focus to the point on the ellipse.
I'm trying to...
Homework Statement
Find the equation of the tangents to the ellipse 4x^2+9y^2 = 36 which are equally inclined to the x and y-axis.
Homework Equations
Quadratic discriminant
The Attempt at a Solution
First I substituted y=mx+c into the ellipse, and determined its discriminant, and got c^2 =...
Homework Statement
Show that the equation of the chord joining the points P(a\cos(\phi), b\sin(\phi)) and Q(a\cos(\theta), b\sin(\theta)) on the ellipse b^2x^2+a^2y^2=a^2b^2 is bx cos\frac{1}{2}(\theta+\phi)+ay\sin\frac{1}{2}(\theta+\phi)=ab\cos\frac{1}{2}(\theta-\phi).
Prove that , if the...
Homework Statement
You have been employed but(sic) the Mathematics Football League (MFL) to design a football. Using the volume of revolution technique, your football design must have a capacity of 5L ± 100mL. You must present a statement considering the brief below. Just a quick side note, I...
What is a foci of an ellipse?
The foci, an ellipse is the locus of points such that the sum of the distance to each focus is constant.
Can someone explain this in simple terms?
This is part of a larger question, but this is the part I am having difficulty with. I have had an attempt, but am not sure where I am making a mistake. Any help would be very, very appreciated.
1. Homework Statement
Let C2 be the part of an ellipse with centre at (4,0), horizontal semi-axis...
Homework Statement
Hello, my friend asked my If I could help him with this problem. However I just can't seem to find a way to solve this.
Ellipse
Focus(2,2)
vertex(2,-6)
Point(26/5,2)
a+e=8
find the equation of the ellipse
Homework Equations
(x-m)^2/a^2+(y-n)^2/b^2=1
Center(m,n)
a=moyor axis...
MENTOR Note: Moved this thread from a math forum hence no template
Is it possible to find this? Really only need the semi major axis or even it's orientation.
In the image below, elements in red are known.