## Lines and conics in

• A line equation in is

• Substituting by homogeneous coordinates we get a homogeneous linear equation

• A line in is represented by a homogeneous 3-vector .
• A point on a line:

or     or

• Two points define a line:

• Two lines define a point:

• Matrix notation for cross products:
The cross product can be represented as a matrix multiplication

where is a antisymmetric matrix of rank 2:

• The line at infinity ( ): is the line of equation . Thus, the homogeneous representation of is .
• The line intersects at the point .
• Points on 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 . To transform to homogeneous coordinates, we substitute and to obtain . This is homogeneous in degree 2. Note that both and 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

Homogenizing this by replacements and , we obtain

which can be written in matrix notation as

where is symmetric and is the homogeneous representation of a conic.

• Five points define a conic.
• The line tangent to a conic at any point is given by .
• (because ). This is the equation of the dual conic.

• The degenerate conic of rank 2 is defined by two line and as

Points on line satisfy and are hence on the conic because . (Similarly for ).

The dual conic represents lines passing through and .

Subhashis Banerjee 2008-01-20