The distance between the cone and the eye, h_{eye}, is found based on a measurement and calculation that was taken on a side-elevation that did not include the lower chapel. So, in order to measure the distance between the cone and the eye a side-elevation of a different scale was used. This side-elevation included the floor which allowed a measurement between the cone and the eye to be taken. However, one needs to adjust the numbers for a change in scale. The measured distance between the base and first layer of the cone is 0.80 cm and the measured distance between the cone and the eye is 10.60 cm. Based on these numbers a calculation was set-up to derive the new distance, h_{eye}:
The new heights, H_{i}, are determined easily by multiplying the layer (1, 2, 3, 4, 5 or 6) by 1.70 cm. Thus, in terms of a formula, let i be the layer number and
In order to compute the new radii, one has to use both the original measured heights, h_{i}, and the formula for the radii that is needed in order to compute the radii of the main cone:
Using the following proportions one can determine R_{i}, the radius of the cone at each respective new height.
h_{eye} + H_{i} |
= | h_{eye} + h_{i} |
R_{i} = r_{i} (h_{eye} + H_{i})/(h_{eye} + h_{i})
This equation determines the following radii at their respective heights:
H_{1} = 1.70 cm, R_{1} = 5.38 cm
H_{2} = 3.40 cm, R_{2} = 4.67 cm
H_{3} = 5.10 cm, R_{3} = 4.33 cm
H_{4} = 6.80 cm, R_{4} = 4.03 cm
H_{5} = 8.50 cm, R_{5} = 3.75 cm
H_{6} = 10.20 cm, R_{6} = 3.47 cm
Optical Illusion & Projection in Domes: A Study of Guarino
Guarini's |