golden ratio is indeed the right answer, but not achieved at the point where n is minimal. For an obvious reason: for any parameterized P(m(t),n(t)) representation, when n'=0, m'<>0 and n>0 therefore (m/n)' = m'n-n'm/n^2 = m'/n <> 0. Hence ratio m/n cannot be extremal when n is.

1st method: O being the center of the circle, take P such as cos(angle(OP,horizontal)) = 2/3.

2nd method: draw a line from bottom right corner through the point where n is minimal (as constructed previously). This line crosses the circle in a second point which is the requested point.

Unfortunately, all these results come from differential calculus and trigonometry. Any idea for a geometric justification?

(1+sqrt5)/2

ReplyDeleteJoin the centre of the circle to the top right corner, then n is the shortest and m/n equals the Golden Ratio (1+sqrt5)/2.

ReplyDeletegolden ratio is indeed the right answer, but not achieved at the point where n is minimal. For an obvious reason: for any parameterized P(m(t),n(t)) representation, when n'=0, m'<>0 and n>0 therefore (m/n)' = m'n-n'm/n^2 = m'/n <> 0. Hence ratio m/n cannot be extremal when n is.

ReplyDelete1st method: O being the center of the circle, take P such as cos(angle(OP,horizontal)) = 2/3.

2nd method: draw a line from bottom right corner through the point where n is minimal (as constructed previously). This line crosses the circle in a second point which is the requested point.

Unfortunately, all these results come from differential calculus and trigonometry. Any idea for a geometric justification?

bleaug

To simplify,

ReplyDelete* The center point of the semicircle is (0,0).

* The square size is 2 x 2.

m/n is represented by:

sqrt((2-cos(a))^2+(-1-sin(a))^2)/sqrt((2-cos(a))^2+(1-sin(a))^2); (-pi/2 < a < pi/2)

Then I got the max value of m/n as follows.

Max: (1 + sqrt(5))/2 (= golden ratio) at a = 0.8410..radians

When m/n reaches the max, the coordinates of P is ((2/3,sqrt(5)/3), assuming the center of the semicircle is (0,0).

ReplyDeletem = PB = approx 2.196

n = PA = approx 1.357

m/n = approx 1.618 (golden ratio)

∠APB = approx 63.45 degrees