Mathematical Modelling of Doraemon

While logistic function exists, it is much complicated than the circle formula to transform and match the shape. Thus, manually adjusting the formula is both time-consuming, and human-eye is never accurate, to begin with.

I sought for a solution where I can use the existing known data that could be reused mathematically in a form of equation. Initially, free mathematics software GeoGebra was used to plot data points on a photograph, but I faced a problem where I was not able to calibrate the plot scale as I desired to, which resulted in exporting a dataset

Before I dive in to calculations, in order to make this investigation intuitive, I have used a 3D Doraemon figure I had since I was small. The 3D model I own is vertically symmetrical, which is significant for this investigation as when using volume of revolution method, it will revolve the shape 2" radians, thus requiring the shape to be symmetrical around the axis of symmetry.

Since Doraemon has a complex shape, it must be split into different sections. Three main section that comprises his body are Head (Face), Body, and feet. Exterior elements such as nose, arms, pocket that stick out of the main body parts will be calculated individually and added on to the original volume, as these elements cannot be integrated to the original equation.

The bump of his neck, in context is a result of him wearing a choker with a bell. Since it is a choker made of strings, it is fair to assume that the cross-section is a perfect circle. Therefore, it does not require enlargement or compression.

Letting g(x) be the function of his neck, we know that:

\(g(x)=\sqrt{(r_2)^2-(x-h_2)^2}+k_2\)

According to the information the software provides, the value for radius is following:

\(r_2=0.27\)

In order to allow multiple equations to seamlessly connect with each other, &-value at the domain limit of Head equation was found and applied as a transformation so that two equations will not overlap.

\(f(5.303)\approx1.610\)

\(Vertical\ translation\ by\ \left(\begin{matrix}0\\1.610\\\end{matrix}\right)\)

\(k_2=1.610\)

The centre of the circle was also moved for the same reason:

\(Horizontal\ translation\ by\ \left(\begin{matrix}0.27+5.303\\0\\\end{matrix}\right)=\left(\begin{matrix}5.5703\\0\\\end{matrix}\right)\)

\(h_2=5.5703\)

Final equation of the neck:

\(g(x)=\sqrt{0.{27}^2-(x-5.5703)^2}+1.610\ \left\{5.303\le x<5.840\right\}\)​​​​​​​

I would be able to utilise my skills in industrial design to simple steps, and to develop modeling skills by first, creating Doraemon’s model graphically, then applying skills in mathematics of transformation and calculus (volumes of revolution) to find out his volume from limited information I am provided of – which is a skill that is required for a professional designer.

By finding out the volume, it would allow to apply this skill when full-scale modeling, which is again an important skill in assessing the ergonomics of a product. Silicon casting is typically a popular option in molding. The found volume would specifically be applicable in knowing the volume of silicon to be used in this process.

I assumed that this error could be because the shape Figure 5 had was mirrored compared to normal shape of logistic function (Figure 4). Therefore, I performed a horizontal reflection for all the datasets extracted and thus would match the shape seen in logistic function.

Additionally, the whole data set was moved to quadrant 1, as few of the plots after reflected invaded quadrant 4. The following translation was applied:

\(Translation\ by\ \left(\begin{matrix}9.322465\\0\\\end{matrix}\right)\)

-9.322... was the smallest value amongst the data plots, and translation had to be greater than this to make everything in quadrant 1. This point now lies at  \(x=0\) .

Doraemon is a manga series and name of a walking cat-shaped robot that came from the future to change youth protagonist’s life using future gadgets that act in magical ways. It is a nationwide known character in Japan and has been airing in Japan for more than 40 years.

As a citizen of Japan, I, too became fond of this series since when I was small. Although now I understand that the technology probably would not be advanced while I am alive, as a child, I genuinely believed that his gadgets are possible sometime soon. It was around that time that I started developing interests in industrial design  which start off as a mere sketch of imaginary items that I wanted to make. Doraemon was a huge source of inspiration in those ideas and eventually led me in studying design principals by myself. This interest is also reflected in my MYP Personal Project ‘Typography’ and taking SL Art in Diploma Programme.

Similar procedure as Body section was used as well for the feet. WebPlotDigitzer allowed me to access the numerical data points of orange line, and the datasets extracted from this software was imported to Geogebra, which gave the following plots and model:

Based on the adjusted height, I have created an overview of Doraemon’s body using Illustrator’s Ellipse tool and Pen tool to visualize the outline of his shape. Ellipse tool can create shapes based of circle, and Pen tool can draw any sort of lines. In both cases, the software will provide the dimensions (width and height) of created shapes.

As stated above, head will use the circle formula. To begin with, following basic transformations must be identified.

\((x-h_1)^2+(y-k_1)^2=r^2\)

While both h and k indicates the coordinate of the circle’s center, ℎ translates the x-coordinate, and: translates the y-coordinate. In this case, we can see that the green line directly extends from the origin (0,0) (Figure 3/) and does not require a vertical translation.

\(k_1=0\)

The dimensions of this green line are also provided by the software that was used to create this model. Since the graphical representation was created by adjusting the circle, semi-circle, the maximum height (y-axis) of this line would provide us the radius.

\(r_1=2.968\ (given)\)​​​​​​​

The value of \(h_1\) is also same as radius, as we want the circumference of the circle to lie on (0,0), not anywhere negative.

Overall, following is the equation of the head after substituting the values back in.

\((x-2.968)^2+y^2=2.968^2\)

\(y=\sqrt{2.968^2-(x-2.968)^2}\\\ \ =\sqrt{8.809024-(x^2-5.936x+8.809024)}\\ \ \ =\sqrt{5.936x-x^2}\)

As previously mentioned, the software provides dimensions for the shape created. From this data, I am also come to aware that the ellipse is not a complete circle, and it is transformed slightly to match the outline. According to the software, width of the circle is 5.764cm, and height is 2.968cm.

Dividing the width by 2 will give enough information to find out the scale factor.

\(\frac{5.764\ (width)}{2}=2.882\)

I now can adjust the width of the equation to 2.882 cm by applying horizontal compression.

\(scale\ factor\ of\ x=\frac{2.882}{2.968}=0.9710242588\)

Applying the transformation:

\(y=\sqrt{5.936\left(\frac{x}{0.9710242588}\right)-\left(\frac{x}{0.9710242588}\right)^2}\ \ (0\le x<5.303)\)

From the calculation above, a function for the head\(f(x)\) is found.

\(f(x)=\sqrt{6.17x-1.08x^2}\ \left\{0\le x<5.303\right\}\)​​​​​​​