Supplementary mathematics/Conical section

In mathematics, a conic section (or simply a conic, sometimes called a quadratic curve) is a curve obtained as the intersection of the surface of a cone with a plane. Three types of conic sections are: hyperbolic, parabolic and elliptical. A circle is a special case of an ellipse, although historically it is sometimes called the fourth type. Ancient Greek mathematicians studied conic sections, culminating in Apollonius Perga's systematic work on their properties around 200 BC.

Definition

Circle

A circle is the set of points in a plane that are equidistant from a given point O . The distance from the center is called the radius, and the point O is called the center. Twice the radius is known as the diameter D=2r . The angle a circle subtends from its center is a full angle, equal to 360° or $2\pi$ radians.

A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.

Ellipse

In mathematics, an ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity $e$ , a number ranging from $e=0$ (the limiting case of a circle) to $e=1$ (the limiting case of infinite elongation, no longer an ellipse but a parabola).

Parabola

In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

The parabola was studied by Menaechmus in an attempt to achieve cube duplication. Menaechmus solved the problem by finding the intersection of the two parabolas $x^{2}=y$ and $y^{2}=2x$ . Euclid wrote about the parabola, and it was given its present name by Apollonius. Pascal considered the parabola as a projection of a circle, and Galileo showed that projectiles falling under uniform gravity follow parabolic paths. Gregory and Newton considered the catacaustic properties of a parabola that bring parallel rays of light to a focus (MacTutor Archive), as illustrated above.

Hyperbola

Each branch of the hyperbola has two arms which become straighter (lower curvature) further out from the center of the hyperbola. Diagonally opposite arms, one from each branch, tend in the limit to a common line, called the asymptote of those two arms. So there are two asymptotes, whose intersection is at the center of symmetry of the hyperbola, which can be thought of as the mirror point about which each branch reflects to form the other branch. In the case of the curve the asymptotes are the two coordinate axes.

Hyperbolas share many of the ellipses' analytical properties such as eccentricity, focus, and directrix. Typically the correspondence can be made with nothing more than a change of sign in some term. Many other mathematical objects have their origin in the hyperbola, such as hyperbolic paraboloids (saddle surfaces), hyperboloids ("wastebaskets"), hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed for use in both relativity and quantum mechanics which is not Euclidean).

Equations of Conics

The conic sections can be described in a suitable x-y coordinate system by 2nd degree equations:

Ellipse with center M at point (0,0) and major axis on x-axis:
${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,\quad b=|MS_{3}|,\qquad a,b\neq 0\quad ,$ (see image). (For $a=b=r$ there is a circle.)
Parabola with vertex at point (0,0) and axis on y-axis:
$y=ax^{2},\quad a={\frac {1}{4|SF|}},\qquad a\neq 0\quad ,$ (see figure).
Hyperbola with center M at point (0,0) and major axis on x-axis:
${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1,\quad b^{2}=|MF_{1}|^{2}-a^{2},\ qquada,b\neq 0\quad ,$ (see figure).
Intersecting pair of lines with intersection at point (0,0):
$a^{2}x^{2}-y^{2}=0,\ a\neq 0.$
Line through the point (0,0):
$x^{2}=0.$
point, the point (0,0):
$a^{2}x^{2}+b^{2}y^{2}=0,\ a,b\neq 0.$

For the sake of completeness, two more cases are added, which do not appear as actual conic sections, but are also described by equations of the 2nd degree:

Parallel pair of lines:
$x^{2}=a^{2},\ a\neq 0.$
The empty set:
$x^{2}+y^{2}=-1$ or $x^{2}=-1$ .

The last two cases can appear as plane sections of a right circular cylinder. A circular cylinder can be viewed as the limiting case of a cone with a cone apex at infinity. Therefore these two cases are included in the conic sections.

Plane sections of the unit cone

conic-cases To determine that the curves/points referred to above as conic sections actually occur when a cone intersects a plane, here we intersect the unit cone (straight circular cone) $K_{1}\colon x^{2}+y^{2}=z^{2}$ with a plane parallel to the y-axis. This is not a limitation as the cone is rotationally symmetrical. Any right circular cone is the affine image of the unit cone $K_{1}$ and ellipses/hyperbolas/parabolas/... go back into the same with an affine mapping.

