1
vote
1answer
27 views

Polar correlation and conics in $\Bbb RP^2$

I'm stuck on a small detail in Proposition 1.2.8 in Geiges' Introduction to Contact Topology. Let $C$ be a conic in $\mathbb{R}P^2$ given by $q^tAq=0$, where $A$ is a nonsingular, symmetric 3x3 ...
-1
votes
1answer
16 views

Calculating rotated half ellipse axis aligned bounding box

I want to find the height and width of a rotated half ellipse axis aligned bounding box. For that, I have the minor and major axes of the ellipse and its rotation angle. I have no idea how to form the ...
0
votes
2answers
59 views

Relation of ellipse semi-axes with rotation angle and projection length

In the following setup, assume $w$ (length of the projection of the ellipse) and $\theta$ (the rotation angle) are known. I want to know what equation(s) do I have here that helps me to derive the ...
1
vote
3answers
37 views

How to find the outermost points in an ellipse?

If an ellipse is given in the form: $$ A(x − h)^2+ B(x − h)(y − k) + C(y − k)^2 = 1 $$ (where A, B, C, h, and k are given) What would be the simplest way of finding the outermost points, by which I ...
1
vote
0answers
15 views

Catenary and parabola minimum comparison

Do the catenary and a parabola that approximates the catenary, have the same minimum (maximum sag)? IF plotted, it looks to me they do, and that they only difer somewhere on the "slope". (sorry for ...
0
votes
2answers
22 views

Ellipse as projection of a disk - function describing ellipse diameter with disk rotation?

Say I have got a disk of radius $r$ and a plane $p$ in $3D$ space. The disk is "aligned" to $p$ and lies at an arbitrary distance, so that its orthogonal projection on $p$ is an identical disk of ...
0
votes
1answer
18 views

$P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ on the line that bisects $\angle F_1PF_2$. Prove $|PF_1-PF_2|>|QF_1-QF_2|$.

$\require{cancel}$ Sorry for the grammatical mistake in the title; it was needed to keep the title under 150 characters. $P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ is ...
0
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0answers
29 views

Predicting Bending of Plywood (Conical Shapes)

I'm building a NASCAR-style banked track for 1/27 scale RC cars, and I'm trying to predict the bends I need to cut to produce the banked track I want. I've already built some simple banked pieces, ...
0
votes
0answers
20 views

How to find the tangent vector for intersection of two cones?

Let two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given vertex of two cones and the generating angle of two cones are $m$. Intersection of two cones could be either an ellipse or ...
2
votes
3answers
112 views

What is the cone of the conic section?

Given the general (real valued) equation of a conic section: $$ A x^2 + B xy + C y^2 + D x + E y + F = 0 $$ Then what is the circular cone associated with it ? Is it unique ? And is there a way to ...
0
votes
1answer
40 views

Finding the area of an equilateral triangle on an ellipse

The question is as follows: Let $E$ be an ellipse with major axis length $4$ and minor axis length $2$. Inscribed an equilateral triangle $ABC$ in $E$ such that $A$ lies on the minor axis and $BC$ is ...
2
votes
0answers
37 views

Can one prove Brianchon's theorem using Ceva's theorem?

Can I prove Brianchon's theorem using Ceva's? I am also wondering if parabola and hyperbola can be inscribed in a hexagon?
0
votes
1answer
27 views

Why does the focus point distances of an ellipse sum up to the length of the major axis diameter

Why does the distances from the focus points of an ellipse to arbitrary point in the ellipse sum up to the length of the diameter of the ellipse in the major axis? In other words, how to prove: I ...
1
vote
4answers
62 views

Find depth of a half-filled parabolic cross-section

Given a cross-section of an object that is parabolic in shape, how do you find the depth of the object when it is "half full". A full example given in an exam: A long trough whose cross-section ...
0
votes
1answer
28 views

Minimal number of points to define a rotated ellipse?

