OVALS AND POLYCENTRIC CURVES

My formulas and procedures to draw ovals, eggs and a class of more general polycentric curves can be implemented in CAD software.

Properties, parameters and Borromini's mysterious construction

(2017, Springer Ed.)

**The following papers have been published by the Nexus Network Journal:**

*A EUCLIDEAN APPROACH TO EGGS AND POLYCENTRIC CURVES*

*(2014, Nexus Network Journal, 10.1007/s00004-014-0189-5)*

*Abstract. *I tackle here the problem of smoothly connecting two arcs of curves with two more arcs, and present the application of this procedure to the drawing of a class of egg-shaped curves and to the generation of some open and closed polycentric curves. I provide solid mathematical background as well as ruler and compass constructions, some of them new. The use of basic, although lengthy, Euclidean proof tools suggests these properties and techniques were within reach – if not known – by architects and scientists long before the invention of analytic geometry.

*WHAT BORROMINI MIGHT HAVE KNOWN ABOUT OVALS. RULER AND COMPASS CONSTRUCTIONS*

*(2014, Nexus Network Journal, 10.1007/s00004-014-0190-z)*

*Abstract. *This paper is about drawing ovals once a sufficient number of certain parameters is given. New constructions are displayed, including the case when the symmetry axes are not given. Many of these make use of a recent conjecture by Ragazzo, for which I found a Euclidean proof, thus suggesting it might have been known at the time Borromini chose the ovals for the dome of San Carlo alle Quattro Fontane.

My personal archive of oval constructions.

All constructions are justified by my own mathematical proofs (see the above cited paper).

*Ovals of given axis lines, given three independent parameters:*

Oval 01 – given the measure of the axes and the centre of the smaller circumference.

Oval 02 – given the measure of the axes and the distance of the connecting point from the minor axis.

Oval 03 - given the measure of the axes and the centre of the bigger circumference.

Oval 04 - given the measure of the axes and the distance of the junction point from the major axis.

Ovals 01 to 04:

Constructing ovals with given axis lines – Ovals 01 02 03 and 04

Oval 05 - given the measure of the major axis, the centre of the smaller circumference and the distance of the junction point from the minor axis.

Oval 06 - given the measure of the major axis and the two centres of the circumferences.

Oval 07 - given the measure of the major axis, the centre of the smaller circumference and the distance of the junction point from the major axis.

Oval 09 - given the measure of the minor axis and the two centres of the circumferences.

Ovals 05 06 07 and 09:

Constructing ovals with given axis lines – Ovals 05 06 07 and 09

Oval 10 - given the measure of the major axis, the centre of the bigger circle and the distance of the connection point to the major axis.

Oval 11a - given the measure of the minor axis, the centre of the smaller circle and the distance of the connection point to the minor axis.

Ovals 10 and 11a:

Constructing ovals with given axis lines – Ovals 10 and 11a

Oval 11b - given the measure of the minor axis, the centre of the smaller circle and the distance of the connection point to the minor axis. SECOND CONSTRUCTION.

Ovale 11b:

Constructing ovals with given axis lines – Oval 11b

Oval 12 - given the measure of the minor axis and the two centres of the circumferences.

Oval 14 - given the measure of the minor axis, the centre of the big circumference and the distance of the
junction point from the minor axis.

Oval 15 - given the minor axis and the distances of the connection point from the two axis.

Oval 16 - given the minor axis, the centre of the bigger circumference and the distance of the
connection point from the major axis.

Ovals 12 14 15 and 16:

Constructing ovals with given axis lines – Ovals 12 14 15 and 16

Oval 17 - given the centres of the two circumferences and the distance of the connection point from the minor axis.

Oval 18 - given the centre of the smaller circumference and the distances of the connection
point from the two axes.

Oval 19 - given the centres of the two circumferences and the distance of the connection
point from the major axis.

Oval 20 - given the centre of the bigger circumference and the distance of the connection
point from the two axes.

Ovals 17 to 20:

Constructing ovals with given axis lines – Ovals 17 18 19 and 20

Oval 21 – given the two axes and the angle of the line of the centres.

Oval 22 - given the centers of the two circumferences and the ratio between the two axes.

Oval 23 - given the centre of the small circumference and the radii of both circumferences.

Ovals 21 to 23:

Constructing ovals with given axis lines – Ovals 21 22 and 23

Oval with the minimum radius ratio:

Oval with minimum ratio of the radii, for any given axis measures

The stadium problem:

*Ovals of*unknown*axis lines, given six independent parameters.*

Oval 106 (U7) of unknown axis lines – given the two vertices on the major axis, the distance of the centre of one of the smaller circumferences to the minor axis and the distance of the connection point to the major axis.

Oval 120 (U21) of unknown axis lines - given two consecutive vertices, the (constrained) connection point and the (constrained) centre of simmetry.

Ovals 106 (U7) and 120 (U21):

Ovals with unknown axis lines – Ovals 106 and 120

Oval 121 (U22) of unknown axis lines – given two consecutive vertices and the centre of the smaller circumference.

Oval 122 (U23) of unknown axis lines – given two consecutive vertices and the centre of the bigger circumference.

Ovals 121 (U22) and 122 (U23):

Ovals with unknown axis lines – Ovals 121 and 122

Oval 132 (U26) of unknown axis lines - given two centres of consecutive circumferences, their (constrained) connection point and the (constrained) centre of simmetry.

Oval 160 (U29) of unknown axis lines - given a vertex on the major axis, its centre and a point on the same half oval along with a tangent through it.

Ovals 132 (U26) and 160 (U29):

Ovals with unknown axis lines – Ovals 132 and 160

* *

**Constructing eggs – Geogebra animation videos**

My personal archive of egg constructions.

All constructions are justified by my own mathematical proofs (see the above cited paper).

**Constructing certain PCCs – Geogebra animation videos**

*Four-arc polycentric curves constructed starting from two external arcs.**Four-arc closed polycentric curves, starting from two connection points and two tangents.*

Drawing four-arc closed polycentric curves, starting from two connection points and two tangents, by means of the Connection Locus:

Constructing a class of four-arc closed polycentric curves by means of the Connection Locus (CL)

Construction of an egg given four independent centres.

Construction of an egg given the same centres taken in a different order.

Construction of an egg inscribed inside a trepezium.

Construction of eggs:

Constructing a six-centre egg from its centres or inscribing it inside a trapezoid

My personal archive of constructions of polycentric curves.

All constructions are justified by my own mathematical proofs (see the above cited paper).

Given two arcs and a point on each one of them I draw the Connection Locus, made of two circles. Choosing then a point on one or the other circle it is possble to draw two tangent circles. Any choice of four consecutive arcs yields a polycentric curve:

Constructing a class of four-arc polycentric curves (PCCSs) by means of the Connection Locus (CL)