In GGB, sign() gives one of three values: {−1, 0, 1}.
It distinguishes +/− from 0. This is extremely useful.
Sometimes a binary sign is necessary:
sign2[s] = If[sign(s) ≥0, 1, -1]
A simple definition, taking 0 as positive.
Object definition is such a time.
The orientation of an object is an intrinsic property; it cannot be given by any logical rule or analytic system.
So first we must allow free choice of direction. For example, above, we are free to choose either left or right of the line. (That is, we can orient the line.) This allows us to relate the line to other objects in space.
Once the intrinsic properties of an object have been given, then we define a system: a common framework in which objects can be consistently related. For simplicity, I will use the familiar rectangular coordinates: distinct directions in space are represented by mutually perpendicular unit vectors. Each object will have its own coordinate space, which I will call local space.
I will call the Geogebra worksheet viewport (or GGB's internal x- y- coordinate grid) global space.
{To Do: worksheets demonstrating relative coordinates. 1. Boxy mcGee! O, boxy. O bliss.}
Example:
Consider a box.
The box is labeled with u-, v-, and w- directions.
I am free to put whatever object in that box that I wish, and orient it how I please.
When I am done turning the object about and deforming it, let us agree that I have a way of fixing its position inside the box, the way --say-- electronics are packaged so they don't move about.
Then I can flip the box around however I want, place it anywhere in space.
To keep track, I will label the space around me with fixed x-y-z directions.
This is precisely how I will orient measured figures in space, in code. The important thing to note is that the coordinates which can be given in advance, by rule are just containers: reference spaces, in which oriented objects are placed, and the individual containers which relate each object to the reference space. They carry no information about the structure of the objects, which is up to us... either to describe or define. And, once boxed, the orientation of the boxes in space is again up to us.
I shall endeavor to make all of this plain, by example.