The approach of
affine geometry, as opposed to the approach of
analytic geometry, uses vector methods to reveal geometric facts in a simple manner. This approach dates back to
Felix Klein (1849-1925) and
Sophus Lie (1842-1899).
To quote my source book: "It gives concrete examples leading to an appreciation of the theory of
groups", hence a knowledge of
Abstract algebra can be useful.
Euclidean geometry can be seen as the geometry associated to the group of
isometries.
Here are some basic definitions to get a feel of
affine geometry.
A point is defined by two values, (
a1,
a2), its
Cartesian coordinates.
A point is also considered to be a vector from the origion to the location of the point. Two vectors are equal if their coordinate values are equal.
If
A = (
a1,
a2) and
B = (
b1,
b2),
the addition of vectors is defined as
A + B = (a1 + b1, a2 + b2).
Multiplication by a scalar is defined as:
rA = (a1, ra2).
When using abstract
vectors,
division is not defined.
However, what is traditionally the division symbol is used. For example, if a point
P lies on line
lAB (line passing both
A and
B), distinct from
B, then (
P -
A)/(
P -
B) =
b/a
is an alternative to writing (
P -
A) = (
b/a)(
P -
B) where
a and
b are
scalers.
Point
P is on the line
lAB iff
there exists a scalar
s:
P = A + s(B - A)
This also means
P = rA + sB (where r + s = 1)
The
midpoint M can be written as
M = (1/2) (
A +
B), where
M is defined as a
midpoint of
A and
B iff A -
M =
M -
B.
In addition to these operations,
dot product is used to define
orthogonality, and the
projection function is used to define reflection
isometries.
Theorems such as the
theorem of Thales,
theorem of Menelaus,
theorem of Ceva, the
nine point circle theorem, etc.
can be proven taking the
affine geometry approach.
Source:
"Vectors and Transformations in Plane Geometry" by Philippe Tondeur, Publish or Perish, Inc. 1993