What's the minimum power k (algebraic solution)?

This is a follow-up looking for an algebraic solution to Scythian’s question:

http://answers.yahoo.com/question/index?qid=2908…

Consider the 3-4-5 triangle with vertices A(0,4), B(0,0), C(3,0)

For any given power 0 < k < ∞, there exists a (unique) point P(x,y) on the plane where the sum:

AP^k + BP^k + CP^k is a minimum.

(and by the way, I think it’s trivial to prove with inequalities that P must lie within ABC for k > 0)

For k = 2, P is at the centroid of ABC.

For k = 1, it’s where lines AP, BP, CP meet at 120° apart (Fermat Center of ABC, when all angles < 120°).

(I’m not aware of any other significant integer values of k and corresponding power-center names, and I did check http://mathworld.wolfram.com/KimberlingCenter.html…

For k → ∞, P falls on midpoint of CA for the trivial reason that it’s where the greater of AP, BP, CP is at a minimum.

For k → 0, P “snaps” towards vertex B, but it is not a smooth progression as k decreases.

At some critical value k~0.899484…, the local minimum in point B becomes equal with minimum inside the triangle, S = 6.16606

i.e. we’re considering the global minimum of the fn

S(x,y) = (x²+(4-y)²)^k/2 + (x²+y²)^k/2 + ((3-x)²+y²)^k/2

over the region 0 ≤ x ≤ 3, 0 ≤ y ≤ 4/3 (3-x)

Can you get an algebraic solution to this?

In particular, can you derive the ALGEBRAIC value of k_min where P snaps to/away from vertex B;

and the algebraic values of x,y @ k=k_min

When P snaps to B(0,0), we know: S(4,0) = 4^k (+ 0^k) + 5^k

I tried by taking partial derivatives and solving for 0, writing K=k/2 for clarity:

∂S/∂x = 2x* (x²+(4-y)²)^(K-1) + 2x* (x²+y²)^(K-1) + 2(3-x)* ((3-x)²+y²)^(K-1)

∂S/∂y = 2(4-y)* (x²+(4-y)²)^(K-1) + 2y* (x²+y²)^(K-1) + 2y* ((3-x)²+y²)^(K-1)

then taking a first-order approximation in the neighborhood of the solution x≈0, y≈0, 0.89946<k<0.899485,

(i.e. -0.55027 < K-1 < -0.550257 )

For homogeneity write y = mx, then it reduces to:

0.29848(x-3) – 0.21748 x = x* (1+m²)^(K-1) (x²)^(K-1)

0.21748(mx-4) – 0.29848 mx = mx* (1+m²)^(K-1) (x²)^(K-1)

Divide the bottom eqn by the top one to get:

(-0.08mx – 0.8699) / (0.89544x -1.1129) = m

-0.08mx – 0.8699 = 0.89544mx -1.1129m

Iterate on m and x alternately.

Step k upwards in small increments until you find out where this breaks down.

That’s not too algebraic or rigorous, so can you do any better?

If you get the algebraic solution you should also be able to explain the discontinuous snapping of P(k_min), since P(k) seems to behave smoothly everywhere else?

If anyone can plot graphs of the locus of P for range of k, that would be awesome, either as

P(x,y) or P(x,m) or P(ρ,θ) (for some choice of origin)

or surface plots of S(x,y) for various choice of k.

I don’t currently have access to MatLab or MayaVi but I’d love to see those plots.

Update:

[Nice plots, Dragan.]

Does anyone have an opinion on whether my algebraic first-order approach is good, bad or awful?

What values did anyone get for the critical location of P(x,y) at k=k_min?

Update 2:

Let’s see some algebra, people.

At least find me the minimum of S and the location of P(x,y) for k=k_min ~ 0.899484.

Just fix k at 0.899484 and that should simplify things.

I already offered one approach.

✅ Answers

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I don’t know if this has any significance, or if it is even close enough, but here’s what I noticed:

phi/2 approx=(k_min)^2

based on Dragan’s value

Edit:

Sorry, didn’t use algebra, but I got the minimums with Mathematica.

For k_min=0.89948393765067

S=13.9119

x=0.985956

y=1.26221

For k_min=Sqrt[GoldenRatio/2]

S=13.9112

x=0.985952

y=1.26218

http://i768.photobucket.com/albums/xx325/nothingav…

Edit2:

sqrt(x^2 + y^2)≈0.99*phi

Edit3:

Sorry, I forgot to divide k by 2, and that completely changed everything. With the k_min=0.899484, Mathematica is saying the min is B, or (0,0).

http://i768.photobucket.com/albums/xx325/nothingav…

Edit4:

Sorry, I was using the wrong syntax. I think these are the correct values for k_min=0.899484:

S≈6.1660642

x≈0.525638

y≈0.479288

where as S(0,0)≈6.16606432

http://i768.photobucket.com/albums/xx325/nothingav…

Edit5

I found a way to manipulate the value of k over a certain range, but I’m not sure how to upload it or if its legal.

http://i768.photobucket.com/albums/xx325/nothingav…

Edit6:

This is the best I could do. Here’s the fixed image:

http://i768.photobucket.com/albums/xx325/nothingav…

Here’s the video where 0.89<k<0.91:

http://s768.photobucket.com/albums/xx325/nothingav…

A hundred thanks to both of you, smci and Dragan K, for really following up on this problem. it started out to be an idle speculation about the locus of P as k varies and I quite naturally assumed that it’d smoothly travel to point B, and I was going to suggest a puzzle based on that. I was disappointed in finding out that the locus ends abruptly instead, jumping to B.

I’m not sure if I can figure out a more graphical way to illustrate this.

I don’t have algebraic solution, but there are some interesting surface plots.

For k slowly decrease from 1 there are minimum in P and local minimums in A, B, C.

Minimums in A and C are always higher then B, so we don’t consider them.

For k=0.94 there is a picture:

http://s524.photobucket.com/albums/cc321/Krejakovi…

Minimum in B is higher then minimum in P.

Very similar is for k=0.92:

http://s524.photobucket.com/albums/cc321/Krejakovi…

For k=0.90 minimums in B and P become approximately the same:

http://s524.photobucket.com/albums/cc321/Krejakovi…

(more accurate value is 0,89948393765067)