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.

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2
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343 views

How to calculate a PHI-ellipse defined by 3 points and its width/length ratio

in the field of technical analysis for stock markets, the usage of so-called Phi-Ellipses is getting popular. One important property of this ellipses is its constant length/width ratio (e.g. 1.618). ...
3
votes
3answers
251 views

Proving that a quartic has exactly 2 real roots

Question: Find the equation of the normal to the hyperbola $xy=c^2$ at $P(ct,\frac{c}{t})$ then prove that exactly two normals can be drawn to the hyperbola $xy=c^2$ from a point $(0,k)$, where $k$ ...
0
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1answer
54 views

Trying to create fitness function

I'm trying to create an idea fitness function using an inverse parabola but I'm struggling with the math. Basically what I have is: ...
0
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1answer
129 views

Finding dual conic

How do you find the dual conic associated with a conic and also a degenerated conic in matrix form? I have been attempting to find the intersection of two conics and the dual conic is a key step ...
2
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0answers
168 views

Equation of an intersection of two cones when the intersection is an ellipse

The two cones with vertex $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ and generating angle of two cones is $\alpha$ given. I need to write the equation of the intersection of two cones ...
2
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2answers
253 views

How to find the vertex of a 'non-standard' parabola? $ 9x^2-24xy+16y^2-20x-15y-60=0 $

I have to find out the vertex of a parabola given by: $$ 9x^2-24xy+16y^2-20x-15y-60=0 $$ I don't know what to do. I tried to bring it in the form: $$ (x-a)^2 + (y-b)^2 = \dfrac {(lx+my+n)^2} ...
2
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2answers
218 views

An equation with graph as a line *segment*: $ \sqrt {x^2 + (y+12)^2} = 13 - \sqrt{(x-5)^2 + y^2} $

The graph of this function is (quite oddly) a line segment. I don't understand why it is so. I mean, is there a way of telling just by looking at this equation that its graph will be a line segment? ...
3
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0answers
456 views

Volume of the intersection of two solid cones

I want to calculate the volume of the intersection (overlap) of two solid cones, as shown below. I imagine this must be a known problem but I'm having a hard time finding anything online, so any ...
5
votes
1answer
884 views

How to maximize the volume of a rectangular parallelepiped in an ellipsoid?

This question comes from an exam about 15 years ago. How to find the maximal volume of a rectangular parallelepiped inscribed in an ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$? ...
5
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1answer
1k views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
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4answers
553 views

Is $x^2-y^2=1$ Merely $\frac 1x$ Rotated -$45^\circ$?

Comparing their graphs and definitions of hyperbolic angles seems to suggest so aside from the $\sqrt{2}$ factor: and:
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0answers
102 views

Conic Sections - Points on a cone

On page 80 of Spivak's Calculus, 4th Edition, he writes: One of the simplest subsets of this three-dimensional space is the (infinite) cone illustrated in Figure 2; this cone may be produced by ...
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2answers
109 views

Show that the vertex is the point on a branch of a hyperbola that is closest to the focus associated with that branch

Given the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1,$$ show that the point closest to focus $F(c, 0)$ where $c^2=a^2+b^2$ is the vertex $V(a, 0)$
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2answers
2k views

Ellipse circumference calculation method?

Actually I know how to calculate the circumference of an ellipse using two methods and each one of them giving me different result. The first method is using the formula: ...
2
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2answers
67 views

Finding the slope at a point $P(x_1,y_1)$ on a parabola

Given a point $P(x_1,y_1)$ on the graph of a parabola $y^2=4px$, prove that the slope at point P is $$\frac{y_1}{2x_1}$$
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vote
1answer
132 views

Horizontal displacement of a falling object from trajectory length

I have an object falling down in a parabolic trajectory. I can estimate the total distance traveled during the time t, i.e. the length of the parabola's arc is known. I need an efficient algorithm ...
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2answers
127 views

Good book introducing Inconics

Is there a book on conic sections or triangle geometry that has a particularly good introduction to inconics of a triangle? I am interested in that subtopic now. Ideally, I would like a book that ...
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1answer
100 views

How many cones pass through a given conic section?

