Questions on conic sections and their properties; the curves formed by the intersection of a plane and a cone. Circles, ellipses, hyperbolas, and parabolas are examples of conic sections.

learn more… | top users | synonyms (3)

0
votes
1answer
1k views

length of the focal chord

Paragraph: $PQ$ is a focal chord of the parabola: $y^2=4ax.$ The tangents to the parabola at the points $P$ and $Q$ meet at point $R$ which lies on the line $y=2x + a.$ Question: Find the length ...
1
vote
2answers
31 views

How to find the surface area of revolution of an ellipsoid from ellipse rotating about y-axis

Suppose the ellipse has equation $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. I understand the way to obtain the surface area of the ellipsoid is to rotate the curve around y-axis and use surface of ...
1
vote
1answer
27 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
-9
votes
1answer
42 views

Algebra 2 help! [on hold]

Write the equation of a circle with the given center and radius. Center (-2,3); Radius 8
0
votes
1answer
91 views

Best fit circular arc to an elliptical arc?

Is there a standard procedure or algorithm for finding the best fit circular arc to an elliptical arc ? Where the ellipse arc is: symmetrical about the minor axis, subtending $[+\theta, -\theta]$ ...
0
votes
1answer
18 views

Polar conversions of coordinates and parametric equations

Express the polar coordinates $P\left(6, -\dfrac{\pi}{4} \right)$ in Cartesian coordinates. $\displaystyle x=r\cos{(\theta)} ,\ y=r\sin{(\theta)} \implies x^2+y^2=r^2 \wedge \theta = ...
1
vote
1answer
65 views

Finding the intersection points of common tangents on a pair of non-intersecting ellipses

I'm having some trouble with this, I don't know why but for some reason it is giving me a lot of trouble. Ultimately I intend to implement it into a program for modelling something, but I cannot even ...
3
votes
1answer
81 views

reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola?

Does anyone have a reference on the equation $$\sqrt{ax}\,+\sqrt{by}=c\ ?$$ Clearing square roots and rearranging gives $$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$ This is the equation of a parabola, so ...
0
votes
0answers
17 views

Finding the eccentricity of ellipse when a line is a normal to the ellipse

Finding the eccentricity of an ellipse when a line joining the foot of the perpendiculars from a point of a known ellipse (having eccentricity e) at 2 perpendicular lines(example the x and y axes ...
0
votes
0answers
12 views

Modeling smoke cloud as expanding Gaussian / ellipse

I am making a simplified model of smoke coming from a train's smokestack. You can imagine that if you want an accurate model you have to think in 3D and use computational fluid dynamics and stochastic ...
2
votes
3answers
224 views

Parametric coordinates of parabola?

Can $(a(\sin(t))^2 , 2a \sin(t))$ be the parametric coordinates of the parabola $y^2 = 4ax$ ? I found that these coordinates satisfy the equation of the parabola but my friend says that although ...
0
votes
1answer
41 views

Parabola equation in Fortune algorithm for building Voronoi diagram

in DeBerg's "Algorithms and Applications", the part about Voronoi diagram, i have encountered the following formula for parabola arising in the beach line for a site point: $$\beta := y = ...
0
votes
0answers
19 views

How to deduce this ellipsis equation?

I was looking in my physics book and it basically gave out the polar equation for an ellipse, as well as a few other definitions (which I assume are the definitions from which this ellipse equation is ...
0
votes
1answer
327 views

Parabola investigation

Edit 4: I added the below picture for clarity I'm trying to figure out how to find the angle between the red line and the blue line, but I have no idea how to start. (I have a feeling that this ...
2
votes
0answers
34 views

Rotating a 3d-ellipse equation?

So I have very limited Linear Algebra knowledge, and I'm trying to program a computer graphics application in Android using OpenGL. I understand my design is not great, so if you have questions as to ...
4
votes
2answers
40 views

Area of a polygon inscribed into an ellipse

I have recently found a paper describing that the percentage area error of a polygon inscribed within a circle can be calculated using the following formula. The output of the algorithm is a set ...
4
votes
1answer
692 views

Centers of the osculating circles along an ellipse

Consider an ellipse on the plane $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We will use the usual parametrization: $P(t)=(x(t),y(t))=(a\cos t,b\sin t)$. Then the tangent vector is $T(t)=(-a\sin t, b\cos ...
3
votes
1answer
188 views

Computing the trajectory of an orbiting body so that it collides with another orbiting body

I am creating a 2D game in which two space ships, orbiting around a planet under the influence of gravity, fire projectiles at each other, which are also under the influence of gravity. I'm creating ...
5
votes
0answers
46 views

Keplerian orbits and closest approaches to Earth.

