# A New Ellipse or Math Porcelain Service

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, then the pencil lead is tossed over through the pin stuck at point F

_{2}and fixed to point F

_{3}. Moving the pencil and thus changing the lengths L

_{1}, L

_{2}, and L

_{3}, but keeping the sum of L

_{1}+ 2L

_{2}+ L

_{3}constant, it is possible to draw the closed curves shown in Figure 2. Authors will not use in the further text the 2 before L

_{2}. This trick was obtained because the authors used a model with a string in Figure 1 with a double string between the second focus and the pencil.

_{1}, f

_{2}, f

_{3}, are drawn on the plate, as well as a trace of a stretched rope, the length of which remains constant when drawing the outer large ellipse of Tschirnhaus, on which one point is marked. To draw the same internal ellipse, the rope was shortened. The distances from the foci to one of the points of the outer ellipse are denoted by L

_{1}, L

_{2}and L

_{3}. These parameters can be estimated in this way:

## 2. Materials and Methods

## 3. Results and Discussion

#### 3.1. New Type of Ellipses

_{1}+ L

_{2}(bifocal ellipse) or the sum of L

_{1}+ L

_{2}+ L

_{3}(three focal ellipse) is sufficiently large. Therefore, the authors conducted a study not only of a circle, but of an equilateral triangle and a square as the focus of the ellipse. The animation of this study is provided within the Supplementary Materials (Video S1).

#### 3.2. Animation and Augmented Reality

#### 3.3. Tschirnhaus Watch

#### 3.4. Parabola and Hyperbola

#### 3.5. Curves of Order Greater Than Two

#### 3.6. 2.5D Printer

_{1}to x

_{2}and from y

_{1}to y

_{2}. Variables x

_{1}, x

_{2}, y

_{1}and y

_{2}, as well as other quantities and functions needed for calculation, are set in advance. The value of the integer variable n that defines the scan step can be changed, achieving a compromise between accuracy and duration of the computation. In the double for loop, the distances from the current point of x and y coordinates to the first focus (L

_{1}), to the second focus (L

_{2}) and to the third focus (L

_{3}) are calculated. Instead of points as foci, segments of curves, circles, squares, triangles, etc. can be put. Distances from a point to these curves can be easily calculated using special functions created by the authors. Available in the Supplementary Materials (File_S2.xmcd). If the sum of the distances L

_{1}, L

_{2}and L

_{3}turns out to be approximately equal to the given variable a, then the coordinates of the current point are recorded in the vectors X and Y, whose length is increased by one (i ← i + 1, where i is the index of the vectors X and Y). Then the vectors X and Y are displayed on the graph in the form of a desired curve consisting of points. If these points are large enough, they merge into a line.

^{n}+ (y/b)

^{n}= 1. For n = 2, we have an ordinary ellipse. Figure 29 and Figure 30 show sketches for the design of porcelain plates in the style of Lame ellipses with different exponents of degree n. In the right parts of the drawing’s information is given, which will need to be placed on the reverse sides of these “mathematical” plates.

## 5. Conclusions

## Supplementary Materials

_{c}+ L

_{s}+ L

_{t}= S), Video S2: Curves of different orders.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Sketch of a porcelain plate with modified ellipses of Tschirnhaus (different blue ovals correspond to different values of the sum L

_{circle1}+ L

_{circle2}+ L

_{circle3}).

**Figure 5.**Sketches of the drawing of two porcelain plates obtained with a circle and two squares as “foci” (different blue ovals correspond to different values of the sum L

_{circle}+ L

_{squate1}+ L

_{squate2}).

**Figure 6.**An Ellipse (

**a**- the blue oval) and an elliptic-hyperbolic oval (

**b**- the blue oval); the brown circle is the second focus, the first focus is a point.

**Figure 7.**Another sketch of a Tschirnhaus plate; the green triangle is a first focus, the dark blue square is the second focus, the red circle is a third focus; blue ovals have different sum L

_{c}+ L

_{s}+ L

_{t}.

**Figure 8.**Frames of the drawing animation on a Tschirnhaus plate at (L

_{c}+ L

_{s}+ L

_{t}= S): (

**a**) S = 1.920 m; (

**b**) S = 1.360 m; (

**c**) S = 0.910 m; (

**d**) S = 0.620 m; (

**e**) S = 0.330 m (the green triangle is a first focus, the blue square is the second focus and the red circle is a third focus).

