Trammel of Archimedes
Encyclopedia
A trammel of Archimedes is a mechanism that traces out an ellipse
. It consists of two shuttles which are confined ('trammelled') to perpendicular channels or rails, and a rod which is attached to the shuttles by pivots at fixed positions along the rod. As the shuttles move back and forth, each along its channel, the end of the rod moves in an elliptical path. The semi-axes a and b of the ellipse are the distances between the end of the rod and the two pivots.
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
.
Wooden versions of the trammel of Archimedes have been produced also as toy
s or novelty items, and sold under the name of nothing grinders or do nothing machines. In these toys the drafting instrument is replaced by a crank
handle, and the position of the sliding shuttles is usually fixed.
These are the in the form of the standard parametric equations for an ellipse in canonical position.
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
. It consists of two shuttles which are confined ('trammelled') to perpendicular channels or rails, and a rod which is attached to the shuttles by pivots at fixed positions along the rod. As the shuttles move back and forth, each along its channel, the end of the rod moves in an elliptical path. The semi-axes a and b of the ellipse are the distances between the end of the rod and the two pivots.
An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in wood
Wood
Wood is a hard, fibrous tissue found in many trees. It has been used for hundreds of thousands of years for both fuel and as a construction material. It is an organic material, a natural composite of cellulose fibers embedded in a matrix of lignin which resists compression...
or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife, router
Wood router
A router is a tool used to rout out an area in the face of a relatively hard workpiece, typically of wood or plastic. The main application of routers is in woodworking, especially cabinetry....
, 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
Proclus
Proclus Lycaeus , called "The Successor" or "Diadochos" , was a Greek Neoplatonist philosopher, one of the last major Classical philosophers . He set forth one of the most elaborate and fully developed systems of Neoplatonism...
and perhaps even to the time of Archimedes
Archimedes
Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and an...
.
Wooden versions of the trammel of Archimedes have been produced also as toy
Toy
A toy is any object that can be used for play. Toys are associated commonly with children and pets. Playing with toys is often thought to be an enjoyable means of training the young for life in human society. Different materials are used to make toys enjoyable and cuddly to both young and old...
s or novelty items, and sold under the name of nothing grinders or do nothing machines. In these toys the drafting instrument is replaced by a crank
Crank (mechanism)
A crank is an arm attached at right angles to a rotating shaft by which reciprocating motion is imparted to or received from the shaft. It is used to change circular into reciprocating motion, or reciprocating into circular motion. The arm may be a bent portion of the shaft, or a separate arm...
handle, and the position of the sliding shuttles is usually fixed.
Mathematics
Let C be the end of the rod, and A, B be the pivots of the sliders. Let p and q 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 byThese are the in the form of the standard parametric equations for an ellipse in canonical position.
Oblique trammel
The trammel of Archimedes is a 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.External links
- Cutting ellipses in wood
- Photo of a Kentucky Do-Nothing
- Instructions on how to build a Kentucky Do-Nothing
- Video of a Do-Nothing made from LegoLegoLego is a line of construction toys manufactured by the Lego Group, a privately held company based in Billund, Denmark. The company's flagship product, Lego, consists of colorful interlocking plastic bricks and an accompanying array of gears, minifigures and various other parts...
bricks - "Wonky Trammel of Archimedes" An exploration of a generalized trammel.
- Patent for ellipse cutting guide allowing small ellipses