Section 1: Plane Conics

Projective plane

$\begin{eqnarray*} \mathbb{R}\mathbb{P}^{2} & = & \mbox{lines of }\mathbb{R}^{3}\mbox{ through origin}\\ & = & \mbox{ratios }X:Y:Z\\ & = & \mathbb{R}^{3}/\sim\mbox{, where }(X,Y,Z)\sim(\lambda X,\lambda Y.\lambda Z),\lambda\in\mathbb{R}-\{0\}. \end{eqnarray*}$

Points $(X,Y,Z)$ with $Z\neq0$ correspond one-to-one to $\mathbb{R}^{2}$ via $(X,Y,Z)\to(X/Z,Y/Z,1)$ . This is the affine piece of $\mathbb{R}\mathbb{P}^{2}$ .
Points $(X,Y,0)$ are points at infinity corresponding to asymptotic directions or pencils of parallel lines in $\mathbb{R}^{2}$ .

Lines, asymptotic directions, and the line at infinity

A line $L$ in $\mathbb{R}\mathbb{P}^{2}$ is $\begin{eqnarray*} aX+bY+cZ & = & 0. \end{eqnarray*}$

Two lines pass through the same point at infinity iff they have the same ratio $a/b$ , so that “parallel lines meet at infinity”.

Points of $\mathbb{R}\mathbb{P}^{2}$ with $Z=0$ form the “line at infinity”, a copy of $\mathbb{R}\mathbb{P}^{1}=\mathbb{R}\cup\{\infty\}$ .

Transformations

An affine transformation is of the form $A\mathbf{x}+B$ where $A$ is an invertible $2\times2$ matrix. If $A$ is orthogonal, this is an Euclidean transformation.

A projective transformation (or projectivity) of $\mathbb{R}\mathbb{P}^{2}$ is of the form $M\mathbf{X}$ where $M$ is an invertible $3\times3$ matrix.

If $M=\left[\begin{array}{cc} A & B\\ \begin{array}{cc} c & d\end{array} & e \end{array}\right],$ then on the affine piece $\mathbb{R}^{2}\subset\mathbb{R}\mathbb{P}^{2}$ $\begin{eqnarray*} \left(\begin{array}{c} x\\ y \end{array}\right) & -\to & \tfrac{1}{cx+dy+e}\left[A\left(\begin{array}{c} x\\ y \end{array}\right)+B\right]. \end{eqnarray*}$

Conics

Conics in the plane

A conic in $\mathbb{R}^{2}$ is given by the equation $\begin{eqnarray*} q(x,y) & = & ax^{2}+bxy+cy^{2}+dx+ey+f\\ & = & 0. \end{eqnarray*}$

Depending on the values of the parameters they can be

ellipse
parabola
hyperbola
single point
empty set (three different cases)
line
line pair
double line
whole plain

Conics in the projective plane

The inhomogeneous quadratic polynomial $q(x,y)$ above corresponds to the following quadratic form (form = homogeneous polynomial) $\begin{eqnarray*} Q(X,Y,Z) & = & aX^{2}+bXY+cY^{2}+dXZ+eYZ+fZ^{2}. \end{eqnarray*}$

A conic in $\mathbb{R}\mathbb{P}^{2}$ is given by the equation $\begin{eqnarray*} Q(X,Y,Z) & = & 0. \end{eqnarray*}$

Classification of conics

A quadratic form $aX^{2}+2bXY+cY^{2}+dXZ+eYZ+fZ^{2}$ is nondegenerate if the corresponding bilinear form given by $\left[\begin{array}{ccc} a & b & d\\ b & c & e\\ d & e & f \end{array}\right]$ is nondegenerate.

