The visual content of a chalkboard lecture can be described by a piecewise continuous curve γ:I = [a,b] → R2, after making some simplifications. The argument t ∈ I can be interpreted as time. Discontinuities in the curve are traversed instantly, which is not true for a real chalkboard lecture. Also, the available space of chalkboard is finite, but by erasing we can effectively artificially extend the available space to R2.
From this curve we get the image of the curve, im(γ) = {γ(t) | t ∈ I} ⊂ R2, and the time-dependent image of the curve: defining γT:[a,T] → R2 as the restriction of γ to [a,T], im(γT) = {γT(t) | t ∈ [a,T]} = {γ(t) | t ∈ [a,T]}.
Admittedly the content conveys more information than just the image, but we can think about the restrictions on the kind of information that can be conveyed in a classical chalkboard lecture, and when these restrictions can be overcome by animations.
Also, being able to visualise using your imagination is better than relying on animations or pictures; the animations are really to guide you to be able to visualise in your head. They are less useful for learning concepts rigorously, but are very useful for building intuition.
What is the visual content of an animation? On a computer screen, for each time t we have an assignment of colour to each pixel. Generalise colour to a space of 'pixel values', V. Each pixel can be represented as an entry of a matrix. Then at each time we have a V-valued matrix. If the screen is m×n then an 'animation' is a map A:I→Matm×n(V). In the limit that pixel size goes to zero, we replace the matrix with a function. Indeed the matrix itself may be viewed as a function (i,j)↦Mij. Now we take K ⊂ R2 a compact subset, and denote the space of functions from K to V by (K→V). Then in the limit, the animation is a map A:I→(K→V). (N.B. which can alternately be viewed as a map A:I×K→V.)
First, let's think about what information a chalkboard lecture can convey, and can convey well.
Next, data that is difficult to convey
While the first two points are still a problem for computer animations, the subsequent ones are handled by computer animations much better.
I want to understand how to use visualisations effectively. This exploits our intuition with how human vision works. Hence to gain a better understanding of how to push the capabilities of visualising, it is useful to think about how vision works.
Pick a point p in R3, and a plane not containing p, Σ. We can draw rays from p through q ∈ Σ, and these are our lines of sight. Exercise: what is the subset of R3 lying on a line of sight? There is a bijection between lines of sight and q. We can also regard Σ as R2 via a map φ, restricting φ to be an isometry and enforcing that the point where the normal to p from Σ hits Σ, call it p0, gets mapped to 0 ∈ R2.
We also need matter, to see. In the simplest model, we could pick a subset of R3, S, and then see whether the lines of sight are obstructed by this subset, or not. A subset of R3 corresponds to the indicator function 1S:R3→{0,1}, with 1S(x) = 1 if and only if x ∈ R3. Then, to add colour or other properties to our matter we instead add a property space P, which is equipped with a distinguished element 0, the vacuum element. Then matter is a function M:R3→P.
Now pick point p such that M(p)=0, and Σ. Consider the line of sight through q, and we will parametrise this γ:R>0→R3, t↦(1-t)p + tq. Given our matter field M, our line of sight gets mapped to an element of P. Recalling that lines of sight correspond to points on Σ, explicitly this map is q↦M(γ(T)) such that for t<T, M(γ(t)) = 0, where such a T exists, and q↦0 otherwise.
Therefore, p and Σ have given us a way to take a 3D matter map M:R3→P to a 2D matter map π(M):R2→P.
Exercise: Set p at the origin, Σ at x = d, and matter to be 1S where S is a vertical line segment of length 1, at x = L. Then move S to x = 2L. By what factor does its length change under the projection?
Lets generalise a little. Instead of Σ being a plane, allow it to be a surface in R3, possibly with boundary, which we call our window. For lines of sight to be in bijection with points of Σ, we get the restriction that Σ must come from a function σ U⊂S2→R>0, and be the image of x(σ):U→R3, n↦σ(n)n (check). Then, together with charts on Σ, we get projections of a 3D matter map M to a 2D matter map M'=π(M), or possibly multiple maps πα(M).
But the important thing is not the shape of Σ, but really the field of view, the union of the image of the lines of sight. The field of view depends only on U, not on σ. There was also something nice about taking a plane to be the window: our chart was an isometry. Exercise: given U⊂S2, when can we find a σ such that Σ lies in a plane? The human field of view is 220°. Is this compatible with your answer?
We can generalise more. Example: isometric projection. Let Σ be the plane z = 0. To each point q=(x,y,0)∈Σ we associate a line of sight γ:t↦(x,y,t). This gives a projection M↦π(M) similarly to above.
We can now define a visualisation. A ray curve is a map γ:R>0→R3. A visualisation is a family of ray curves indexed by V⊂R2. This gives projections M↦π(M) in the now familiar way. This admits a large class of visualisations. We can define conditions that will make our visualisation 'nice'.
For example, if W⊂V is homeomorphic to a disc, then the image of W under the ray curves should be homeomorphic to a disc times the half line R>0.
We can also define funky visualisations that allow us to see round corners or past obstructions. Or, motivated by physics, we could make these rays be the projection of null geodesics onto a spacelike slice.
Let's now go for a different generalisation. Instead of visualising R3, we try to visualise 3-manifolds. We begin by discussing the video game Portal 2.
In Portal 2, you run around in a 3D bounded room and solve problems using a portal gun. The portal gun can shoot a blue portal and an orange portal, and going through one takes you to the other. Mathematically, they define an equivalence relation which turns the room from a 3-manifold with boundary to a 3-manifold with 'second-order' boundary. Second-order refers to the fact that the boundary of the portal is S1, a 1-manifold. To simplify things, we'll make portals occupy the entire wall that they're on.
Example: in a cubic room, shoot portals onto opposite walls. The ways the portals can join correspond to elements of D8, but only one preserves both the direction of gravity and orientation. This one turns the room into the interior of a torus.