Lines and conics in ${\cal P}^2$

• A line equation in $\mathbb{R}^2$ is
  $$a_1 x_1 + a_2 x_2 + a_3 = 0$$

• Substituting by homogeneous coordinates $x_i = X_i/X_3$ we get a homogeneous linear equation
  $$(a_1,a_2,a_3) \cdotp (X_1,X_2,X_3) = \sum_{i=1}^{3} a_i X_i = 0,$$     $${\bf X} \in {\cal P}^2$$

• A line in ${\cal P}^2$ is represented by a homogeneous 3-vector $(a_1,a_2,a_3)$ .
• A point on a line:
  $${\bf a} \cdotp {\bf X} = 0$$    or $${\bf a}^T {\bf X} = 0$$    or $${\bf X}^T {\bf a} = 0$$

• Two points define a line:
  $${\bf l} = {\bf p} \times {\bf q}$$

• Two lines define a point:
  $${\bf x} = {\bf l} \times {\bf m}$$

• Matrix notation for cross products:
  The cross product ${\bf v} \times {\bf x}$ can be represented as a matrix multiplication
  $${\bf v} \times {\bf x} = \left[ {\bf v} \right]_{\times} {\bf x}$$
  where $\left[ {\bf v} \right]_{\times}$ is a $3 \times 3$ antisymmetric matrix of rank 2:
  $$\left[ {\bf v} \right]_{\times} = \left[ \begin{array}{ccc} 0 & -v_z & v_y \\ v_z & 0 &-v_x \\ -v_y & v_x & 0 \end{array}\right]$$

• The line at infinity ( ${\bf l}_\infty$ ): is the line of equation $X_3 =0$ . Thus, the homogeneous representation of ${\bf l}_\infty$ is $(0,0,1)$ .
• The line $(u_1,u_2,u_3)$ intersects ${\bf l}_\infty$ at the point $(-u_2,u_1,0)$ .
• Points on ${\bf l}_\infty$ are directions of affine lines in the embedded affine space (can be extended to higher dimensions).
• Consider the standard hyperbola in the affine space given by equation $xy = 1$ . To transform to homogeneous coordinates, we substitute $x = X/T$ and $y = Y/T$ to obtain $XY = T^2$ . This is homogeneous in degree 2. Note that both $(0,\lambda,0)$ and $(\lambda,0,0)$ are solutions. The homogeneous hyperbola crosses the coordinate axes smoothly and emerges from the other side. See the figure.
  

• A conic in affine space (inhomogeneous coordinates) is
  $$ax^2 + bxy + cy^2 + dx + ey + f = 0$$
  Homogenizing this by replacements $x = X_1/X_3$ and $y = Y_1/Y_3$ , we obtain
  $$aX_1^2 + bX_1X_2 + cX_2^2 + d X_1X_3 + e X_2X_3 + f X_3^2 = 0$$
  which can be written in matrix notation as
  $${\bf X}^T{\bf C}{\bf X} = 0$$
  where $C$ is symmetric and is the homogeneous representation of a conic.
  $$C = \left[ \begin{array}{ccc} a & b/2 & d/2 \\ b/2 & c & e/2 \\ d/2 & e/2 & f \end{array}\right]$$

• Five points define a conic.
• The line ${\bf l}$ tangent to a conic ${\bf C}$ at any point ${\bf x}$ is given by ${\bf l} = {\bf C}{\bf x}$ .

• $${\bf x}^t{\bf Cx} = 0 \implies ({\bf C}^{-1}{\bf l})^t{\bf C}(({\bf C}^{-1}{\bf l}) = {\bf l}^t{\bf C}^{-1}{\bf l} = 0$$
  (because ${\bf C}^{-t} = {\bf C}^{-1}$ ). This is the equation of the dual conic.
• The degenerate conic of rank 2 is defined by two line ${\bf l}$ and ${\bf m}$ as
  $${\bf C} = {\bf lm}^t + {\bf ml}^t$$
  Points on line ${\bf l}$ satisfy ${\bf l}^t{\bf x} = 0$ and are hence on the conic because $({\bf x}^t{\bf l})({\bf m}^t{\bf x}) + ({\bf x}^t{\bf m})({\bf l}^t{\bf x}) = 0$ . (Similarly for ${\bf m}$ ).

  The dual conic ${\bf x}{\bf y}^t + {\bf y}{\bf x}^t$ represents lines passing through ${\bf x}$ and ${\bf y}$ .