Saturday, May 10, 2014

Skew Cut Profiles

I recently was surprised (anew) at what happens when you cut a prism which has a uniform cross-section with a plane which is not orthogonal to the length of the prism. Especially if the plane must be described in terms of two angles. A cross-section through a square prism is (by definition) a square. If you make your cut at an angle, you will get a rectangle if you have a simple angle cut. But if you make your cut not only a miter, but also a bevel you end up with a parallelogram.

The simplest way to visualize this is to imagine a plane that intercepts the prism at a single edge first and has its steepest slope in the direction of the opposite edge:
The resulting section when viewed in the intersecting plane would be something like this:

where the blue square is the cross-section, the red parallelogram is the intersection of the above described plane through the square prism when viewed in that plane. (A few thought experiments of your own will probably serve you better than any further illustration of this phenomenon I could drum up at the moment.)

It is interesting to see the result in an I-beam shape if you use a 45° miter and 45° bevel:



As a matter of where this observation fits into mathematics generally, this is a projective or affine transformation. Such transformations preserve straight lines but they do not, in general, preserve right angles.

Aside: Convex Hulls

The fact that they preserve straight lines is a boon to the would-be-writer of a convex hull algorithm in an arbitrary plane since you can project the 3D coordinates onto a simple 2D plane and use the 2D algorithm (being careful to preserve or restore the other coordinate at the end). If you do so, you should probably choose which set of 2 coordinates to use for the algorithm, based on the direction of the plane. For example, if the normal of the plane has a larger z-coordinate than x- or y-coordinates, then the (x,y) pairs are best to use for the information to run the 2D convex hull algorithm on. There are two reasons to be picky about this:
  1. If the points fed to the routine are in thy yz-plane, you will get incorrect results (if you assumed the (x,y) coordinates should always be used).
  2. It may reduce the effect of rounding errors in the calculations (in terms of deciding whether points "near the line" of the hull are inside or part of the hull).
Code

The maxima code that I used to produce two of the illustrations in this post can be found here:
  1. DXFert1.lisp: contains some lisp routines for writing DXF files. Very basic text output for producing a bunch of DXF lines. Used by 2.
  2. Double-Skewed Cross-section.wxm: uses matrix multiplication to do rotations, uses the solve routine on simple linear equations, defines an I-beam cross-section from basic metrics, etc. For DXF file output, name the output file appropriately (use a full path name with fore slashes (/) to be sure it goes where you want, unless you know your maxima setup well) and identify the path to DXFert1.lisp.
The code files are free to use on an "as is" basis.  

See the Maxima project for download information for Maxima.

Saturday, May 3, 2014

What Do You Mean, "Equal"?

Mathematics makes use of the terms equal, equivalent, similar and congruent quite frequently—we could stretch ourselves out a little further and include analogous. They all have something in common: they refer to some kind of "sameness."  But they are not interchangeable.

By similar, when speaking of shapes in basic geometry, we mean that all of the angles between adjacent lines are the same between the two shapes being considered and those angles can be construed as being in the same order (say that three times fast).  We have to sharpen our pencils to deal with curved shapes, but we can in fact speak precisely about similarity between curves.  We might say, A is similar to B if and only if there is some function f that can be applied to A to turn it into B which is (strictly) a composition of translations, rotations, and (universal) scaling.

By congruent, when speaking of shapes in geometry, we are referring to similarity, but nix the scaling. Scaling not allowed. This is a more restrictive requirement. Two things might be similar, but not congruent.

The word "equivalent" is often used in the context of logic. (Logic, by the way, is what many math geeks really like about math—not mainly numbers! Mind you, it is generally hard to argue with the "logic" of numbers when they have been handled rightly. But numbers aren't really useful without logic being applied to their interpretation.) Two propositions are equivalent to each other if they have the same "truth table". When we say propositions P and Q are equivalent we mean that whenever P is true, Q is true, when Q is true, P is true, when P is false Q is false, when Q is false, P is false.

Equivalence is defined for other applications as well.  Equivalence classes are an overarching concept that can be applied in describing similarity, congruence, and various types of equality.  What constitutes equivalent has to be spelled out either by context, convention, or explicit definition for clear communication to occur.

Suppose I said triangle A is equal to triangle B? If you've taken a bit of geometry, you know that you are supposed to frown when people say rubbish like that.
  • Do you mean the triangles have equal area?
  • Do you mean the triangles have equal perimeter?
  • Do you mean the angles are all equal? (similar)
  • Do you mean the corresponding angles and sides are equal? (congruent)
When we talk about equality in whatever form, we should take some thought to what we mean. In well established fields, this is communicated by using the relevant term which has been defined in that field for the "kind of sameness" you wish to refer to. However, when the field is not so well established—or if the audience might not have the same understanding of the meaning, it's best to be explicit.

Application

What, for example, does it mean that "all men are created equal"? I do not intend to discuss the matter at length (as I do not pretend to have a complete answer) but only to point out that our discussions and claims of the properness of equality amongst human beings are fraught with the difficulty of differing understandings of what "equal" and "equality" refer to in such a discussion. For the time being I will content myself with one great, confusing ideal prevalent in my culture (western): meritocracy. (I mean the term in a loose sense—there probably is no such a thing, in reality, on planet earth.)

You may well be shocked at my so (apparently) deriding such a principle. To judge by merits is, in many circumstances, to be contrasted with partiality or prejudice. I heartily recommend the phrase "all men are create equal", but on what basis? Surely merit will not undergird this recommendation. Merit is a basis for distinguishing, which I also recommend!

Perhaps you do not believe that the two issues (merit vs inherent equality) are commonly confounded. I am satisfied that they are. When I hear the merits of representatives of one group (demarcated by ethnicity, gender, or whatever) touted for the vindication of that group, I normally listen quietly, but I do not need such things. Nor do I have much appreciation for the comparing of statistical averages of various metrics in an effort to prove equality. I realize such things are sometimes done in an effort to silence and defend against bigotry, and I do not deride that intent. But I have also seen the defended group touted implicitly as superior (better at this, better at that), in which case I take it as a betrayal of a pretense—appearing to seek equality but with dominance of some kind as the true objective. I do not see inherent equality as based on merit. My key word here, as you may guess, is "inherent". There may be value in recognizing differences, but merit is the "poor man's" rubric for deciding the matter of inherent equality.

Whatever you think of these things, you'll have to deal in your own mind with what you mean by "equal" and what your basis for it is. But I'll tell you where I start:
So God created man in his own image, in the image of God created he him; male and female created he them. (Genesis 1:27)