In those universes of high school geometry discussions are made wwithout any origin. (Ignore graphing and plotting points.) Unless specificly mentioned otherwise, discussions involving the basic universes do not have origins. Therefore, position vectors do not exist. However, vectors from one point to another do exist. AB is a vector from point A to a point B.
Recall that points are collinear if they lie on a common (straight) line. And they are coplanar if they lie in a common plane. Recall also that a function F carries a set S onto a set F(S) which is the totality of all points that are images of points in S. Lines and planes are sets of points. If lineo denotes a line then it is a collections of points (a subset), so that F(lineo) is a set (subset) of images. What F does to lineo geometrically is important; in other words, what is the shape of F(lineo) ? The function F could "bend" the line into a curve such that F(lineo) it has the U shape of a parabola or the wavy shape of a sine curve. Also a geometric function should preserve the flatness of a plane.
[1.1] (Geometric functions)
(a) A geometric function carries any line entirely onto a line or onto a point.
(b) If the universe is a plane, then a geometric function carries any it entirely onto itself or entirely onto a line or entirely onto a point.
(c) If the universe is space, then a geometric function carries it entirely onto itself, or onto a plane or onto a line or onto a point.
Statement [1.1a] means that a geometric function carries collinear points onto collinear points. If all the images form a line, then the function is non-singular. If the function carries any entire line onto a single point then the function is singular.
Even some geometric functions may do "weird" things. The following statements are intended to reduce weird behavior.
In the basic universes, segments of lines (connected subsets of finite length) of lines exist. Consider any point as a segment whose endpoints coincide, and has zero length.
[1.2] (Preserving segment structure) A geometric function preserves segment structure if it carries every segment onto a segment and both endpoints onto endpoints of the image segment
Intuitively speaking, the function picks up a segment, during the process of carrying it the segment may be stretched, shrunk or left unchanged, then put down somewhere but always in the form of a segment.
Preserving segment structure means that a geometric function preserves betweenness. If A and B are points on a line and P is a point between them then P is in segment AB. If F preserves segment structure then F carries all points of segment AB onto segment A'B', where A'=F(A) and B'=F(B). Therefore F(P) must be in segment A'B' and so must be between F(A) and F(B).
A non-singular geometric function that preserves segment structure carries triangles onto triangles, in fact poliygons onto polygons of the same number of sides, polyhedra onto polyhedra. However, presrving segment structure does not mean that a segment and its image have the same length. (Preserving length is a special property to be discussed later.) This means that the triangle and its image may have very different shapes. For example, the image of an equilateral triangle may be scalene. Then angle measurements may or may not be preserved. A 60° angle in the equilateral triangtle appears nowhere in the scalene triangle.
[1.3] A geometric function preserves parallelograms if it preserves segment structure and carries all parallelograms onto parallelograms.
It is not to difficult to show that [1.3] implies that such geometric functions carry parallel lines onto parallel lines. But to make this statement true for singular geometric functions that may carry distinct parallel lines onto a single parallel line, a line may be parallel to itself. Also a plane may be parallel to itself and a line in a plane may be parallel to that plane. (This makes parallelism to be an equivalence relation among lines and among planes.)
Also [1.2a] also presents a problem if the geometric function carries a genuine parallelogram in a plane (or space) onto a line to obtain a "collapsed parallelogram." This figure is essentially two pairs segments of the same lengths on a common line.
The ratio of a first segment to a second segment is a number λ such that the length of the first segment is equal to the product of λ and the length of the second segment. For example, if segment AB is four times as long as segment CD then λ = 4. But if λ = 1/3 then AB is only 1/3 the length of CD. If λ = 0 then length of AB is zero, which means A=B. This would be true no matter what the length of CD is.
Lengths are never negative. But it will be necessary for λ to be any real number, including negatives. If λ is positive, then the vector equation AB = λCD implies vectors AB and CD point in the same direction. For a negative λ they point in opposite direction.
[1.2b] A geometric function preserves proportionality if it carries two parallel line segments, the ratio of lengths of which is λ, onto parallel line segments, the ratio of lengths of which is λ. The real number λ is the proportionality factor of the function. λ is negative if as vectors the segments point in opposite directions.
For example if F is a geometric function that preserves proportionality and if segment AB is 4 times as long as segment CD, and the segments are parallel, then F(AB) is a segment that is 4 times as long as F(CD) and segments F(AB) and F(CD) are parallel. Here length(AB)/length(CD) = 4 = λ and length F(AB)/lengthF(CD) = 4 = lambda;.
The real number λ may be negative due to the opposite orientations of the parallel segments. Click here for a brief discussion of orientation. If λ is negative then the absolute value |λ| = ratio of lengths.
[1.3] (Affine transformations) A geometric function is affine if it preserves parallelograms and preserves proportionality of segments. Affine functions are usually called affine transformations.
Intuitively speaking, a property is affine if no affine transformation exists to carry a figure with that property onto a figure without that property. Parallelism is an affine property by definition of all affine transformations. But perpendicularity is not an affine property. It is possible to construct a linear transformation that carries the three points (1,0), (0,0), (0,1) onto the three points (1,0), (0,0), (1,1). This carries the perpendicular lines that are the x- and y-axis onto lines x-axis and a line through (0,0) and (1,1) which is not perpendicular to the x-axis. Therefore, perpendicularity is not an affine property.
If F is an affine transformation from a basic universe to a basic universe both with origins, then F may or may not carry the first origin onto the second origin (where both universes and both origins may be the same) may or may not be an origin: F(O) may or may not be the second origin. If the fixed image F(O) is not an origin then F cannot be a linear transformation. However it is possible to "extract" a linear transformation from any affine transformation if origins exist.
[1.4] Lemma (Getting a linear transformation from an affine transformation) For basic universes with origins, an affine transformation minus its image of the origin is a linear transformation.
Notation: for any affine transformation F, the transformation L, defined by L(p) = F(p) -- F(0) for any point P, is linear.
Click here to see a discussion supporting this lemma.
[1.5] (Form of an affine transformations in terms of a linear transformation) For any affine transformation there is a unique linear linear transformation such that the affine transformation is equal to the linear transformation plus the image of the origin under the affine transformation.
Notation: If F is an affine transformation, then there is a linear transformation L such that F = L + F(O). (This means that F(P) = L(P) + F(O) for every point P which really means F(p) = L(p) + F(0) for every position vector p.)
From [1.4] L, defined by L(p) = F(p) -- F(0) for any point P, is linear. Solving this equation for F(p produces F(p) = L(p) + F(0).
[1.6] All linear transformations are affine transformations that carry origins onto origins.
Intuitive argument: if F(O) = O then F = L + F(O) becomes F = L. A more formal argument uses the equations
[1.y] All linear transformations are affine. Click here for a discussion supporting this statement. The converse is false: later an affine transformation called translation is not a linear transformation. A further study of affine transformations, not done in this chapter, leads to a consistent geometry called "affine geometry."