Given a conic section in the $xy$-plane, how many cones (infinite double cone) in the surrounding 3D space intersect the $xy$-plane at that conic? Is the family continuous, with a nice parametization? ...
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2answers
689 views

find length of semi major axes of ellipse

suppose that equation of ellipse is given by $4x^2+3*y^2=25$ we should length of major axes ,first let us transform this equation into standard form or divide by $25$ $4*x^2/25+3*y^2/25=1$ if ...
3
votes
1answer
1k views

Determining the angle degree of an arc in ellipse?

Is it possible to determine the angle in degree of an arc in ellipse by knowing the arc length, ellipse semi-major and semi-minor axis ? If I have an arc length at the first quarter of an ellipse and ...
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2answers
83 views

If $A=(-4,0)$ and $B=(4,0)$, what is the locus of points $P$ such that $|AP-BP|=16$? Does it even exist?

I am stuck in this question for about a week: If there are points $A$ and $B$ such that $A(-4,0)$ and $B(4,0)$ then what is the locus of points $P$ such that $|AP-BP|=16$? I think this is a ...
3
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1answer
50 views

How can I transform this equation in a conical?

In this equation $$2x²+y²-4x-6y+11=0$$ I got the result $(1,3)$ completing squares $2(x - 1)² + (y - 3)² = 0$   But on my list exercises, demanded that determine the foci, straight guideline ...
3
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1answer
558 views

How to know the flattening factor for a ellipse?

I want to know how can I get the flattening factor for a ellipse by knowing its semi-major and semi-minor axes ? Actually I tried this formula: $f=\left(\frac{a}{b}-1\right)$ While $f$ is the ...
3
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2answers
110 views

How to find the minute points of an ellipse clock, knowing the minor axis and the major axis?

I want to make an analogic clock, not circle, but ellipse. So the distance between minute points is not constant. I guess it grows proportionally with the division of major axis with minor axis. How ...
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3answers
16k views

How to determine the arc length of ellipse?

I want to determine the arc length of a ellipse. So what data should I know ? And what law should I use ? For example I have this ellipse on picture below: How can I determine the $d$ length of ...
6
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2answers
6k views

How to get the radius of an ellipse at a specific angle by knowing its semi-major and semi-minor axes?

How to get the radius of an ellipse at a specific angle by knowing its semi-major and semi-minor axes? Please take a look at this picture :
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0answers
107 views

Circle Geometry and Conic Section textbook

I seek a textbook for good conic section and circle geometry questions. Slightly above introductory level. - slightly. But I wouldn't mind introductory level questions to consolidate my knowledge. I ...
0
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2answers
101 views

Given two ellipses, does there exist an explicit transformation between them?

Say you have the standard parameters (axes lenghts, angle of rotation) of two different ellipses. Is there a swift way of transforming the points of one ellipse to the points of the other? Thanks!
2
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1answer
1k views

Derivation of the equations for ellipse and ellipsoid

Could someone perhaps explain / prove / guide to a source where I can find the derivations of the general equations of ellipse and ellipsoid? I'm trying to understand where these formulas come from: ...
2
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1answer
447 views

Locus of vertex of parabolas through three points

Consider all parabolas through three given points $A,B,C$. What is the locus of the vertex? Qualitatively it traces three branches with the lines through the mid-points of the sides of the triangle ...
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0answers
131 views

calculate the angles for an ellipse to get nearly equal (if not equal) arc lengths based on major and minor axes lengths

I am trying to get equidistant points on an ellipse. I have the major axis and minor axis. Is there a way to calculate the angles in a loop, where the points are at equal distance on teh ellipse? I ...
0
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1answer
45 views

Verifying the vertices of a Hyperbola

In this exercise it is required to verify that the points O & A(4,0) are the vertices of the Hyperbola H , as you can see in the marked part. Can someone help verify that ?
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2answers
2k views

Finding Intersection of an ellipse with another ellipse when both are rotated

Equation of first ellipse=> $$\dfrac {((x-xFirstEllipseCenterPoint)\cdot \cos(A)+(y-yFirstEllipseCenterPoint)\cdot \sin(A))^2}{(a_1^2)}+\dfrac{((x-xFirstEllipseCenterPoint)\cdot ...
3
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1answer
635 views

Enlarging an ellipses along normal direction

Given an ellipses, enlarge it along normal direction a fixed length say 1cm. Do we get another ellipses? If so, how to prove ?
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1answer
727 views

Trajectory of a projectile.

