- 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 - 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*homogeneous representation*of a**conic**. - Five points define a conic.
- The line tangent to a conic at any point is given by .
*dual conic*.- The
*degenerate conic*of rank 2 is defined by two line and asThe dual conic represents lines passing through and .