Wikipedia:Reference desk/Archives/Mathematics/2008 January 2

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January 2

Angular Diameter

How is the equation for angular diameter derived? Imaninjapirate^{talk to me} 01:35, 2 January 2008 (UTC)

Try drawing a triangle whose vertices are at either extent of the object in question (for the first equation, you can assume that the object is a flat circle facing the observer, so that the base of the triangle is the diameter of the circle), and at the observer. The triangle should be isoceles, and so the line from the observer to the midpoint of the object will be a right-angled triangle, and will bisect the angle at the observer. From that you can use the definition of tan to express the half-angle in terms of the various lengths involved, and then double it to get the required formula.

For the second equation, instead of letting the object be flat, assume it's a perfect sphere, and draw the lines as follows: (1) A tangent to the sphere, passing through the observation point, (2) The radius of the sphere from the point of contact of the tangent, and (3) the line from the centre of the sphere to the observer. Use the fact that a tangent to a circle is perpendicular to the radius at the point of contact, and again some basic trigonometry should give you the half-angle formula. Confusing Manifestation(Say hi!) 02:33, 2 January 2008 (UTC)

Ahhh.. I see. Thanks for the quick response! Imaninjapirate

^{talk to me} 03:09, 2 January 2008 (UTC)

Minimum Number of Meets in a Tournament

Say you are the administrator of a tournament with x number of teams. You can fit y number of teams in each meet. There are z number of rooms to hold meets in. What's the minimum number of meets needed to establish one winner (assuming all losing teams are elimanated)? Also, what's the minimum number of rounds needed?

This isn't a homework question. I was helping my mom design a quizbowl tournament and it got me to wondering. Thanks! 160.10.98.106 (talk) 15:52, 2 January 2008 (UTC)

You'll have to clarify a little. What does "You can fit y number of teams in each meet" mean? Does it mean you can have a single game in which y teams participate and one of them comes out as the winner? Does each such game take a fixed amount of time to finish? We are trying to minimize the number of "meets" - does that mean that each time unit in which a game takes place in every one of the rooms counts as one "meet"? -- Meni Rosenfeld (talk) 16:04, 2 January 2008 (UTC)

(After edit conflict) If I understand the question correctly, then only one winning team in each meet survives, so each meet eliminates y-1 teams (or possibly less, if there are less than y teams in the meet). To find a single overall winner from x teams, you must eliminate x-1 teams. So your first guess at the minimum number of meets might be (x-1)/(y-1) - but what do you do if this is not a whole number ? Hint: see floor and ceiling functions.

To work out the minimum number of rounds, notice that you can handle up to y teams in just 1 round, up to y² teams in 2 rounds, up to y³ teams in 3 rounds etc. So the minimum number of rounds is given by the next power of y that is greater than or equal to x. You can write an expression for this in terms of x and y using logarithms (and the ceiling function again). I haven't used the number of rooms, z, because this does not affect the number of meets or rounds - although it might limit how many meets can take place at the same time.

Using the minimum number of meets is not always the best way to organise a tournament. For example, if you have 13 teams with up to 4 teams in a meet, then you could organise a tournament with just four meets, with three 4-team meets in the first round followed by a final, and one lucky team getting a bye straight to the final. However, it is probably fairer to have four meets in the first round of the tournament - three 3-team meets and a 4-team meet - and so a total of five meets altogether. Gandalf61 (talk) 16:23, 2 January 2008 (UTC)

Roulettes

Hello everyone, I've made this animation for the page on roulettes, but I'm not entirely sure it's correct. The created (red) curve is a tractrix, is that right ?

By the way, could someone explain how to find equations for the curve that is rolling on the other and for the curve created ? I sort of messed my way through it with arctangents... The only thing explained on the Roulette article is the parametric equation of the curve (but I find it not very convenient to use complex numbers here). The information on roulettes on the internet seems to be very sparse.

Thanks. -- Xedi (talk) 17:37, 2 January 2008 (UTC)

To be honest, it took me some time to figure out what kind of roulette you were talking about.

I think the term tractrix may have a more restricted meaning that the one you are implying here, but the similarities in the inherent sense of the curves' derivations are clear.

If I were in your shoes, I would take a look at the article on conics, and study the Cartesian expression of parabolae (I assume these are the conics depicted in your animation) to tackle the derivation of the red curve. That said, it may not be a simple task at all. Pallida  Mors 19:45, 2 January 2008 (UTC)

This external site gives insights into rotation and translation of conic sections given in Cartesian coordinates. Pallida  Mors 19:59, 2 January 2008 (UTC)

Yes sorry, it's one parabola rolling on another.

The thing is, I sort of hacked my way through the animation and have an overly complex description of the red curve :

$\left(t-t\cos(2\arctan(4t))-t^{2}\sin(2\arctan(4t))\quad ,\quad t^{2}+1-t\sin(2\arctan(4t))+t^{2}\cos(2\arctan(4t))\right)$

Anyway it's not really bothering me, I'd just like confirmation it is actually a roulette and I have got my equations right, so I can put it in the article that is quite lacking an image to show what a roulette is.

Thanks. -- Xedi (talk) 22:43, 2 January 2008 (UTC)

Those parabolas are the same size, as in same leading coefficient, right? If so, the red curve is tracing out the reflection of the blue parabola's vertex across each line tangent to the blue parabola. You can let t be a parameter and find the reflection of the vertex across the line tangent to the curve at (t,t²+1). I tried it, and got x and y as rational functions of t:

$x={\frac {4t^{3}}{1+4t^{2}}}\,,\,y={\frac {1+2t^{2}}{1+4t^{2}}}$

(You may want to double check my algebra.) Those might or might not be the same as your equations... you could have a computer graph both, and see whether they line up, unless you know some clever trig substitutions that will make the equivalence clear. If they both produce the same curve, then it's probably correct, because it's unlikely we'd make the same error with totally different methods. -GTBacchus^(talk) 23:09, 2 January 2008 (UTC)

The calculation, by the way, is much easier to do if you put the vertex at the origin, and then just translate your y-coordinate up by 1 when you're finished. -GTBacchus^(talk) 23:35, 2 January 2008 (UTC)

If you put t=1/4 in your equation, you get the point (3/16,13/16), which lies on the line y=1-x. If you intersect that line with my equation, you get two solutions, one at the origin, and one at the point (1/4,3/4). I don't think they're quite the same curve, although they're close. I think it still illustrates what a roulette is, anyway. -GTBacchus^(talk) 00:02, 3 January 2008 (UTC)

Aha. You've got yourself a Cissoid of Diocles there. Your picture's better than the one in that article now. -GTBacchus^(talk) 00:40, 3 January 2008 (UTC)

Oh yes, indeed, roulette of a parabola on another, that's it. Yes, I remember now. (And that image is incredibly hard to understand !). By the way, the equation I wrote down here is wrong I think - the graph of it doesn't seem to agree with my picture. Anyway, thanks a lot, I suppose my animation is correct then. -- Xedi (talk) 00:43, 3 January 2008 (UTC)

That's it, uploaded another image (had to reverse top and bottom because of the lengthy explanation underneath.). Hope everything is good now ! -- Xedi (talk) 00:53, 3 January 2008 (UTC)