What is the minimal number of points $N$ to uniquely define the semi-major axis $a$, the semi-minor axis $b$ and the rotation angle $\omega$ of an ellipse whose the center is known/fixed (this is ...
0
votes
1answer
35 views

Equation for the length of a chord parallel to either the minor or major axis in an ellipse

I am looking for a way to compute the length of any chord parallel to the minor (or major) axis of an ellipse. In all cases I know the lengths of both axes, and the distance between the chord and axis ...
1
vote
1answer
22 views

intersection of cone axis with plane

So when the plane intersect the cone, the intersection is a conic. Is (or when is) the axis (of the cone) intersection with the plane the focus of the conic?
2
votes
1answer
88 views

Why $b^2-4ac$ as determinant?

I am curious why $b^2-4ac$ is used as a determinant of a conic section? Like why this specific expression is chosen, why the value is always greater, lesser or equal to zero for hyperbola, ellipse ...
0
votes
1answer
42 views

Central angle of an ellipse

If I have an ellipse centered at the origin and know the length of $a$ and $b$ and was given the length of an arc, how can I find the angle that is between the two radius from the center of the ...
1
vote
1answer
47 views

Show that the ellipse and the hyperbola are convex

In Spivak's chapter on differentiation, he asks the reader to prove that the tangent line to an ellipse or a hyperbola intersect the figure at exactly one point. How is this done most elegantly? I ...
1
vote
2answers
32 views

Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = ...
0
votes
1answer
40 views

Equation of normal to an ellipse

Show that the equation of the normal at the point $x = a\cos(t)$, $y = b\sin(t)$ of the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$ is $$\frac{2a^2 - b^2}{a}$$ Hi, I am not sure how ...
4
votes
0answers
61 views

A surprising locus of points

I was playing with the setup for Desargues Theorem in Geogebra today and got a very odd-looking (to me) result. I imagine I could grind through this analytically and get an ugly-looking parametric ...
5
votes
2answers
88 views

Hyberbolic and Circular (Trig) Functions: Why no parabolic? [duplicate]

There are circular (trig) functions which determine all the points on a unit circle: and which relate to the area swept out by an angle subtended on the circle. -- These functions can of course be ...
1
vote
2answers
39 views

Orientation of rectangle on conic section

Consider a conic section. There are 2 rectangles such that all of the 8 vertices of the 2 rectangles lie on the conic section. Further assume that the 2 rectangles have different orientation (ie. a ...
6
votes
1answer
73 views

Group of Points on an Ellipse

I did some tooling around to find an abelian group operation for the set of points on the ellipse $\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2=1$, given by ...
1
vote
1answer
65 views

Given a skewed ellipse, how to determine its axis lengths?

I am mentoring a student who is working on a library to import Adobe Pagemaker documents into LibreOffice. Pagemaker represents ellipses as a bounding box (of the original, untransformed ellipse) and ...
26
votes
3answers
1k views

Fascinating Lampshade Geometry

Today, I encountered a rather fascinating problem in a waiting room, which is embodied in the image below. Notice how the light is being cast on the wall? There is a curve that defines the ...
1
vote
1answer
32 views

What is the approximation equation for making the day/night wave

Basically, I have a program that will graph the day/night shade similar to this page: http://www.timeanddate.com/worldclock/sunearth.html Could any of you give me the equation for graphing a line ...
0
votes
1answer
37 views

Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$

Find the equations of the circles that have centre $(0,0)$ and touch the circle $x^2 + y^2 - 8x - 6y + 24 = 0$ So far I have said: As the circles have centre $(0,0)$ their equations are of the form ...
1
vote
1answer
47 views

Calculating tangent on ellipse

I want to calculate the slope of the tangent at one point of an ellipse whose centre is shifted towards the coordinates $(x_c;y_c)$ and also rotated by an angle $\alpha$ around its centre. Now, I have ...
0
votes
2answers
13 views

HELP: find the type of a conic from the given equation

However, I am not sure what conic type it is. Should it be divided by 4 in order to get a standard form of a hypebola? Any help will be appreciated. Thank you!
7
votes
1answer
66 views

ellipse circumference

Here is a Wikipedia article about the circumference of an ellipse: http://en.wikipedia.org/wiki/Ellipse#Circumference I don't know how Ramanujan developed the following approximation for the ...
2
votes
3answers
161 views

Focus of parabola with two tangents

A parabola touches x-axis at $(1,0)$ and $y=x$ at $(1,1)$. Find its focus. My attempt : All I can say is that as angle subtended by this chord at focus is $90^\circ$ as angle between tangents is ...
0
votes
1answer
29 views

How to identify any point inside or outside the given cone?