This question arose out of a discussion on Space.SE, but I think it will appeal to mathematicians more than astronomers: Let's consider a small astronomical object following an ideal elliptic ...
0
votes
3answers
16 views

Finding equation of parabola when focus and equations of two perpendicular tangents from any two points on the parabola are given

If the focus of a parabola and the equations of two perpendicular tangents at any two points $P$ and $Q$ on the parabola are given, can we find the equation of the given parabola? If not, what ...
0
votes
1answer
502 views

Find locus of points relating to an ellipse

I would like to find the equation of the following locus. For a big circle C centered at (0,0), the locus of points that the sum of distances to Y-axis and to C is 1, say in the first quadrant, is ...
4
votes
1answer
192 views

Focus-Focus Definition for a Parabola

I am looking at the focus-focus definitions of the conics, i.e. defining them as the locus of points with the property that some function of the distance from the point to two foci is a constant. ...
2
votes
3answers
162 views

How to construct the point of intersection of a line and a parabola whose focus and directrix are known?

I found this problem in Polya's "How to solve it". It goes as follows Using only a straight edge and a compass, construct the point(s) of intersection of a given line and a parabola whose focus ...
0
votes
2answers
1k views

Find angle at given points in Ellipse

I have Ellipse's center-points, minor-radius and major-radius. I can find, how to check if given point(x, y) exists in Ellipse or not. Now, I want to find given point(x,y) exists at which angle in ...
-1
votes
1answer
29 views

The outer parallel (offset) curve of an ellipse [closed]

The inner edge of a track has equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. The track has uniform width $d$. What is the equation of the outer edge?
1
vote
1answer
427 views

Finding the volume of a cone by integration of parabolic conic sections

I am working on a purely academic way of finding the volume of a right circular cone of height $h$ and radius $r$, (assume $h > r$), using integration of parabolic conic sections (conic sections ...
1
vote
0answers
127 views

Plotting an elliptical arc given 3 points, radius ratio and angle

I'm trying to plot an elliptical arc. I know the starting point $P_1$, ending point $P_2$ and a control point $P_3$. I'm also given the ratio of radii $\frac{a}{b}$ and the angle $\theta$ of the ...
1
vote
2answers
60 views

How to get the foci / focus Hyperbola

How to find the lower and upper focus? Hyperbola I started with this $$ 9x^2 + 54x - y^2 + 10y + 81 = 0 $$ and broke it down to $$ \frac{9(x+3)^2}{25} - \frac{(y-5)^2}{25} = -1 $$ center = (-3,5) ...
2
votes
2answers
336 views

Equation of a parabola in 3D space

I have two points with coordinates A(x1,y1,z1) and B(x2,y2,z2). There is a third point which is vertex(lowest point) of the parabola. I only know z-coordinate of this point. I need to find coordinates ...
0
votes
2answers
44 views

$2x^2-16x+28$ into standard form

I think I'm just doing something stupid here, because I know it's not hard. Here's what I did: $$y-28+{\_\_\_}=2x^2-16+{\_\_\_}$$ $$y-28+{\_\_\_}=2(x^2-8+{\_\_\_})$$ $$y-28+16= 2(x^2-8+16)$$ ...
3
votes
1answer
44 views

How to find ellipse circumference using 5 points?

I want to find ellipse circumference using 5 points. I have 5 point of an arc of the ellipse. To reach my goal i know that i have to do the following things: First, I have to find the general ...
2
votes
1answer
337 views

Volume of ellipsoid bounded by two planes.

I need to find the volume of ellipsoid: $$5x^2 + {y^2\over25} + {3z^2\over4} = 1$$ if the ellipsoid is bounded by $z={-1\over2}$ and $z=1$ planes. I was able to find the total volume of the ellipsoid ...
0
votes
0answers
14 views

Parabolae intersection

when in some textbook they say "the right angled intersection of two parabolae", do they mean the intersection of their symmetry axes? If not what do they mean?
1
vote
1answer
652 views

find the center of an ellipse given tangent point and angle

I have an ellipse with known major radius $r_x$ and minor radius $r_y$, aligned with the x- and y-axis. Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ ...
3
votes
3answers
1k views

Normal to Ellipse and Angle at Major Axis

I've tried to detail my question using the image shown in this post. . Consider an ellipse with 5 parameters $(x_C, y_C, a, b, \psi)$ where $(x_C, y_C)$ is the center of the ellipse, $a$ and $b$ are ...
0
votes
1answer
56 views

Matrix representations of parabola.

