Date: Tue, 8 Apr 1997 09:52:48 -0600 (MDT) From: [Permission pending] To: rusin@math.niu.edu Subject: The disco mirror ball Hi Dave -- Thanks for your informative sphere FAQ. In question Q4 you describe an approximation which looks alot like the mathematical description of a disco mirror ball (that is, if each point on the sphere represented the center of a single mirror). Do you know of any references which explore this technique further? Thanks. [Permission pending] > Q4. Can you give a quick approximation to a good distribution? > > Of course. Mark off K equal segments along a meridian using K-1 > points plus the two endpoints. These segments have length pi/K. If you > swing this around the sphere, you'll mark off K-1 circles. Now, in > spherical coordinates, these circles are the points with > ui=-pi/2 +i*(pi/K), where i = 1, 2, ..., K-1. They have radius cos(ui), > hence length 2pi*cos(ui). Thus you can divide them into about > [2pi*cos(ui)]/[pi/K] segments as long as the segments on the meridian. > So take Mi to be an integer near 2*K*cos( -pi/2 + i*(pi/K)) = > 2*K*sin( i * (pi/K) ), and mark off points that have u=ui and > v=j*(2*pi/Mi) for j = 1, 2, ..., Mi. This will give you > 2+Sum{i=1,...,K-1} M_i points on the sphere. > > To within an error less than K-1, this is the same as > 2+Sum 2*K*sin( i* pi/K ) = 2 + 2*K*cot( pi / 2K ). For large values of K, > this is about (4/pi) K^2. (With an area of 4pi on the whole sphere, > this leaves an area of (pi/K)^2 around each point, which is not > surprising since initially the points are about pi/K apart on a > roughly square grid). > > If you have the luxury of placing _approximately_ N points on the > sphere, begin with K an integer roughly equal to sqrt((pi/4)N), and > compute the number of points which would be placed with the method > above; you should get about N points. Here are the values I get > for the first few K using round() to convert to integers when necessary: > for K=1, N=2; for K=2, N=6; then for subsequent K (thru K=22), I get N= > 12,22,34,46,82,106,128,156,184,214,248,288,328,368,412,460,510,562,616,... > > Notice that the first layer around the north pole has about 2Ksin(pi/K) > points. For K large, this is about 2*pi = 6.28, so we would take M1=6, > and arrange the first circle of points in the "kissing pennies" pattern. > > This arrangement of points on the sphere has an obvious regularity, > but the points are not spaced as far apart as possible by any reasonable > measure -- for example, roughly sqrt(K/5) of the rows nearest the > equator will have about the same number of points in them, roughly in > a square-looking grid; clearly a honeycomb distribution will place the > same number of points with a greater spacing. This leads to question Q5. > > I have included as a separate file a BASIC program which carries out this > distribution. In view of the previous paragraph, this code can be > improved when N is large, but I'm not really sure of an optimal way to > do so, so I won't. ============================================================================== Date: Tue, 8 Apr 1997 11:11:16 -0600 (MDT) From: [Permission pending] To: rusin@math.niu.edu (Dave Rusin) Subject: Re: The disco mirror ball Dave Rusin writes: > >the center of a single mirror). Do you know of any references which > >explore this technique further? > > I'm not really sure what you seek. There is a large body of work discussing > optimal methods of spacing points around a sphere (of many dimensions); one > of the big players is Sloane (at Bell Labs or whatever they call it these > days). I think he has a recent book cataloguing some of this. Is that > what you want? I'm sorry, I should've been more specific. I'm trying to see if anyone has done extensive work on describing, mathematically, how mirrors would be placed on a disco mirror ball. (This seems very similar to what your FAQ describes in Q4.) Does Sloane's book go into any detail on this? So I'm really only interested in S^2, though info on other dimensions always makes for interesting reading. Thanks, [Permission pending] "Aha, that's a herbivorous quadruped of a dissimilar chromatic pigmentation." ============================================================================== Date: Tue, 8 Apr 97 13:24:49 CDT From: rusin (Dave Rusin) To: [Permission pending] Subject: Re: The disco mirror ball Ah, yes, sorry. No, I doubt Sloane covers this. We had a little Christmas tree ornament which resembled the Disco Ball. As I recall, the mirror segments were trapezoids arranged in circular rings more or less as I described in the "Q4" you quoted. It doesn't seem too hard in principle: 1. Decide on the nominal width of the trapezoids 2. Determine the true number to be placed on the equator by rounding (circumference)/(nominal width). Glue this many true squares to the equator. 3. Locate the next circle on which trapezoids would be glued (Probably OK to simply space it from the equator by an amount equal to nominal widths) 4. Determine number of trapezoids to be glued along this circle as in #2. 5. Continue in this way until reaching the poles. I guess the final ring of trapezoids will reduce to a ring of triangles with common vertex at the pole. Or you could glue one hexagon (say) right on the pole and make the last ring have 6 trapezoids on it. I suppose one could even work out in advance the dimensions of the trapezoids and their points of contact with the sphere, but it's not clear why this would be done: the vertices of trapezoids in adjacent rows do _not_ line up in this construction (here I have the _number_ of trapezoids per row decrease rather than the _widths_) so a DB thus constructed has a fair amount of "imperfection" already. You could make a true polyhedral sphere simply by using polygonal approximations to the lines of lattitude and longitude, but the trapezoidal faces in such a model will have markedly different widths near the poles from the widths near the equator; I don't believe that's the "standard model" DB which is presumably made from (nearly?) identical bits of glass glued to a spherical base. It might be interesting to construct a computer simulation of the two competing schemes of DBs, then simulate the reflection patterns from a typical dance-hall lighting array. I wonder if the difference would be noticeable? dave ============================================================================== Date: Tue, 8 Apr 1997 13:22:17 -0600 (MDT) From: [Permission pending] To: rusin@math.niu.edu (Dave Rusin) Subject: Re: The disco mirror ball Thanks for the feedback Dave. Your summary description pretty much matches my line of thinking. It's good to know I'm on the right track. Dave Rusin writes: > I suppose one could even work out in advance the dimensions of the > trapezoids and their points of contact with the sphere, but it's not clear > why this would be done: I think it would be quite interesting to know, given a spherical coordinate, which mirror contains that point? That is, if all the mirrors were numbered. I've made quite a lot of progress on this, but not quite done yet. When I finish it off, should I send it to you for inclusion in the FAQ? > the vertices of trapezoids in adjacent rows do _not_ > line up in this construction (here I have the _number_ of trapezoids > per row decrease rather than the _widths_) so a DB thus constructed has > a fair amount of "imperfection" already. Yeah. Calculating the average or maximum error gets tricky. Since we're dealing in integral units, there is a fair amount of it. Though it reduces to insignificance as the number of mirrors on the ball increases. > It might be interesting to construct a computer simulation of the two > competing schemes of DBs, then simulate the reflection patterns from a > typical dance-hall lighting array. I wonder if the difference would be > noticeable? You're on to me: my background is computer graphics, not math. :) The simulation idea is intriguing. Not sure when I'll have time to work on that. Ray tracing might work... Anyhow, thanks for the info and I'll let you know when I have a solution that works for me and might be useful to others. [Permission pending] "An individual of stunted intelligence and the negotiable currency in his possession soon take divergent routes."