Interesting Graphs: Conics

Conic sections: intersection curves of a plane and a right circular conical surface

The four conic sections (hyperbola, parabola, ellipse, circle) are produced when the plane does NOT pass through the vertex. When the plane passes through the vertex, degenerate conics (two intersecting lines, a line, a point) will be produced.

Conics are generally given by a second degree equation: $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$
$Δ>0$ hyperbola, pair of intersecting lines
$Δ<0$ ellipse, circle, point or no graph
$Δ=0$ parabola, line, pair of parallel lines or no graph

Demonstration:

Green: Hyperbola $Δ=5^2-4(2)(2)=9>0$
Orange: Ellipse $Δ=1^2-4(2)(2)=-15<0$
Blue: Circle $Δ=0^2-4(2)(2)=-16<0$
Purple: Parabola $Δ=4^2-4(2)(2)=0$

 

Hyperbola: the set of points in a plane whose distances from two fixed points (foci) in the plane have a constant difference.
Implicit form: $\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1$
Parametric form: $x=h+a\sec\theta$, $y=k+b\tan\theta$

 

Ellipse: the set of all points in a plane whose distances from two fixed points (foci) in the plane have a constant sum.
Implicit form: $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$
Parametric form: $x=h+a\cos\theta$, $y=k+b\sin\theta$

Ellipses and hyperbolas are called central conics because they have a centre of symmetry, while parabolas are non-central. For both ellipses and hyperbolas, a and b are the axis lengths. The larger one of a and b is the major axis while the smaller one is the minor axis.

Circle:
Implicit form: $(x-h)^2+(y-k)^2=r^2$
Parametric form: $x=h+r\cos\theta$, $y=k+r\sin\theta$

 

Parabola:
Implicit form: $x^2=4py$
Parametric form: $x=t$, $y=\dfrac{t^2}{4p}$

How to distinguish between non-degenerate and degenerate conics?
[pending]

Conics in matrix form
Each point $\vec{x}=(x,y)$ is considered to be a column vector with 1 as its third component, i.e. $\vec{x}=\begin{pmatrix} x\\y\\1\end{pmatrix}$ and $\vec{x}^T= (x, y, 1)$. The six coefficients of the general second degree polynomial are then used to construct a 3x3 symmetric matrix as follows: $\vec{Q}=\begin{pmatrix} A&B&D\\B&C&E\\D&E&F\end{pmatrix}$
$\vec{x}^T\vec{Q}\vec{x}=\vec{0}$

Application:
Although conics and quadric surfaces existed around 2000 years ago, they are still the most popular objects in many computer aided design and modeling systems.