Archimedes' ellipsograph

Archimedes' ellipsograph as a 3D animation

Adjustable ellipsograph, manufactured circa 1900

The ellipsograph , Archimedes elliptical circle or stucco circle is a mechanism that creates the shape of an ellipse .

It essentially consists of three different components:

1. a base plate with two guide grooves at right angles to each other (other configurations are technically possible, but uncommon),
2. a drawing arm with the holder for the drawing pencil at point and two joint eyes and , ${\displaystyle E}$ ${\displaystyle A}$ ${\displaystyle B}$
3. two sliding blocks with bearing bolts, pushed into the guide grooves of the base plate, connect the base plate at the point and with the drawing arm that is movable with it. ${\displaystyle A}$ ${\displaystyle B}$

The distance between the pen and the first joint eye is the distance between the joints . By varying and , ellipses of different sizes and shapes can be drawn. So is the length of the semi-major axis and the length of the semi-minor axis . ${\displaystyle A}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle p+q}$ ${\displaystyle q}$

The history of this mechanism is not certain. It is believed that Proclus knew the mechanism, but the mechanism may have been known as early as Archimedean times.

There is a British patent for this mechanism from 1894.

The mechanism is also known as:

• Archimedean ellipsograph
• Proclus's ellipsograph

Mathematical basics

Sketch of the ellipsograph with the parameters and drawn quarter of the ellipse, animation, start after a 30 s break

As can be seen in the adjacent sketch, the segment has the same length as the semi-axis and the segment the same length as the semi-axis of the ellipse line . Since the two right triangles and are similar to each other , the angle is consequently the Z angle of . ${\displaystyle {\overline {BE}}=p+q}$ ${\displaystyle {\overline {MA_{0}}}}$ ${\displaystyle {\overline {AE}}=q}$ ${\displaystyle {\overline {MB_{0}}}}$ ${\displaystyle E_{L}}$ ${\displaystyle BEY}$ ${\displaystyle EAX}$ ${\displaystyle \theta '}$ ${\displaystyle \theta }$

For the general determination of the point in the Cartesian coordinate system , the Pythagorean theorem applies ${\displaystyle E}$

${\displaystyle {\frac {x}{p+q}}=\cos \theta \,}$, it follows

${\displaystyle x=(p+q)\cdot \cos \theta \,}$,

${\displaystyle {\frac {y}{q}}=\sin \theta \,}$, thus is

${\displaystyle y=q\cdot \sin \theta }$.

The line that can be generated at the point with the mechanism of the ellipsograph and the pen is a so-called ellipse in the 1st main position , because if the length is used for the major semiaxis and the length for the minor semiaxis, the equation found corresponds to that for the ellipse in 1. Main location: ${\displaystyle E}$ ${\displaystyle a}$ ${\displaystyle p+q}$ ${\displaystyle b}$ ${\displaystyle q}$

${\displaystyle {\frac {x^{2}}{(p+q)^{2}}}+{\frac {y^{2}}{q^{2}}}=1={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}}$.

Equivalence to the Cardan circles

Cardan circles (black) and ellipsograph (gray). The momentary pole is marked with a pink dot. Two exemplary ellipses appear red and light blue.

As Tusi couple is called a geometric arrangement in which a small circle rolling in a twice as large fixed circle. The movement performed is the same as that performed by the drawing arm. The distance here lies on a diameter of the small circle. Thus, an ellipse can be created with a spirograph if the internal gear has half as many teeth as the ring gear in which it rolls. This analogy also illustrates that the instantaneous pole of the drawing arm moves on the outer circle with the radius . ${\displaystyle {\overline {AB}}}$ ${\displaystyle p}$

Applications

The mechanism was sold as a physics toy for children.

A US patent uses the principle of the ellipsograph for an elliptical cutter.

variants

• 

  A similar mechanism with only one slider and one crank

• 

  Elliptical compass with three gliders

literature

Chris Sangwin: The wonky trammel of Archimedes, This article provides brief notes on an ancient problem: the ellipsograph of Archimedes ( Memento of April 9, 2017 in the Internet Archive ).

Web links

Commons : Circle of ellipses  - collection of images, videos and audio files

Individual evidence

1. ↑ Elliptical circle. Zeno.org MY LIBRARY; from Meyers Großes Konversations-Lexikon , Volume 5. Leipzig 1906, pp. 720–721, accessed on April 6, 2017 . 
2. ↑ Franz Reuleaux : Theoretical Kinematics: Fundamentals of a theory of mechanical engineering . 1875, p.  318 ( full text in Google Book Search).
3. ^ Franz Reuleaux , Alexander Kennedy : Kinematics of Machinery: Outlines of a Theory of Machines . 1876, p.  318 , Fig. 248 (see footnote) (English, full text in the Google book search).
4. ^ Lueger, Otto: Ellipsenzirkel or Ellipsograph. Zeno.org MY LIBRARY; from Lueger, Otto: Lexicon of entire technology and its auxiliary sciences , Vol. 3. Stuttgart, Leipzig 1906, pp. 436–437, Fig. 2, accessed on April 8, 2017 . 
5. ↑ ^{a b} Lueger, Otto: Ellipsenzirkel or Ellipsograph. Zeno.org MY LIBRARY; from Lueger, Otto: Lexicon of all technology and its auxiliary sciences , Vol. 3. Stuttgart, Leipzig 1906, pp. 436–437, accessed on April 6, 2017 . 
6. ↑ Patent GB189402496 : Ellipsograph. Registered on May 5, 1894 , inventors: Heinel Gustav, Barth Carl. Text, drawing (p 3)
7. ^ Archimedean ellipsograph. Do-it-yourSciences, the platform for scientific and educational work , March 16, 2010, accessed on April 6, 2017 . 
8. ^ The ellipsograph of Proclus. Kotsana's Museum of Ancient Greek Technology from The Kinematic Geometric Mechanisms of the Ancient Greeks , accessed April 6, 2017 . 
9. ^ Lueger, Otto: Ellipsenzirkel or Ellipsograph. Zeno.org MY LIBRARY; from Lueger, Otto: Lexicon of all technology and its auxiliary sciences , Vol. 3. Stuttgart, Leipzig 1906, pp. 436–437, Fig. 1, accessed on April 8, 2017 . 
10. ^ Lueger, Otto: Cardan circles. Zeno.org MY LIBRARY; from Lueger, Otto: Lexicon of the whole technology and its auxiliary sciences , Vol. 2. Stuttgart, Leipzig 1905, → Von De la Hire was proven ... S. 423–424, Fig. 1, accessed on April 13, 2017 . 
11. ↑ Entertainment Center: Kentucky Do-Nothing. Scott Kraft, March 22, 2009, accessed April 6, 2017 . 
12. ↑ Patent US4306598 : Ellipse cutting machine, see Fig. 1 and Fig. 2 . Applied on June 26, 1980 , published December 22, 1981 , inventor: David G. Peot. ‌