**Figure 9.**Watch with an elliptical three-focal dial (blue lines are the hours hand and black lines are for the minutes hand).

**Figure 10.**Profile of the vase in the form of a parabola (red curve), a directrix (bottom black straight line), a focus (top ball), and a vertical bar connecting all parts.

**Figure 11.**Parabola with a focus and a directrix (Dir) L

_{1}= L

_{2}= 2.235 (the red line is the parabola, the black point is the focus and the dashed line is the directory).

**Figure 12.**Frames of the animation of the parabola with a circular focus (the red lines are the parabolas, the black circles are the foci and the blue lines are the directories).

**Figure 14.**Hyperbola with two circular foci |L

_{1}−L

_{2}| = a, a = 1.7 m, r = 0.5 m, r

_{1}= 0.7 m (the red circles are the foci, the blue curves is two blanch of the hyperbola).

**Figure 15.**Hyperbola with two square foci |L

_{1}− L

_{2}| = a, a = 0.940 m (the red squares are the foci, the blue curves is two blanch of the hyperbola).

**Figure 17.**Three frames of animation of the three-foci (L

_{1}·L

_{2}·L

_{3}= a) blue Cassini oval with red circles in the role of foci: (

**a**) a = 0.073 m

^{3}; (

**b**) a = 0.317 m

^{3}; (

**c**) a = 1.003 m

^{3}.

**Figure 18.**Extended ellipse, hyperbola, Cassini oval and Apollonius circle (red ovals and curves): (

**a**) L

_{1}+ L

_{2}= s, s = 0.5 m; (

**b**) |L

_{1}+ L

_{2}| = s, s = 0.15 m; (

**c**) L

_{1}·L

_{2}= s, s = 0.05 m

^{2}; (

**d**) L

_{1}/L

_{2}= s or L

_{2}/L

_{1}= s s = 0.7 (blue lines are the first focus and the black points are the second focus).

**Figure 21.**Sketches of the Tschirnhaus plates with a: (

**a**) Constant sum or (

**b**) product of distances from a point to three straight lines.

**Figure 22.**Curves of different orders: (

**a**) 2; (

**b**) 3; (

**c**) 4; (

**d**) 5; (

**e**) 6; (

**f**) 7; (

**h**) 13; (

**i**) 16; (

**j**) 17.

**Figure 24.**A Mathcad function that returns the coordinates of a point on the circumference of a circle with radius r and centre at the point (x

_{c}, y

_{c}) closest to a given point with the coordinates (x, y).

**Figure 25.**A Mathcad function that returns the coordinates of a point on the contour of a square with a “radius” r (half the length of the side of the square) and centred at the point (x

_{s}, y

_{s}) closest to the given point with the coordinates (x, y).

**Figure 26.**A Mathcad function that returns the coordinates of a point on the contour of an equilateral triangle with a “radius” r (the distance from the center of the triangle to its vertex) and centered at the point (x

_{t}, y

_{t}) closest to the given point with coordinates (x, y).

**Figure 27.**The procedure for forming the super-ellipse shown in Figure 28.

**Figure 28.**Super-ellipse with “elliptical” foci, L

_{1}+ L

_{2}= const, L

_{3}+ L

_{3}+ L

_{5}= const.

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**MDPI and ACS Style**

Ochkov, V.; Nori, M.; Borovinskaya, E.; Reschetilowski, W.
A New Ellipse or Math Porcelain Service. *Symmetry* **2019**, *11*, 184.
https://doi.org/10.3390/sym11020184

**AMA Style**

Ochkov V, Nori M, Borovinskaya E, Reschetilowski W.
A New Ellipse or Math Porcelain Service. *Symmetry*. 2019; 11(2):184.
https://doi.org/10.3390/sym11020184

**Chicago/Turabian Style**

Ochkov, Valery, Massimiliano Nori, Ekaterina Borovinskaya, and Wladimir Reschetilowski.
2019. "A New Ellipse or Math Porcelain Service" *Symmetry* 11, no. 2: 184.
https://doi.org/10.3390/sym11020184