Wikipedia:Reference desk/Archives/Mathematics/2014 March 5

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March 5

Where are the constraints violated?

It is well known that, using only a compass and an unmarked straightedge, it is impossible to trisect an arbitrary angle.

Here are three methods to trisect an arbitrary angle - where are the constraints violated?

Let DCE, or gamma, be an arbitrary acute angle. (An arbitrary obtuse angle can be made acute by lopping off one or multiples of 90 degrees, and an angle of 90 degrees is trisectable using a compass and unmarked straightedge.)

Let line segments CB and BA have the same length as DC, both CB and BA lie on the same side of EC as does point D, and point A is constrained to lie on the extension of EC.

One method is to move point A (which moves point B along the arc of a circle) along the extension of EC, until the extension of BA strikes point D. The angle BAC (=DAC), or alpha, is 1/3 of gamma (proof shown below).

A second method is to draw the extension DBA' (where A' is also on the extension of EC), and to move point A (not A' !!) along the extension of EC, until line BA is colinear with DBA'.

A third method is to initially let segment BA=DC (as above) but to let CB have arbitrary length, and to change the length of CB by moving point B (which moves point A along the extension of EC) until CB=BA (or CB=DC).

(Proof of trisection (for all three methods):

Angle BAC of isosceles triangle ABC is equal to alpha, so that the other isosceles angle ACB is equal to alpha, so that the third angle ABC is 180-2*alpha.

Angle ABC (=180-2*alpha) plus angle CBD is a straight angle, so that angle CBD is 2*alpha, as is the other isosceles angle BDC of triangle BCD, thus the third angle BCD of triangle BCD is 180-4*alpha.

Angle DCE plus angle BCD plus angle ACB = straight angle, or gamma+(180-4*alpha)+alpha=180, or gamma-3*alpha=0, or alpha=gamma/3.)

My question: Where are the constraints (of using only a compass and an unmarked straightedge) violated?Bh12 (talk) 00:05, 5 March 2014 (UTC)

First, compass and straightedge constructions don't involve moving points. You're restricted to a very specific set of operations: given two points, you can draw the line containing them; given two points, you can draw the circle centered at one and containing the other; given intersecting lines/circles, you can draw the point of their intersection.

Also, while I don't understand what you're saying with your second method, in your first and third method, B = D. This causes your later argument to break down.

Ignoring that, I don't see why you claim DCE + BCD + ACB = 180. It seems that DCE - (BCD + ACB) = 0. Or perhaps BCD - (DCE + ACB) = 0, depending on the relative positions of B and D.--80.109.80.78 (talk) 06:23, 5 March 2014 (UTC)

You might be interested in our article on Neusis construction which allows methods giving accurate trisections. Dbfirs 08:33, 5 March 2014 (UTC)

Constant second quotients property

We know that some functions have special properties on how $x$ and $f(x)$ are related:

Add-add property

Linear functions have an add-add property. This means that adding a constant to $x$ will add a constant to $f(x)$.

Add-multiply property

Exponential functions have an add-multiply property. Adding a constant to $x$ will multiply $f(x)$ by a constant.

Multiply-add property

Logarithmic functions have a multiply-add property. Multiplying $x$ by a constant will add a constant to $f(x)$.

Multiply-multiply property

Power functions have a multiply-multiply property. Multiplying $x$ by a constant will multiply $f(x)$ by a constant.

Constant second difference property Quadratic functions have a kind of property called constant second difference. Is there a similar kind of function that has a power called constant second quotient?? Please reveal a general equation for this kind of function. Georgia guy (talk) 20:52, 5 March 2014 (UTC)

When you say "constant second difference" do you mean as in (x+1)^2 - x^2 = 2x + 1, 2(x + 1) + 1 - (2x + 1) = 2? I'm not sure what exactly "constant second quotient" is as a property, could you elaborate? Are these your terms, or are they from some source?Phoenixia1177 (talk) 05:55, 6 March 2014 (UTC)

Constant second difference is a property of sequences. You make the difference sequence, and then the difference sequence of that, and you get a constant sequence.--80.109.80.78 (talk) 06:03, 6 March 2014 (UTC)

We can make constant second quotient easily enough by working backwards. Let's start with the quotient sequence: $a_{i+1}/a_{i}=c$. So $a_{i}=c^{i}a_{0}$. Then let's make the original sequence: $b_{i+1}/b_{i}=a_{i}=c^{i}a_{0}$. So $b_{i+1}=c^{i}a_{0}b_{i}$. So $b_{i}=c^{i(i-1)/2}a_{0}^{i}b_{0}$. There's your general form.--80.109.80.78 (talk) 06:03, 6 March 2014 (UTC)

A slight modification of above. Note $a_{0}=c^{\alpha }$ for some $\alpha$ and $b_{0}=c^{\beta }$ for some $\beta$. Hence

$b_{i}=c^{\,i^{2}/2-i/2}c^{\alpha i}c^{\beta }=c^{\,{\frac {1}{2}}i^{2}+(\alpha -{\frac {1}{2}})i+\beta }=e^{Ai^{2}+Bi+C}$

For some A, B, C. Basically the exponential of any quadratic function will have the constant second quotient property. (I've reforatted the question to remove sub-headings, hope thats OK)--Salix alba (talk): 06:45, 6 March 2014 (UTC)

Perhaps the OP is thinking of Finite differences? A math-wiki (talk) 14:49, 9 March 2014 (UTC)

Turning radius

Our article on the turning radius of a vehicle doesn't give a formula :( I'm trying to work out the turning circle (i.e. the turning diameter) of a car. This website (http://www.ehow.co.uk/how_7225784_calculate-turning-circle.html) does:

(TrackLength/2)+(Wheelbase/sin(TurningAngle))

Other websites say it is

2*(Wheelbase/sin(TurningAngle))

Which do you think is correct? I just need an approximate. Thanks! 143.210.123.60 (talk) 21:02, 5 March 2014 (UTC)

Probably the former is intended to give the radius and the latter the diameter of the circle. The website refers to the average steer angle which I take to mean the average between that for the wheel on the inside of the bend and for the wheel on the outside (the wheel on the outside travels around a bigger circle of course). So (wheelbase / sin(average steer angle)) is an estimate of the turning radius for the mid-point of the axle, and adding half the axle length gives the turning radius for the outside wheel. For the latter expression, the turning angle is not precisely defined (whether for the inside wheel, the outside or the average), but in any case, the factor of 2 means it gives a diameter rather than a radius. --catslash (talk) 01:37, 6 March 2014 (UTC)

You are simply AWESOME! Thanks for explaining that to me; it makes so much more sense when someone explains it. 143.210.123.113 (talk) 14:08, 6 March 2014 (UTC)

Also note that, as a practical matter, you may not want to turn at the tightest radius possible. Many cars will make a squealing noise at the sharpest turn possible, and that sure doesn't sound like it's good for the car. I will mark this Q resolved. StuRat (talk) 14:29, 6 March 2014 (UTC)

Resolved

StuRat (talk) 14:30, 6 March 2014 (UTC)

See Ackermann steering geometry (also steering) --catslash (talk) 17:46, 6 March 2014 (UTC)