Ellipsograph

Key takeaways

Ellipsograph belongs to physics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Ellipsograph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Ellipsograph from memory before moving on to harder problems.

Reference excerpt

An ellipsograph is a mechanism that generates the shape of an ellipse. One common form of ellipsograph is known as the trammel of Archimedes. It consists of two shuttles which are confined to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at adjustable positions along the rod. As the shuttles move back and forth, each along its channel, all points on the rod move in elliptical paths. The motion of the rod is termed elliptical motion. The semi-axes a and b of the ellipses have lengths equal to the distances from the point on the rod to each of the two pivots. The straight lines described by the pivots are special cases of an ellipse, where the length of one axis is twice the distance between the pivots and that of the other is zero. All points on a circle with a diameter defined by the two pivots reciprocate in such straight lines. This circle corresponds to the smaller circle in a Tusi couple. The point midway between the pivots orbits in a circle around the point where the channels cross. This circle is also a special case of an ellipse. Here the axes are of equal length. The diameter of the circle is equal to the distance between the pivots. The direction of travel around the orbit is opposite to the sense of rotation of the trammel. Thus, if a crank centred on the crossing point of the channels is used to engage the trammel at the midway point to drive it, the rotation of the crankpin and the trammel are equal and opposite, which in practical applications results in extra friction and accelerated wear. This is compounded by high forces owing to the short throw of the crank of only one quarter the travel of the pivots.

Versions are also made as toys or novelty items (sold under the name of Kentucky do-nothings, nothing grinders, do nothing machines, smoke grinders, or bullshit grinders). In these toys the drafting instrument is replaced by a crank handle, and the positions of the sliding shuttles along the rod are usually fixed.

Mathematics

Let C be the outer end of the rod, and A, B be the pivots of the sliders. Let AB and BC be the distances from A to B and B to C, respectively. Let us assume that sliders A and B move along the y and x coordinate axes, respectively. When the rod makes an angle θ with the x-axis, the coordinates of point C are given by

x = ( A B + B C ) cos ⁡ θ y = B C sin ⁡ θ {\displaystyle {\begin{aligned}x&=(AB+BC)\cos \theta \,\\y&=BC\sin \theta \,\end{aligned}}}

These are in the form of the standard parametric equations for an ellipse in canonical position. The further equation

x 2 ( A B + B C ) 2 + y 2 ( B C ) 2 = 1 {\displaystyle {\frac {x^{2}}{(AB+BC)^{2}}}+{\frac {y^{2}}{(BC)^{2}}}=1}

is immediate as well. The trammel of Archimedes is an example of a four-bar linkage with two sliders and two pivots, and is special case of the more general oblique trammel. The axes constraining the pivots do not have to be perpendicular and the points A, B and C can form a triangle. The resulting locus of C is still an ellipse.

Ellipsographs

An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife, router, etc.) attached to the rod. Usually the distances a and b are adjustable, so that the size and shape of the ellipse can be varied. The history of such ellipsographs is not certain, but they are believed to date back to Proclus and perhaps even to the time of Archimedes.

See also Beam compass Bourke engine John Farey Jr. Hypocycloid Hypotrochoid Tusi couple Useless machine Scott Russell linkage

Notes

References J. W. Downs: Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas. Courier Dover 2003, ISBN 978-0-486-42876-5, pp. 4–5 (restricted online copy, p. 4, at Google Books) I. I. Artobolevskii Mechanisms for the Generation of Plane Curves. Pergamon Press 1964, ISBN 978-1483120003.

External links

Video of various trammel designs in action Cutting ellipses in wood Photo of a Kentucky Do-Nothing Video of a Do-Nothing made from Lego bricks "Wonky Trammel of Archimedes" Archived 2012-02-20 at the Wayback Machine An exploration of a generalized trammel. US-Patent 4306598 for ellipse cutting guide allowing small ellipses Secrets of the Nothing Grinder YouTube video by Mathologer

Illustrations

Worked examples

Example 1 — a first encounter with Ellipsograph

Start with the simplest possible case. Write down what Ellipsograph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Ellipsograph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Ellipsograph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Ellipsograph

In research: Ellipsograph appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Ellipsograph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Ellipsograph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Educational toys, Ellipses, Linkages (mechanical), so understanding it makes those chapters shorter.
In everyday life: Look for Ellipsograph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Ellipsograph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ellipsograph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Ellipsograph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ellipsograph in simple terms?

An ellipsograph is a mechanism that generates the shape of an ellipse. One common form of ellipsograph is known as the trammel of Archimedes.

Why does Ellipsograph matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Ellipsograph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Ellipsograph.