Functions on the plane with some geometric properties Intuitively speaking a plane is a portion of space and is very thin and extends in all directions. For any pair of distinct points in a plane the line through them lies entirely in the plane. That makes the plane flat and thin. A function on the plane is a function that carries all points in the plane back onto points in the plane. If P,Q,R are any points in that plane, then images F(P), F(Q), F(R) are also points in that plane. By definition the identity function I on the plane carries each point back onto itself:
I(P)=P,   I(Q)=Q,   I(R)=R
Another function carries all points of the plane onto a single point:, say A:
F(P)=A,   F(Q)=A,   F(R) = A
The image of every point in the plane is A. But a single point is much less interesting than what a plane can hold. Therefore, such functions carrying the entire plane onto a single point are trivial in nature and are not very interesting.

Every function on a plane carries any subset of points onto some subset of points. The second subset is called the image of the first subset. However, the image may or may not look line the original subset. In Fig 1, a "nice" function Fcarries a subset which is a straight line onto another subset which is also a straight line. But in Fig 2 the function G carries the straight line onto something not a straight line.This carrying produces a distortion of the original subset which is a line. (In all discussions, a "line" means a "straight line.")

[1.1a] (Preserving lines) A function on the plane preserves lines if it carries every line onto a line.
Notation: If F preserves lines, then F carries the line through any points P and Q onto a line through points
F(P) and F(Q).

Recall that collinear points all line on the same line. Therefore a definition equivalent to [1.1a] is that a function on the plane carry collinear points onto collinear points. Intuitively speaking, such functions preserve straightness, the visible property of lines. A (line) segment is a piece of some line and is therefore straight. A segment has two end points both on the line.

[1.1b] (Preserving segments) A function on the plane preserves segments if it carries every segment onto a segment and end points onto end points.
Notation: if F preserves segments, then F carries any segment with end points P,Q onto a segment with end points
F(P),F(Q).

Let PQ denote segment joining points P and Q, and let F(PQ) be the image of segment PQ. The end points of F(PQ) are F(P) and F(Q).

Strictly speaking, it should be said that the function preserves the structure of a segment. Preserving segments is a stronger condition than preserving lines. It implies the condition that if point R is between points P and Q on one line, then image F(R) is between F(P) and F(Q) on the other line. In other words, the function preserves "betweenness." Suppose now that R is the midpoint of segment PQ. It seems natural to ask if F(R) is the midpoint of segment F(PQ) ?

[1.2] (Preserving midpoints) A segment preserving function preserves midpoints if it carries the midpoint of every segment onto the midpoint of every segment.
Notation: If point R is the midway between two points P and Q on some line, then image F(R) is midway between points F(P) and F(Q) on their line.

Recall that a quadrilateral is a figure with four sides and four vertices. Theorems from plane geometry state that the diagonals of a quadrilateral bisect each other if and only if the quadrilateral is a parallelogram.

[1.3] (Preserving parallelograms) If a function on the plane preserves segments and midpoints then it carries parallelograms onto parallelograms.
Notation: Let A,B,C,D be vertices of a parallelogram with diagonals AC and BD. Then F(A), F(B), F(C), F(D) are vertices of a parallelogram with diagonals F(A),F(C) and F(B),F(D). If AC and BD bisect each other, then their images F(AC) and F(BD) also bisect each other.

If M is the intersection of segments AC and BD, then F(M) is the intersection of segments F(AC) and segments F(BD). But M is the mid point of both AC and BD since the diagonals of a parallelogram bisect each other. Then F(M) is the midpoint of segments F(AC) and F(BD) bccause F carries midpoints onto midpoints. Therefore F(A), F(B), F(C), F(D) are vertices of a parallelogram because the diagonals bisect each other.

It is important to notice that the two parallelograms need not be identical in size or even have congruent corresponding angles. They are simply parallelograms. But since the opposite sides of a parallelogram are parallel and congruent segments, then it follows that a function that preserves segments and midpoints must carry parallel and congruent segments onto parallel and congruent segments. Also lines containing these sides must be parallel.

[1.4] (Preserving parallelism) A function preserves parallelism if and only if it carries parallel lines onto parallel lines.
Notation: If line1 is parallel to line2 then F(line1) is parallel to F(line2).

Return now to the midpoint of a line segment. If M is the midpoint of segment AB then M is 1/2 the way from A to B. This means that segment AM is is 1/2 the length of segment AB. This can be written two ways:

(1) |AM|/|AB| = 1/2      (2) |AM| = (1/2)|AB|
But if point P is a point of trisection of segment AB nearest then
(3) |AP|/|AB| = 2/3      (4) |AP| = (2/3)|AB|
If P is any point between A and B, then let λ equal the quotient |AP|/|AB|. Then
(5) |AP|/|AB| = λ      (6) |AP| = λ|AB|
Let F be a function on the plane, and assume that F preserves segments. The question here concerns the relative position of a point in one segment and the relative position of the image of that point in the other segment.
If P is the point of trisection of AB nearest B, then is F(P) the point of trisection of segment F(AB) nearest F(B)?
If P is 2/5 the way from A to B, is F(P) 2/5 the way from F(A) to F(B)?
Both of these questions can be restated as a question of equality of ratios:
is |AP|/|AB| = |F(AP)|/|F(AB)| ?
Another revision of this question is the following:
if |AP|/|AB| = λ then does |F(AP)|/F(AB)| = λ ? Of course λ is a real number. Another way of phrasing this question is:
if |AP| = λ|AB| then does |F(AP)| = λ|F(AB)| ?
This question leads to the following definition:

[1.5] (Preserving local proportionality) Let F be a function on the plane, and suppose F carries segments onto segments. If P is any point on a line through distinct points A and B, such that P satisfies the vector equation

AP = λAB
and the same value of λ satisfies the vector equation
F(AP) = λF(AB)
then F is said to preserve local proportionality.