First attempt to try a catenary as a target shape.
(let me use this post like a notebook, so including errors and wrong directions)
Don’t know… maybe.
This shape is thick 1.2 and large 2.0. Almost there as a real stone in cm.
Let me try to work on ears to increase the slope at around |x|=0.75 with a square root function.
Taking apart proportions that can be distorted by axis scales not even, it seems to me that square root function can work. Now the game is to find the right x for the junction.
Let me check the derivative of both curves to check for which x value they are equal.
There are two points that I should investigate.
The first one is too far from the edge, so I will focus on the second one.
To better see where the derivatives are equals it is more useful to plot the difference between the two and equal it to zero:
To find the exact value we must solve computationally a system of two equations:
Solution over the reals:
x1 = 0.450972
x2 = 0.766747
x2 value confirms that 0.75 can be a good candidate for the junction of the two curves but doesn’t means that x2 itself is the exact point of the junction. In fact, for x2 the derivatives are the same but y2 for this value of x is not the same for the two functions:
y2(x2) for catenary = 0.253150…
y2(x2) for square root = 0.255527… so there is still a gap. It’s small, yes (2 thousandths of a millimeter) but it is a gap.
Let zoom-in the original picture in the x2 point with the two functions running close one to the other (same derivative) but with a gap, just to make this visible.
I want remove the gap.
Someone can think that it would be enough scale a little bit the square root function to adjust this curve to the point where the two functions have the same derivative. But stretching one curve with a scale factor would change also the derivative in the point we found experimentally. It doesn’t work.
I should arrange a system of two equations and two variables to find the exact values of the variables (if they exist) that fix this gap.
First of all, which variables I must choose?
One is the x2 point (the x value for which the difference of the two derivatives is equal to zero) since the value I found is a mere experimental value found looking to a plot that seemed to be working for my purposes.
The other variable is the parameter of the square root function that allows me to adjust its shape with the objective to create the junction in the right place without any gap. Since the 0.9 constant determines the intersection with the x-axis, I don’t want to change it because I want a stone 1.8 cm wide. It isn’t the candidate. So, the only other parameter I can play with is 0.7 which multiply the square root. This is the y scale factor of such function so it is what I’m looking for…
Secondarily, which equations should I put in the system?
The first equation is simple, the difference between the two derivatives equaled to zero.
The second is the difference between the two equations equaled to zero because this is equivalent to say that the gap doesn’t exist anymore in correspondence with x2 (the solution of x in the first equation of the system).
Let me start with the second equation because it can be written straightforward. We need simply to equal it to zero.
For the first equation, let me write the plain form for the derivative and equal it to zero:
Now let see if WolframAlpha can find a solution for this system considering x and a variables.
Computational time exceeds (I don’t have a pro account) so I must invent something to reduce the computational effort.
Let me simplify the cosh and sinh part of our equations, since I think this is the more complex function to manage for the engine.
We are interested in a good representation of such functions around the point x=0.75, so let expand those functions in two Taylor series neglecting orders superior to 5:
we will check later if the approximation is good enough in x2 (solution of our system).
Now I can write the system using this simplified versions of the hyperbolic functions and let the engine work for me:
Now it works and the numerical solution found by WolframAlpha is:
So, the parameter a has changed in order to cancel the gap between the catenary and the square root in the junction point as expected. But the point for which the derivatives are the same now has moved forward respect to x2=0.75 envisaged initially. Now we have a solution with x2=0.770978.
Let see if the junction is smooth enough.
y2(x2) for catenary is 0.24907(9)02584617236
y2(x2) for square root is 0.24907(6)43015856845
we got a pretty good approximation. The error is of the order of 3 millionths of a centimeter. To be precise 26 nanometers. Or if you prefer the gap in the junction is meters.
Ok, I like it.
Let see again the two curves in a closeup just in the junction point as we made earlier:
So we checked even visually that the curves fit very well at x=0.77.
Now we can take a look to our stone’s shape:
I cannot find the way to represent the plot with even axis units using WolframAlpha, so the shape appears distorted in the picture above. Using Google instead, I can represent better the real shape that I have obtained with the given parameters.
If we assume the axis units like real centimeters, we have to do here with a stone 12mm thick (quite thick for a Go stone, even if there are thicker stones out there) and 18mm in diameter. We know that the standard diameter for a stone is around 22mm, so we can play with parameters of the catenary to make the sone wider.
We can also play (more easily) with the square root function moving the intersect of x-axis from 0.9 to 1.1 in order to obtain a 2.2cm stone in diameter. Obviously we must adjust also the scale factor and repeat all the calculations with the above derivatives to find the new juncture point. I would expect that - in this case - a smooth junction will fit the curves around 0.5 or 0.6.
I leave this job to someone more patient than me but crazy enough like me.
I would like to see a 3D rendering of such stone, but this is much beyond my capacity. I don’t have 3D software and I don’t know how to transport math functions in it. Let me leave this game to someone so crazy to try this challenge.
In principle, we can repeat this approach looking for other candidate functions. I like the way in which square root terminate the edge (or ear, 耳) of the stone (it is a parable at the end of the day).
I leave also the game to try other functions to someone else.
What did I say somewhere before?
"Beauty is simple"