The base cone is a surface of revolution and so,
construction of the base cone is similar to that of the
main cone and uses the same
side-elevation. This cone also is truncated. Let
*y* be the height of the complete cone, a quantity needed in order to
find its equation.

The base cone has a different base radius and vertex height than the main
cone. The base cone is a cone that is truncated at height 1.70 cm. At
this height, the radius must match the radius of the main cone, which is
computed to be 5.38 cm. Its base radius is the same radius needed to
construct triangle layer 1, and is 5.97 cm.

As a result, another equation of proportions is set up to find the total height:

y5.97 |
= | y - 1.705.38 |

5.38*y* = 5.97(*y* - 1.70)

*y* = 17.20

Now, let *r* be the radius of this cone at any given height *v*.
Since one knows the total height, the same proportions that are set up for
the main cone can be used here:

17.20 |
= | r17.20 - v |

17.20*r* = 5.97(17.20 - *v*)

*r* = 5.97 - .35*v*

Thus, the equation for the base cone becomes

where *u* varies from 0 to 2pi and *v* from 0.00 to 1.70.

Optical Illusion & Projection in Domes: A Study of Guarino
Guarini's Santissima Sindone |