Mœud deux

August 7, 2018

Pairing functions are fun. Last week, we had a look at the Cantor/Hopcroft and Ullman function, and this week, we’ll have a look at the Rosenberg-Strong function—and we’ll modify it a bit.

Read the rest of this entry »


Mœud

July 31, 2018

Pairing functions are used to reversibly map a pair of number onto a single number—think of a number-theoretical version of std::pair. Cantor was the first (or so I think) to propose one such function. His goal wasn’t data compression but to show that there are as many rationals as natural numbers.

Cantor’s function associates pairs (i,j) with a single number:

…but that’s not the only way of doing this. A much more fun—and spatially coherent—is the boustrophedonic pairing function.

Read the rest of this entry »


The Well-Tempered Palette (Part 2)

July 17, 2018

Last week, we’ve had a look at how to distribute maximally different colors on the RGB cube. But I also remarked that we could use some other color space, say HSV. How do we distribute colors uniformly in HSV space?

Read the rest of this entry »


Paeth’s Method (Square Roots, Part VII)

March 13, 2018

In Graphics Gems [1], Paeth proposes a fast (but quite approximate) method for the rapid computation of hypotenuse,

\displaystyle h=\sqrt{x^2+y^2}.

The goal here is to get rid of the big bad \sqrt{} because it is deemed “too expensive”—I wonder if that’s still actually true. First, he transforms the above equation:

Read the rest of this entry »


Random Points on a Sphere (Generating Random Sequences III, Revisited)

February 27, 2018

While searching for old notes—that I haven’t found anyway—I returned to an old blog entry and I thought I was kind of unsatisfactory, with be best part being swept under the carpet with a bit a faery dust, and very handwavingly.

So let’s work-out how to uniformly distribute points on a sphere in a more satisfactory fashion.

Read the rest of this entry »


Square roots (Part VI)

February 20, 2018

I’ve discussed algorithms for computing square roots a couple of times already, and then some. While sorting notes, I’ve came across something interesting: Archytas’ method for computing square roots.

Archytas’ method is basically the old Babylonian method, where you first set

a=1,

b=n,

and iterate

\displaystyle a'=\frac{a+b}{2},

\displaystyle b'=\frac{n}{a'}=\frac{2n}{a+b},

until desired precision is achieved (or the scribe is exhausted).

Read the rest of this entry »


Unary numbers.

February 13, 2018

A positional number system needs a base that is either greater than one, or smaller than minus one—yes, we can have a negative base for a number system. The system, however, seems to break down if the base we chose is base 1.

If the base is 1, then there are no permissible digits since the digits d, in a base b system, must be 0\leqslant{d}<b. But we can still represent numbers using just 1s. That's the unary numeral system, and numbers are just represented as repeated 1s. 15? Fifteen ones: 111111111111111. Operations? Not very complicated, just… laborious.

Read the rest of this entry »