A tetrahedron has four vertices and four faces.
A cube has eight vertices and 12 triangular faces.
I used pure red $(1, 0, 0)$ and pure blue $(0, 0, 1)$.
In the first case we simply rotate the camera about its Z axis. Because of the transposed convention, this rotation is opposite what we expect.
$$R = \begin{bmatrix}0 & 1 & 0 \\-1 & 0 & 0 \\0 & 0 & 1\end{bmatrix}\text{, } t = \begin{bmatrix}0 \\ 0 \\0 \end{bmatrix}$$
In the second case we simply move the camera away from the subject along its Z axis. This creates a zoomed out effect.
$$R = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1& 0 \\0 & 0 & 1\end{bmatrix}\text{, } t = \begin{bmatrix}0 \\ 0 \\3 \end{bmatrix}$$
In the third case we shift the object left and down, so we are looking at it from above and to the right.
$$R = \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix}\text{, } t = \begin{bmatrix}0.5 \\ -0.5 \\0 \end{bmatrix}$$
Finally, we rotate the camera right about the up axis. Since the camera is not at the origin, we need to compensate for this rotation by translating right and back.
$$R = \begin{bmatrix}0 & 0 & 1\\0 & 1 & 0 \\-1 & 0 & 0\end{bmatrix}\text{, } t = \begin{bmatrix}-3 \\ 0 \\3 \end{bmatrix}$$
It is computationally more expensive to generate a mesh in the manner we described because rather than simply generating points and the surface of the mesh, we need to generate them for a dense grid in the space the object occupies. However, once a mesh has been generated, there a a number of advantages to it. The first is there is an explicit notion of ray-object intersection. This can be used to texture a mesh with observations from a camera while respecting occlusions or check whether a point is in- or outside a mesh. Furthermore, in regions where curvature is low, a mesh can be represented by only a few vertices compared to the number of points it would take to densely sample the region. This holds true for rendering especially, where a point cloud must be sampled very densely to obtain a solid-looking representation. One final downside to meshes is they cannot allways be created easily. In data-driven applications, where point clouds were observed, it can be very challenging to fit a watertight mesh to the (potentially noisy) observations. This renders the benefits of meshes moot, if they cannot be generated for a given application.
I was interested in volume rendering so I explored volume rendering a simple scene. I took the distance to the center of a torus, inverted it, and appropriately scaled it so the structure was visible in the volumetric render. Below is a visualization of the result.