For any vector space $V$ over a field $k$ of characteristic $\neq2$ , for every quadratic form $Q:V\to k$ , there is a basis of $V$ such that $\begin{eqnarray*} Q & = & \sum_{i=1}^{n}\epsilon_{i}x_{i}^{2}. \end{eqnarray*}$
Therefore, every (non-zero) conic in $\mathbb{P}\mathbb{R}^{2}$ is one of the following $\begin{eqnarray*} (\alpha) & \mbox{non degenerate conic} & X^{2}+Y^{2}-Z^{2}=0;\\ (\beta) & \mbox{empty set} & X^{2}+Y^{2}+Z^{2}=0;\\ (\gamma) & \mbox{line pair} & X^{2}-Y^{2}=0;\\ (\delta) & \mbox{one point} & X^{2}+Y^{2}=0;\\ (\epsilon) & \mbox{double line} & X^{2}=0. \end{eqnarray*}$

Parametrization of conics

If $C$ is a nondegenerate nonempty conic, then in suitable coordinates $C$ is given by $\begin{eqnarray*} XZ & = & Y^{2}. \end{eqnarray*}$ This is a curve parametrized by $\begin{eqnarray*} \Phi:\mathbb{P}\mathbb{R}^{1} & \to & \mathbb{P}\mathbb{R}^{2}\\ (U,V) & \to & (U^{2}:UV:V^{2}). \end{eqnarray*}$

Intersection of conics

Homogeneous forms in two variables

If $F(U,V)$ is a nonzero form of degree $d$ with coefficients in a field $k$ , then it has $\le d$ zeros, and $=d$ zeros if $k$ is algebraically closed, where roots are

$(\alpha,1)$ where $\alpha$ is a root of $f(\alpha)=F(\alpha,1)$ , counted with multiplicities;
$(1,0)$ if $\deg f<d$ , counted with multiplicity $d-\deg f$ .

Intersection of a line or conic with a curve

If $A$ is a line or nondegenerate conic in $k\mathbb{P}^{2}$ , and $B$ is a curve defined by a form $G(X,Y,Z)=0$ of degree $d$ such that $A\not\subset B$ , then the number of points of intersection of $A$ and $B$ counted with multiplicity is $\le d$ or $\le2d$ , respectively, where there is a natural definition of multiplicity of points of intersection; if $k$ is algebraically closed, then equality holds.

For every $5$ distinct points such that no $4$ are collinear, there is at most one conic containing these points.

Space of conics

Let $\begin{eqnarray*} S_{2} & = & \mbox{\{quadratic forms on }\mathbb{R}^{3}\}\\ & = & \{3\times3\mbox{ symmetric matrices}\}. \end{eqnarray*}$ Let, for $P_{1},\ldots,P_{n}\in\mathbb{R}\mathbb{P}^{2}$ , $\begin{eqnarray*} S_{2}(P_{1},\ldots,P_{n}) & = & \{Q\in S_{2}|Q(P_{k})=0\mbox{ for }k=1,\ldots,n\}. \end{eqnarray*}$ Then $\dim S_{2}(P_{1},\ldots,P_{n})\ge6-n.$ Equality holds if $n\le5$ and no $4$ points are collinear.

Degenerate conics in a pencil

A pencil of conics is a family of the form $\begin{eqnarray*} C_{(\lambda,\mu)} & : & (\lambda Q_{1}+\mu Q_{2}=0). \end{eqnarray*}$ Then, $\begin{eqnarray*} C(\lambda,\mu)\mbox{ is degenerate} & \Leftrightarrow & \det(\lambda Q_{1}+\mu Q_{2})=0. \end{eqnarray*}$

Either all conics in a pencil of conics are degenerate, or there are at most $3$ degenerate conics. If $k=\mathbb{R}$ , then there is at least $1$ degenerate conic.

Subpages

Section 1: Plane Conics

Section 1: Plane Conics

Projective plane

Lines, asymptotic directions, and the line at infinity

Transformations

Conics

Conics in the plane

Conics in the projective plane

Classification of conics

Parametrization of conics

Intersection of conics

Homogeneous forms in two variables

Intersection of a line or conic with a curve

Space of conics

Degenerate conics in a pencil