# Why call “Affine” Geometry ?

The term “Affine” was coined by Euler  (1748b), motivated by the idea that images related by affine transformation  have an affinity with one another.

Note 1: The term “Affine Geometry” is never used in GCE A-level Math, but commonly taught in French Baccalaureate.

Note 2:  “Affinity” 亲和力 => 模仿
Affine transformation => 仿射 变化

In geometry, an affine transformationaffine map[1] or an affinity (from the Latin, affinis, “connected with”) is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line.

Examples of affine transformations include translationscalinghomothetysimilarity transformationreflectionrotationshear mapping, and compositions of them in any combination and sequence.

# Does Abstract Math belong to Elementary Math ?

The answer is : “Yes” but with some exceptions.

Most pedagogy mistake made in Abstract Algebra teaching is in the wrong order (by historical chronological sequence of discovery):

[X] Group -> Ring -> Field

It would be better, conceptual wise, to reverse the teaching order as:

Field -> Ring -> Group

or better still as (the author thinks):

Ring -> Field -> Group

• Reason 1: Ring is the Integers, most familiar to 8~ 10-year-old kids in primary school arithmetic class involving only 3 operations: ” + – x”.
• Reason 2: Field is the Real numbers familiar in calculators involving 4 operations: ” + – × ÷”, 1 extra division operation than Ring.
• Reason 3: Group is “Symmetry”, although mistakenly viewed as ONLY 1 operation, but not as easily understandable like Ring and Field, because group operation can be non-numeric such as “rotation” of triangles, “permutation” of roots of equation, “composition” of functions, etc. The only familiar Group is (Z,+), ie Integers under ” +” operation.

Some features which separate Advanced Math from Elementary Math are:

• Proof [1]
• Infinity [2]
• Abstract [3]
• Non Visual [4]
•

Note [1]: “Proof” is, unfortunately, postponed from high-school Math to university level. This does not include the Euclidean Geometry axiomatic proof or Trigonometry Identity proof, which are still in Secondary school Elementary Math but less emphasized since the 1990s (unfortunately).

Note [2]: However, some “potential” infinity still in Elementary math, such as 1/3 = 0.3333…only the “Cantor” Infinity of Real number, ${\aleph_{0}, \aleph_{1}}$ etc are excluded.

Note [3]: Some abstract Algebra like the axioms in Ring and Field  (but not Group) can be in Elementary Math to “prove” (as in [1]): eg. By distributive law
$(a + b).(a - b) = a.(a - b) + b.(a - b)$
$(a + b).(a - b) = a^{2}- ab + ba - b^{2}$
By commutative law
$(a + b).(a - b) = a^{2}- ab + ab- b^{2}$
$(a + b). (a - b) = a^{2} - b^{2}$

Note [4]: Geometry was a “Visual” Math in Euclidean Geometry since ancient Greek. By 17 CE, Fermat and Descartes introduced Algebra into Geometry as the Analytical Geometry, still visual in (x, y) coordinate graphs.

20 CE Klein proposed treating Geometry as Group Transformation of Symmetry.

Abstract Algebra concept “Vector Space” with inner (aka dot) product is introduced into High School (Baccalaureate) Elementary Math – a fancy name in  “AFFINE GEOMETRY” (仿射几何 , see Video 31).

eg. Let vectors
$u = (x,y), v = (a, b)$

Translation:
$\boxed {u + v = (x,y) + (a, b) = (x+a, y+b)}$

Stretching by a factor ${ \lambda}$ (“scalar”):
$\boxed {\lambda.u = \lambda. (x,y) = (\lambda{x}, \lambda{y})}$

Distance (x,y) from origin: |(x,y)|
$\boxed {(x,y).(x,y) =x^{2}+ y^{2} = { |(x,y)|}^{2}}$

Angle ${ \theta}$ between 2 vectors ${(x_{1},y_{1}), (x_{2},y_{2})}$:

$\boxed { (x_{1},y_{1}).(x_{2},y_{2}) =| (x_{1},y_{1})|.| (x_{2},y_{2})| \cos \theta}$

Ref: 《Elements of Mathematics – From Euclid to Gödel》by John Stillwell (Princeton University Press, 2016) [NLB # 510.711]