Questions tagged [conic-sections]

For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.

A conic section is a smooth planar curve that is the result of intersecting a cone with a plane. There are commonly four conic sections: circles, ellipses, parabolas, and hyperbolas.

We can construct these conic sections analytically. The solutions to the equation $x^2+y^2=z^2$ give us a cone in three-dimensional space. An plane in three-dimensional space that goes through a point $p=(x_0,y_0,z_0)$ and has normal vector $\langle a,b,c \rangle$ is given by the equation

$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0\,.$$

Finding the common solutions to the equation of the cone and equation of the plane for various choices of $p$ and normal vector will—after a change in coordinates to write them as planar curves—lead you to the (potentially) familiar equations

  • Circle: $x^2+y^2 = r^2$
  • Ellipse: $ax^2 + by^2 = r^2$
  • Parabola: $ax^2 +by = r^2$
  • Hyperbola: $ax^2 - by^2 = r^2$

There are also geometric constructions of the conic sections. For example, a circle is the set of all points that are a fixed distance from a given points. An ellipse is the set of all points .... The construction of Dandelin spheres (see Wikipedia) unifies the analytic and geometric constructions of conic sections.

4347 questions
91
votes
2 answers

Modelling the "Moving Sofa"

I believe that many of you know about the moving sofa problem; if not you can find the description of the problem here. In this question I am going to rotate the L shaped hall instead of moving a sofa around the corner. By rotating the hall…
newzad
  • 4,683
  • 23
  • 49
70
votes
9 answers

Do "Parabolic Trigonometric Functions" exist?

The parametric equation $$\begin{align*} x(t) &= \cos t\\ y(t) &= \sin t \end{align*}$$ traces the unit circle centered at the origin ($x^2+y^2=1$). Similarly, $$\begin{align*} x(t) &= \cosh t\\ y(t) &= \sinh t \end{align*}$$ draws the right part…
Argon
  • 24,247
  • 10
  • 91
  • 130
67
votes
4 answers

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by the ellipse?
65
votes
9 answers

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 motivates this is the following example. The…
Mr Bingley
  • 848
  • 7
  • 11
63
votes
8 answers

Using Gröbner bases for solving polynomial equations

In my attempts to understand just how computer algebra systems "do things", I tried to dig around a bit on Gröbner bases, which are described almost everywhere as "a generalization of the Euclidean algorithm and Gaussian elimination". I've tried to…
59
votes
6 answers

What is the general equation of the ellipse that is not in the origin and rotated by an angle?

I have the equation not in the center, i.e. $$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1.$$ But what will be the equation once it is rotated?
andikat dennis
  • 849
  • 3
  • 10
  • 13
56
votes
4 answers

Parabola is an ellipse, but with one focal point at infinity

While I was reading about conic sections, I came across the following statement: A parabola is an ellipse, but with one focal point at infinity. But it is not clear to me. Can someone explain it clearly?
Kumar
  • 2,077
  • 3
  • 19
  • 29
45
votes
3 answers

Lampshade Geometry Problem

Today, I encountered a rather interesting problem in a waiting room: $\qquad \qquad \qquad \qquad$ Notice how the light is being cast on the wall? There is a curve that defines the boundary between light and shadow. In my response below, I will…
Kaj Hansen
  • 31,861
  • 4
  • 53
  • 94
43
votes
6 answers

What do cones have to do with quadratics? Why is $2$ special?

I've always been nagged about the two extremely non-obviously related definitions of conic sections (i.e. it seems so mysterious/magical that somehow slices of a cone are related to degree 2 equations in 2 variables). Recently I came across the…
D.R.
  • 5,864
  • 3
  • 17
  • 44
38
votes
3 answers

Why are certain PDE called "elliptic", "hyperbolic", or "parabolic"?

Why are the Partial Differential Equations so named? i.e, elliptical, hyperbolic, and parabolic. I do know the condition at which a general second order partial differential equation becomes these, but I don't understand why they are so named? Does…
37
votes
7 answers

Looking for a function that approximates a parabola

I have a shape that is defined by a parabola in a certain range, and a horizontal line outside of that range (see red in figure). I am looking for a single differentiable, without absolute values, non-piecewise, and continuous function that can…
36
votes
14 answers

Derivation of the formula for the vertex of a parabola

I'm taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the $x$ coordinate of the vertex. I asked her why, and…
36
votes
3 answers

How to determine the arc length of ellipse?

I want to determine the length of an arc from the ellipse in the picture below: How can I determine the length of $d$?
Mohammad Fakhrey
  • 1,275
  • 3
  • 14
  • 23
30
votes
5 answers

Is this an ellipse?

Is this parameterisation an ellipse: \begin{align}x(t) &= \frac{2 \cos(t)}{1 + a \sin(t)}\\ y(t) &= \frac{2 \sin(t)}{1 + a \sin(t)}\end{align} where $a$ is a real positive parameter. I tried to do it the naive way but couldn't find a…
29
votes
4 answers

The locus of the intersection point of two perpendicular tangents to a given ellipse

For a given ellipse, find the locus of all points P for which the two tangents are perpendicular. I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry proof.
Matt
  • 8,545
  • 5
  • 25
  • 43
1
2 3
99 100