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Serlio's Ovals

The oval was an essential pattern in baroque architecture and later periods. Well-known examples of architecture employing ovals are Saint Peter's Square in Rome and the Oval Office in the White House. In his famous work "Tutte l'Opere d'Archittetura", Sebastiano Serlio declared, "In diversi modi si possono fare delle forme ovali, ma in quattro modi ne darò la regola." [As a rough translation: "There are many ways to draw an oval, but I will give you four ways to do it."] In this Demonstration you can regulate not only four ovals but a huge variety of other shapes.
Two centers are marked by a red locator and a blue locator. The straight lines from the centers of the circles to the points of contact are radii and the dashed lines are the tangents. With the green locator you can change the radius of the blue circles. You have chosen good parameters if the tangents coincide. If the tangents are identical then the two circles join smoothly.

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"Serlio I" shows ovals with constant distance; one may say "parallel" ovals. The construction in this Demonstration uses three famous theorems of Euclid (ca. 360–280 BC) that were not known to Serlio (1475–1554?). The edges of the ovals consist of four segments of circles that are joined smoothly. This is easily done by using the theorems of Euclid:
"If two circles touch one another, their midpoints and the point of contact lie on a straight line (III, 11 and III, 12)."
"The straight line from the center of a circle to the point of contact of a tangent and the tangent are perpendicular (III, 18)."
The plan of Saint Peter's Square is based on the configuration in the image Serlio IV.
In architectural literature the words "oval" and "ellipse" are often used synonymously, which is incorrect in a mathematical sense.
References
[1] T. K. Kitao, Circle and Oval in the Square of Saint Peter's: Bernini's Art of Planning, New York: New York University Press, 1974.
[2] P. L. Rosin, "On Serlio's Construction of Ovals," Mathematical Intelligencer, 23(1), 2001 pp. 58–69, 2001.
http://users.cs.cf.ac.uk/Paul.Rosin/resources/papers/oval2.pdf.
[3] S. Serlio, Tutte l'opere d'architettura di Sebastiano Serlio Bolognese, Venice: Presso Francesco de'Franceschi Senese, 1584.
http://digi.ub.uni-heidelberg.de/diglit/serlio1584.
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