A Geometric Chuck (sometimes called an Epicycloidal Chuck) is used for cutting patterns made up of rolling centres. Basically, it is achieved by rolling one object around another, forming a composite of two perfectly circular motions (see also the graphic below of a planet in epicyclic motion).
This chuck is so complicated that John Jacob Holtzapffel never got around to putting it into his last published work, Turning and Mechanical Manipulation, vol. 5 - The Principles and Practice of Ornamental or Complex Turning (1884).
There is an article about one in the "English Mechanic and Mirror of Science", Vol VII, No. 171 (03 July 1868). And Thomas Bazley published a book, Index to the Geometric Chuck about this chuck. The Bazely book also has some nice pictures showing what can be produced on such an apparatus.
The details are difficult for me to explain, and are far better explained by John Edwards' document. John captured this and more in what he calls the "Holtzapffel Volume 6", which you can get from the The Society of Ornamental Turners.
The geometric mathematics behind this chuck's movement are explained by Dr. Frank Farris in his 1996 paper, "Wheels on Wheels on Wheels-Surprising Symmetry". There is also a nice article on this at Wolfram MathWorld.
This is a YouTube video from Chuck Bommarito (aka, outsidescrewball) showing a Leinhard Rose Geometric Chuck in use. The video is a bit long, but does show the chuck well, as well as a piece made on it.
This video shows an Ibbetson Geometric Chuck on a Holtzapffel rose engine lathe.