Given: plane $\varepsilon \colon ax+cz=d\ ,$ cone $K_{1}\colon x^{2}+y^{2}=z^{2}$ .

Wanted: Intersection $\varepsilon \cap K_{1}$ .

Case I: $c=0$ $c=0$ In this case the plane is perpendicular and $a\neq 0$ $a\neq 0$ and $x=d/a$ $x=d/a$ . Eliminating $x$ $x$ from the cone equation yields $z^{2}-y^{2}=d^{2}/a^{2}$ $z^{2}-y^{2}=d^{2}/a^{2}$ .
- Case Ia: $d=0$ . In this case the intersection consists of the pair of lines $t(0,1,\pm 1),\ t\in \mathbb {R} .$ .
- Case Ib: $d\neq 0$ . The above equation now describes a hyperbola in the y-z plane. So the intersection curve $\varepsilon \cap K_{1}$ is itself a hyperbola.
Case II: $c\neq 0$ $c\neq 0$ . If you eliminate $z$ $z$ from the cone equation using the plane equation, you get the system of equations $(1)\quad (c^{2}-a^{2})x^{2}+2adx+c^{2}y^{2}=d^{2},\qquad (2)\quad ax+cz=d.$ $(1)\quad (c^{2}-a^{2})x^{2}+2adx+c^{2}y^{2}=d^{2},\qquad (2)\quad ax+cz=d.$
- Case IIa: For $d=0$ the plane goes through the apex of the cone $(0,0,0)$ and equation (1) has now the form $(c^{2}-a^{2})x^{2}+c^{2}y^{2}=0$ .
  For $c^{2}>a^{2}$ the intersection is the point $P_{0}=(0,0,0)$ .
  
  For $c^{2}=a^{2}$ the intersection is the line $t(c,0,-a),\ t\in \mathbb {R} .$
  
  For $c^{2}<a^{2}$ the intersection is the pair of lines $t(c/\pm {\sqrt {a^{2}-c^{2}}},1,-a/\pm {\sqrt {a^{2}-c^{2}}}),\ t\in \mathbb {R} .$
- Case IIb: For $d\neq 0$ the plane does not go through the apex of the cone and is not perpendicular.
  For $c^{2}=a^{2}$ , (1) goes into $x=-{\frac {c^{2}}{2ad}}y^{2}+{\frac {d}{2a}}$ across and the intersection curve is a parabola.
  
  For $c^{2}\neq a^{2}$ we transform (1) into ${\frac {(c^{2}-a^{2})^{2}}{d^{2}c^{2}}}\left(x+{\frac {ad}{c^{2}-a^{2}}}\right)^{2}+{\frac {c^{2}-a^{2}}{d^{2}}}y^{2}=1$ .
  
  For $c^{2}>a^{2}$ the intersection curve is an ellipse and
  
  for $c^{2}<a^{2}$ there is a hyperbola.

Parametric representations of the intersection curves can be found in Weblink CDKG, pp. 106-107.

Summary:

If the cutting plane does not contain the apex of the cone, the non-degenerate conic sections result (see figure for Ib, IIb), namely a parabola, an ellipse or a hyperbola , depending on whether the axis of the cone is intersected by the cutting plane at the same, greater, or lesser angle than the generatrices of the cone.
If, on the other hand, the apex of the cone is in the section plane, the degenerate conic sections are created (see picture for Ia, IIa), namely a point (namely the apex of the cone), a straight line ' (namely one surface line) or a intersecting pair of straight lines, (namely two surface lines).

Pencil of conics

A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics. The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.

In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.

A pencil of conics can be represented algebraically in the following way. Let $C 1$ and $C 2$ be two distinct conics in a projective plane defined over an algebraically closed field $K$ . For every pair $λ, μ$ of elements of $K$ , not both zero, the expression:

\lambda C_{1}+\mu C_{2}

represents a conic in the pencil determined by $C 1$ and $C 2$ . This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation defining the object.) Thinking of $C 1$ , say, as a ternary quadratic form, then $C 1 = 0$ is the equation of the "conic $C 1$ ". Another concrete realization would be obtained by thinking of $C 1$ as the 3×3 symmetric matrix which represents it. If $C 1$ and $C 2$ have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they differ by a non-zero multiplicative constant.