## Cheating a little bit, and teasing a shaded surface, out of a Computer Algebra System.

One of the subjects which I posted about some time ago was, how easy it is for a Computer Algebra System – a CAS – to output a single variable, in order to colour an Iso-Surface, and how hard it is in contrast, to output a normal vector, from whatever Geometry Shader computes the Iso-Surface, such that this normal vector can be used to shade the surface, in a later Fragment Shader invocation.

What needs to be done in ‘3D Game Design’ and (other) ‘CGI’, is essentially that.

But, given that a CAS can be used both, to plot a 3D surface, as well as, to define what the colour-range of this surface is supposed to be, a bit of a trick can be used, to bypass the need actually to compute a normal vector, but to achieve an equivalent result. And this posting will begin with an example which is slightly simpler, than what my earlier posting had assumed. Iso-Surfaces tend to smack of ‘implicit’ functions, while this example is going to start with an ‘explicit’ plot, in which (X) and (Y) are parameters of the function, but where it was already easy to achieve, that a single (Z) value results, such that (X, Y, Z) are in fact the coordinates plotted.

Because we have a Computer Algebra System in the first place, for continuous functions, it’s easy to compute the derivative with respect to one of the parameters. That derivative can be used, just to modulate the brightness of the surface. The simplest example is shown blow:

(Updated 7/10/2020, 6h10… )

## A simple 3D animation created with Maxima.

One of the things which I find myself doing quite often is, to be undertaking some sort of task on the computer(s), that I know is possible, but, not knowing in advance what the correct syntax and semantics are, to perform this task. This tends to take me on some sort of search on the Web, and I’ll find that other people have undertaken similar tasks, but not, a task with the same combination of parameters, as my task.

Thus, Web pages can be found according to which 2D animations have been created using a free, open-source Computer Algebra System named “Maxima”. Other Web pages may explain how to create various types of (static) 3D plots. But there may just be lacking examples out there, on how to create the 3D plot, but to animate it.

Using Maxima, there may be more than one way, such as, to keep refreshing the 3D plot over a time interval. But I find that such solutions tend to be second-rate, because of their use of busy-wait loops, as well as the possibility that they may otherwise be wasteful of computing power. I think that the best way, perhaps, to get Maxima to generate an animated 3D plot, could be, in the form of an animated GIF File (of course, as long as there isn’t an excess of frames to this animation).

Thus, the recipe that seems to work is as such:


load(draw)$scenes: []$

for i thru 20 do (
scenes: append(scenes,
[gr3d(explicit(sin(%pi*(x+(i/10)))*cos(%pi*y),
x, -1, 1, y, -1, 1))]
)
)$draw( delay = 10, file_name = "wavy", terminal = 'animated_gif, scenes )$



The script outputs a file named ‘wavy.gif’ in the same folder, as whatever folder it was originally stored in. In some cases, the GIF File may appear in the user’s home directory, or even, in a temporary directory that’s difficult to find, unless the user also gives a full-path name for the file.

And, this is the GIF File that results:

Caution:

My most recent posting had to do, with a version of Maxima that had been ported to Android. The example above will not work with that version of Maxima. In fact, I can really only be sure that this feature works under Linux, which is the O/S that Maxima was mainly designed to run on. Any directives to ‘plot()’ or ‘draw()’ open a separate GNU-Plot window, which behaves in the predictable way under Linux, including the user’s ability to rotate the 3D plots interactively. AFAIK, commands to change to a non-default ‘terminal’ (for drawing and/or plotting) will fail on other platforms.

But, There is also a Windows or Mac alternative to using this platform, which mainly presents itself in the application ‘wxMaxima’. Here, the functions ‘wxdraw2d()’ and ‘wxdraw3d()’ replace those that open a separate window, and both embed their results in the wxMaxima worksheet. In order to make this more versatile, wxMaxima also offers the functions ‘with_slider_draw()’, ‘with_slider_draw3d()’, ‘wxanimate_draw()’, and ‘wxanimate_draw3d()’.

Potential ‘wxMaxima’ users will find the documentation for how to script that Here.

(Updated 7/06/2020, 15h35… )

## A better way to use “Maxima” to solve cubic equations…

One of the facts which I’ve pointed out in earlier postings was, that if I give the free Computer Algebra System (‘CAS’) named ‘Maxima’, a cubic equation to solve, as in, to find the exact analytical solutions to, it will fail to produce intelligible output, if the usage was naive.

More specifically, cubic equations exist that have 3 distinct, real, irrational roots, and which a ‘CAS’ should be able to solve, just because their general solution is publicly known. That solution boils down, to deriving a second cubic, which is called a ‘depressed cubic equation’, and then performing a trigonometric substitution. (:1)

A fact which I’ve also known for some time is that, especially if a person is using a free or open-source CAS, then in some cases its behaviour has not been made particularly user-friendly, in that work needs to be done by the user, to set up his or her problem for the ‘CAS’ to solve. This latter observation casts a shadow of doubt, over the question of whether a ‘CAS’ will ultimately lead experienced Mathematicians to new discoveries in Algebra, or whether this can only reduce the workload in certain situations.

In this posting I’m going to show, how ‘Maxima’ can be coerced into giving correct answers, by users who know how. What I tend to use is a Graphical Front-End to ‘Maxima’, that is itself named ‘wxMaxima’, but which has equal capabilities, except for the abilities to typeset its solutions, as well as to export its Worksheets to PDF as well as HTML format, using LaTeX.

The following embedded worksheet will only display properly in the reader’s browser, if

• The reader has allowed JavaScript from my blog to run on his browser, and
• The reader has also allowed JavaScript to run from a CDN named ‘mathjax.org’.

What’s observable here is the fact that the package ‘odes’ can be loaded, which is mainly used to solve Ordinary Differential Equations, and that afterwards, the function ‘solvet()’ can be used, even to solve certain polynomials – better than what Maxima can solve on its own, with the built-in ‘solve()’ function. (:3)

(Updated 6/14/2020, 0h30… )

## How 3D-plotted, implicit functions are often inferior, to ISO-Surfaces rendered for 3D Gaming.

One of the subjects which I revisited in recent weeks has been, that either Computer Algebra Systems, or other numeric toolboxes, may plot functions. And a fact that should be pointed out is, that to plot a function, either as a 2D or a 3D plot, is always numeric, even if it’s being offered as part of what a ‘CAS’ can do (a “Computer Algebra System”). And so, a subcategory of what is sometimes offered, is a 3D plot, of an implicit function, kind of like this one:

This is a plot, of complementary hyperboloids, which are the 3D counterparts to 2D hyperbola.

What some people might just wonder is, how the refined toolbox works, that plots this type of implicit function. And one way in which this can be done, is by generating an ISO-Surface, which is a derived mesh, along which a Density that has been computed from X, Y and Z parameters, crosses a threshold-value, which can just be named (H) for the sake of this posting.

And, in turn, such an ISO-Surface can be computed, by using the ‘Marching cubes algorithm‘. If it gets used, this algorithm forms a geometry shader, which accepts one Point as input topology, and which outputs a number of triangles from (0) to (4).

The question which this posting poses is, whether the mesh which is output by such an algorithm, will always include vertex-normals. And the short answer is No. Applications exist, in which normals are computed, and applications exist where normals are not computed. And so, because some users are used to high-end gaming, and used to seeing shaded surfaces, which can only really be shaded if normals have been made available to a fragment shader, those users might find themselves asking, why Mathematical plotting algorithms might exist, which never compute real normals.

(Updated 5/07/2020, 16h15… )