Continuing the epic quest on finding matrix representations from here: Representation of hyperbolas. with a last part, the only conic section left: the parabola. I will present one idea of how to ...
4
votes
1answer
40 views

Start and end point of a rotated ellipse

I have the data of an incomplete ellipse and I need to retreive the minimun information in order to describe an elliptical arc. In particular following are my ellipse data: Major axis vector (x, y) ...
3
votes
5answers
52 views

Showing that certain points lie on an ellipse

I have the equation $$r(\phi) = \frac{es}{1-e \cos{\phi}}$$ with $e,s>0$, $e<1$ and want to show that the points $$ \begin{pmatrix}x(\phi)\\y(\phi)\end{pmatrix} = ...
3
votes
2answers
24 views

The Reason for different Forms of Equations

I recently started learning about conic sections and saw people writing the equations for the different figures (circle, parabola, ellipse, and hyperbola) in different forms. (standard form, vertex ...
50
votes
9answers
953 views

Why does $\cos(x) + \cos(y) - \cos(x + y) = 0$ look like an ellipse?

The solution set of $\cos(x) + \cos(y) - \cos(x + y) = 0$ looks like an ellipse. Is it actually an ellipse, and if so, is there a way of writing down its equation (without any trig functions)? What ...
0
votes
2answers
71 views

Determine the locus of a equation Quickly[Mental Math]

if $Z=X+iy$ then determine the locus of the equation $\left | 2Z-1 \right | = \left | Z-2 \right |$.I can tell that it a circle equation and it is $x^2 + y^2 = 1$.There are a lot of equation in my ...
0
votes
0answers
30 views

How to scale x- and y- axes equally in Maple?

I have the ellipse $\frac{25}{36}x^2+\frac{5}{36}y^2=1$. Maple draws it as a circle: How can I change the coordinates, to make it look like an actual ellipse?
1
vote
0answers
26 views

Plane and Ellipse Intersection

Short Version: If some can solve the easier to read form as follows, I would be thankful. \begin{equation} B = \frac{1 - d^{T}Bd}{ K_{1} } A \end{equation} \begin{equation} B^{T}d = \frac{1 - ...
1
vote
1answer
558 views

Maximum/Minimum of Curvature - Ellipse

Find the sum of the maximum and minimum of the curvature of the ellipse: $9(x-1)^2 + y^2 = 9$. Hint( Use the parametrization $x(t) = 1 + cos(t)$) Tried to use parametrization like that, but then ...
1
vote
0answers
13 views

Projective conic generated by a set of tangent triangles.

I need to proof the following result: Let C be a real projective conic and P, Q two points interiors to C then there is another real projective conic such that every triangle inscribed on that conic ...
1
vote
1answer
894 views

How to compute the chord length of an ellipse?

How do I calculate the chord length of ellipse? I need to design a blade vane rotating inside an elliptical profile. The blade is supposed to be in contact with ellipse with a maximum clearance of ...
0
votes
1answer
37 views

Attempting to put the following conic into standard form: $y^2-2x^2+8y-8x-4=0$

Put the following conic into standard form: $y^2-2x^2+8y-8x-4=0$ I ended up with $$ -\frac {(x-2)^2}{12} + \frac{(y+8)^2}{24} = 1, $$ but I'm not sure if this is right.
1
vote
0answers
22 views

Strong duality in conic programmig

Let $K$ be a convex cone which is not closed. The look at a probem of the from $$\min <C,X>, \,s.t\; <A_i,X>=b_i,\, X\in C.$$ Now suppose I now that both this program and its dual have a ...
0
votes
1answer
736 views

Calculating Intersection of an Ellipse and a Line

I found this page which gave me some equations on solving the intersection of a line with an ellipse given a point on the line and the slope of the line: There Isn't much explanation but ...
0
votes
0answers
18 views

Relationship between polar angle and tangent angle of a conic section

I'm trying to define the relationship between the polar angle (or gradient) of a conic section and the tangent angle (or gradient) without resorting to x,y coordinates of the tangent point, i.e. given ...