...or an environmentalist.

left eye

right eye

The Mona Plane How go fastest?

Abstract. What’s the hardest math I’ve ever used in real life? Me, myself, directly - not by using a cellphone or a GPS device that somebody else designed? And in “real life” — not while studying or teaching mathematics? I use addition and subtraction daily, adding up bills or calculating change. I use percentages often, though mostly it is just “add 15 percents”. I seldom use multiplication and division: when I buy in bulk, or when I need to know how many tiles I need to replace my kitchen floor. I’ve used powers twice in my life, doing calculations related to mortgages. I’ve used a tiny bit of geometry and algebra for a tiny bit of non-math-related computer graphics I’ve played with. And for a long time, that was all. In my talk I will tell you how recently a math topic discovered only in the 1800s made a brief and modest appearance in my non-mathematical life. There are many books devoted to that topic and a lot of active research. Yet for all I know, nobody ever needed the actual formulas for such a simple reason before. Hence we’ll talk about the motion of movie cameras, and the fastest way to go from A to B subject to driving speed limits that depend on the locale, and the “happy segway principle” which is a the heart of the least action principle which in itself is at the heart of all of modern physics, and finally, about that funny discovery of Janos Bolyai’s and Nikolai Ivanovich Lobachevsky’s, that the famed axiom of parallels of the ancient Greeks need not actually be true.

...or an art historian...

I could be a mathematician ...

Goal. Find the least-blur path to go from Mona’s left eye to Mona’s right eye in fixed time. Alternatively, fix your blur-tolerance, and find the fastest path to do the same. For fixed blur, our camera moves at a speed proportional to its distance from the image plane:

100 112 131 167 252 576

103 112 103

Flatlanders airline route map

0 \sqrt{10} \sqrt{20} \sqrt{30} \sqrt{40} \sqrt{50}

The Happy Segway Principle

A Segway is happy iff both its wheels are

happy unhappy

Happy Segways Unhappy Segway

Picture credits. Mona: Leonrado; Al Gore: Futurama; Map 1: en.wikipedia.org/wiki/Greenhouse gas;
Smokestacks: gbuapcd.org/complaint.htm; Penguin: brentpabst.com/bp/2007/12/15/BrentGoesPenguin.
aspx; Map 2: flightpedia.org; Segway: co2calculator.wordpress.com/2008/10; Lobachevsky: en.
wikipedia.org/wiki/Nikolai Lobachevsky; Eschers: www.josleys.com/show gallery.php?galid=325;

Two parallels through one point

B

O?

A

θ

θ sin θ

The Brachistochrone

Bernoulli on Newton. “I recognize the lion by his paw”.

= \frac{1}{2}mv^{2}

mgh

The Least Action Principle. Everywhere in physics, a system goes from A to B along the path of least action.

With small print for quantum mechanics.

p3.y = p2.y + b*x3p;
x = p1.x-p2.x; y = p1.y-p2.y;
d1 = p1.d; d2 = p2.d;
norm = sqrt(x*x + y*y);
a = x/norm; b = y/norm;
x1p = a*x + b*y;
x0 = (x1p + (d1*d1-d2*d2)/x1p)/2;
r = sqrt((x1p-x0)*(x1p-x0)+d1*d1);
x1pp = (x1p-x0)/r; x2pp = -x0/r;
theta1 = acos(x1pp);
theta2 = acos(x2pp);
t1 = log(tan(theta1/2));
t2 = log(tan(theta2/2));
t3 = t1 + s*(t2-t1);
theta3 = 2*atan(exp(t3));
x3pp = cos(theta3);
d3pp = sin(theta3);
x3p = x0 + r*x3pp;
p3.d = r*d3pp;
p3.x = p2.x + a*x3p;

Ops used. +, −, ×, ÷, √ , cos, sin, tan, arccos, arctan, log, exp.

c ∼ 300, 000 c ∼ 250, 000

Fermat’s Principle

Happy camera-carrying Segways above the Mona Plane

y 0.9y

r

0.9r

The Mona Plane

Some further basic geometry also occurs:

The Bolyai-Lobachevsky Plane

Further Fun Facts. • In small scale, π^{H} → π^{E}. In large scale, π^{H} → ∞.

• The sum of the angles of a triangle is always less than π. In fact, sum+area= π, so the largest possible area of a triangle is π. • If your friend walks away, she’ll drop out of sight before you know it. • There are so many places just a stone throw away! But you’d better remember your way back well!

The Actual Code

θ′(t) = sin θ(t)

⇓

θ = 2 arctan e^{t}

Parametrization Finding the Centre