We have already studied the "Ellipsograph" or "Trammel of Archimedes" and we know that every point on the moving rod traces an ellipse.
In this page we are going to think only in a line segment of fixed length whose ends slide along two perpendicular axes. You can imagine that this segment is an sliding ladder.
Following Apostol and Mnatsakanian we will see another proof that every point in the segment draws an ellipse.
We can start with the simplest case: the midpoint M of the segment traces a circle:
Consider the coordinates of midpoint M of a segment of length 2m
OM and the two perpendiculars from M to each axis divide the large right triangle into four congruent smaller right triangles, then:
The moving midpoint M always lies on a circle with center O and radius m.
The coordinates of M verify:
This is the Cartesian equation of a circle.
We can write this equation to notice that the circle is a special case of an ellipse:
We are going to use this equation later.
Now we consider a different point Q on the segment
Q divides the trammel into segments of lengths a and b.
Using similar triangles we can relate the coordinates of Q and of the midpoint M:
Using the cartesian formula of the circle as an special case of the ellipse
we have prove that the coordinates of Q verify equation
and then Q lies on an ellipse.
The same analysis holds if Q is any point on the segment (even if Q is in the line, outside the segment).
In all cases:
The semiaxes of the ellipse are the distances from Q to the endpoints of the segment.
Tom Apostol and Mamikon Mnatsakanian, 'New Horizons in Geometry' (Chapter 9. Trammels, ), Mathematical Association of America, 2012.
H.Martin Cundy and A.P. Rollet, 'Mathematical Models', Oxford University Press, Second Edition, 1961.
Hilbert and Cohn-Vossen, Geometry and the Imagination. Chelsea Publishing Company. pag.278.
Robert C. Yates, 'A Handbook on curves and their properties', J.W.Edwards-Ann Arbor, 1947.
Elliptic Trammel in The National Museum of American History and a comment about ellipsographs.
J.L. Coolidge, The Mathematics of great Amateurs. Second Edition. Claredon Press. Oxford. Jan de Witt's proof in page 124.
Historical Mechanisms for Drawing Curves by Daina Taimina. With De Witt's proof why the trammel describes an ellipse.