The Disclaimer
This is NOT homework... I designed the story. I thought a name MIT would be funny, while something along the line of TULSA or SU will still be decent. I know the algorithm to Q3 and 3D version of Q4, and my major problem are the name of the local coordinate system, the terminology, the formulation of this mathematical question and possible generalization. Insights are welcome.
I'm with $10^3$ rep even though it's pretty weak, but at least I know when to tag homework questions and such. I'm not even immune to homework investigators?
The Story
Long long ago, in the far, far away mystical wonderland of $3$ dimensional Cartesian coordinate system lived many geometric creatures. Nobody knew who first named the wonderland the Mystical Itchy Territory, but its geometric residents would from time to time complain the random itchiness concurring inside their body or on their surface.
The MIT creatures live under a strange rule: Every creature can see all geometric shapes and know the coordinates of any point, but when conveying information about a $n$-dimensional coordinate to others, e.g. $(3,4,5)$, it is only allowed to communicate with at most $n$ numbers in the interval $[-1,1]$.
_Edit: The creatures want a solution like depicted in the picture below and do not prefer bijections drastically different to what's depicted. A trivial answer like mapping every coordinate in $\mathbb R$ to $[-1,1]$ using $\frac2\pi\arctan(x)$ is not preferred by the creature, though it's appealing to me.
The 1D Ritual
I just met this quadrilaterally-faced hexahedral creature, now my friend (photograph in picture (a)), who told me that his species have came up with a ritual to help locating the origin of itchiness on the edges so that others could scratch them.
The ritual is a simple convention plus an equation:
The Convention: Tell others the itchy edge $AB$, let $-1$ represent a vertex $A$, let $1$ represent the other vertex $B$, any point $P(x,y,z)$ on that edge will be represented by a number in $[-1,1]$, by the following equation.
The Equation: $\displaystyle a=\frac{x_A+x_B-2x}{x_A-x_B} \text{or} \frac{y_A+y_B-2y}{y_A-y_B} \text{or} \frac{z_A+z_B-2z}{z_A-z_B}$, whichever is defined.
To get the actual coordinates, they perform a simple linear operation (combination):
$$P = \frac{1-a}{2}A + \frac{1+a}{2}B$$
The Questions
Q1: What term describes this kind of mapping? What's the combination called?
Q2: Relaxing $a$ to $\mathbb R$, what's this mapping called? What's the combination called?
Similarly, for itches on the surface, he suggests that he could tell others which surface to scratch. He wants to design another set of ritual similar to what's shown in Picture (c), however, he doesn't know how to represent points on his quadrilateral face in $[-1,1]^2$. Neither does he know how to represent any interior points in $[-1,1]^3$.
Q3: Please help the creature-- given 4 vertices of a quadrilateral, map a point on the quadrilateral as the creature intended.
Q4: Given all 8 vertices of this creature, upgrade the ritual. If possible, generalize the result to $n$ dimensions following similar conventions.
Q5: What about other geometric shapes?
Illustration of the Mystical Itchy Territory
Q1: If you change $[-1,1]$ to $[0,1]$, you've basically invented 1D Barycentric coordinates.
Q2: Then you have affine combinations.
Q3: If the quadrilateral is not a parallelogram then I'm not so sure there's a completely canonical answer, but several things may be satisfying. The algorithm you linked for Q4 is just one possibility. Assuming the quadrilateral is convex, I might be inclined to say something like "convert the $(x,y)$ to four coordinates for a convex combination of the vertices where basically we have a weighted version of something like $(2-y+x,x+y+2,2-x+y,2-x-y)$ (the x-forward vertices get the benefit of $x$: $x+1$ and the others get $1-x$, similarly for $y$).
Q4: In this case, I would similarly convert to a convex combination in a way like that if the hexahedron is convex.
Q5: Even these generalized barycentric coordinates depend on convexity; If a shape is not convex, I'm really not certain what I'd want an answer to look like...