From the definition of a parabola can we prove that the trajectory of a projectile is parabolic? And can this be proved by calculus?
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3answers
1k views

Finding the major and minor axis vertices for an ellipse given two conjugate diameters?

I've been googling, searching forums and looking in my old algebra/trig books to try to understand how to find the end points to the major and minor axis of an ellipse given the end points of two ...
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0answers
91 views

egg curve estimation

Let $p_{1...3}$ be three points on an ellipse, and $t_{1...3}$ be their tangent lines. For $i={1..2}$, let $M_i$ be the point of intersection of $t_i$ and $t_{(i+1)\%2}$, and $K_i$ be the midpoint of ...
0
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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 ...
5
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0answers
695 views

To construct an ellipse, being a projection of a great circle, given two points on it

I'm looking for a geometric construction which would allow me to draw an ellipse, which is supposed to be an orthographic projection of a great circle of a sphere, given two points on it. The ...
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0answers
34 views

How can i evaluate a parabolic region in 2-Dimensions onto n-Dimensions?

I am working on the K-Nearest Neighbor algorithm and have the solution to a problem as a parabola in 2-dimensional space. Should I continue this solution onwards to n-dimensions (and thereby create ...
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vote
2answers
104 views

Minimizing area of a triangle with two fixed point and a point on parabola

A triangle is made up of three points, $A, B$, and $P$. $A(-1, 0)$ $B(0, 1)$ $P$ is a point on $y^2 = x$ Minimize the area of Triangle $ABP$. My approach is far too complicated, which ...
0
votes
2answers
54 views

Finding equation of parabola

I have a group of points from a graph. When I connect the points I get a shape which looks like the one's of the function f(x) = a / x . How can i precisely find the equation of the shape ?
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0answers
205 views

Sample Code to Generate Points on the Rim of a Randomly Rotated Cone : What's Going On Here?

Related to this question: http://math.stackexchange.com/questions/407897/randomly-generate-point-on-shell-from-3-points-2-angles-with-uniform-angle-dis I'm trying to reverse engineer the ...
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2answers
1k views

Finding the point on a rotated ellipse corresponding to a given tangential angle

I was going to initially ask for the solution to this problem here, but I have come upon a solution by some hand derivation and wanted to verify it here. Please note that after high-school I have had ...
16
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4answers
486 views

A geometric reason why the square of the focal length of a hyperbola is equal to the sum of the squares of the axes.

This may be a phenomenally stupid question, so apologies in advance. But when I teach conics, I show why $c^2=a^2-b^2$ for ellipses geometrically, just by drawing the obvious isosceles triangle from ...
19
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0answers
687 views

How many points of intersection between an ellipse and an $L_p$-circle?

Consider an ellipse $E$ in the plane, centered at the origin. (In my case, the minor axis points into the nonnegative quadrant.) Let S be an "$L_p$-circle": $S = \{(x,y) : |x|^p + |y|^p = 1\}$, ...
3
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1answer
233 views

shadow cast by a circle

A point source emits light at a circular disc (thickness negligible), and a shadow is left on a wall (XY plane) behind and parallel to the disc. The Z component of distance between the point source ...
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vote
2answers
501 views

Can I find the equation of an ellipse with these points?

How can I solve an ellipse with its major axis on the $x$-axis, given one focus, and two points on the ellipse, one of which I know to be on the major axis?
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2answers
2k views

hyperbola curve formula in 3 dimensions

Cartesian formula for 2d hyperbola curve is $x^2/a^2-y^2/b^2 = 1$. What is the formula for a 3d hyperbola curve?
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3answers
435 views

Is it possible to find out $x^2$ parabola and function from 3 given points?

I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple $y = ax^2$ parabola to draw the falling ball. I have given two points, the ...