The equation of a double circular cone with a vertex $p=(a,b,c)$ with the generating angle $t$ is given by $(x-a)^2+(y-b)^2= \frac{(z-c)^2}{t^2}$ How do I identify the point ...
1
vote
1answer
30 views

need explanation of what exactly is a directrix & focus?

((I'm not asking why do we need to know conic sections etc.) Like other similar questions.) I actually love math & currently learning conic sections in class, neither my textbook or teacher ...
2
votes
2answers
70 views

Orthogonal tangents to an ellipse [duplicate]

This is the problem I found back in the first year in the university. Suppose we have a non-degenerate (i.e. not a point and not an empty set) ellipse $E\subset \Bbb R^2$. Now define a set $D$ by a ...
3
votes
1answer
69 views

Locate a point a given distance from another point on an ellipse

Similar to Point on circumference a given distance from another point, but for an ellipse. Unfortunately, the difference is non-trivial. I have an ellipse and a point (C) that is somewhere on the ...
1
vote
1answer
66 views

Identify the locus.

Let $A,B,C$ lie on a straight line. $B$ is lying between $A$ and $C$. Consider all circles passing through $B$ and $C$. The point of contact of the tangents from $A$ to these circles lies on ..... We ...
1
vote
0answers
151 views

Equation of intersection of two cones

The equations of two cones are given; $(x-x_{0})^2+(y-y_{0})^2=\frac {(z-z_{0})^2}{m^2}$ and $(x-x_{1})^2+(y-y_{1})^2=\frac {(z-z_{1})^2}{m^2}$ How to find the equations of intersections 1) ...
2
votes
2answers
72 views

Why does the “T=0” method to calculate tangent work?

Given a random equation of a curve: $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$. Suppose we need to find the tangent to this curve at any point $A(x_1, y_1)$. A method given to me by my professor was the ...
0
votes
1answer
21 views

How do I find the width of a given section of an ellipse?

How would I be able to find the width of a horizontal ellipse (with a major axis of 120 and a minor axis of 5) at any given point along the major axis?
0
votes
0answers
102 views

Conic equation from cone/plane intersection

In an orthonormal cartesian frame $(O; \vec{x}, \vec{y}, \vec{z})$ consider: an infinite plane $P$ defined by: a point $p = (p_x, p_y, pz)$ an normal vector $\vec{n} = (n_x, n_y, n_z)$ a cone $C$ ...
4
votes
1answer
112 views

Minimum eccentricity of ellipses around another ellipse

Six circles can surround another circle of equal size, with each circle touching both the central circle and its two neighbouring outer circles. For sufficiently eccentric ellipses, it is possible to ...
1
vote
1answer
38 views

Problem in conics question

A vertical line passing through the point ($h$,0) intersects the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1$$ at the points P & Q.Let the tangents to ellipse at P & Q meet at the point R.If ...
0
votes
1answer
66 views

Is this a correct way to derive the equation of an ellipse/hyperbola?

I was just testing to see if I could derive the equation of an ellipse (and consequently a hyperbola) with the least amount of information to remember. The small amount of information I chose to use ...
0
votes
1answer
105 views

Ellipse Tangents in 3D

I know that we can find the tangent of the ellipse in 2D by taking the derivative of the equation defining the ellipse. But I'm little bit confused about finding the ellipse tangent in 3D. Where the ...
0
votes
1answer
81 views

Homography between ellipses

This is a spin-off from a comment on Stack Overflow. How can I find a homography between two ellipses in the plane?
0
votes
0answers
28 views

How are the sine functions along with the hyperbolic functions visualized with imaginary rotations?

Since we know that: cos(t)=cosh(it) and isin(t)=sinh(it) I've been thinking about this, and obviously this is referring to how if you move at a right angle from a circle on a conic section, you end ...
5
votes
2answers
167 views

Area under parabola using geometry

We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without ...