I received from Mr. Joshua Gottdenker the message in 2011 that he wants to make a model of shape similar to the spherical spiral onto an egg instead of a sphere. Then, I make an equation of spiral on egg surface under the collaboration and instruction given by Mr. SuugakuDokugakujuku Jukuchou (whose correct name is Mr. Yasuyuki ASAI), and obtain the coordinates data of the spiral by computer calculation.
Soon after I sent the date written in an Excel file to Mr. Joshua and also to Mr. Svein Daniel Solvenus who is contact to me in chance at the same time, they create several 3D figures of both egg and apple, and send me them.
1. General formulation
General 3D display method of spiral, which is indicated by Mr. Yasuyuki ASAI, is described below. It is considered general if the original 2D closed curve is symmetric around a coordinate axis.
We start from the following equation which is rewritten in (z, x) plane from (x, y) plane as the symmetric axis is in the vertical direction.
where t is an intermediate variable. When a closed curve revolves around the symmetry axis z, the following equation expressed such a displayed closed surface is written as the follows;
where indicates the phase angle in (y, z) plane and is refined in 0 < < 2.
In the next, some relationship of the intermediate variable t and the phase angle is needed. The most simple relation is the following by using some constant as
2. Spiral on Egg
Though the equation of egg shaped curve in (x, y) plane is given in the Eq.(9b) in the page of "egg shaped curve", it is written as the following in the same as Eq.(1).
We may rewite the followings in the same as Eq.(2) according to the procedure described in the previous section.
where indicates the phase angle in (y, z) plane and is refined in 0 < < 2. As the relationship of the intermediate variable t and the phase angle , we will treat the case that
In the result of the calculation of Eq.(5), 3D displays of Figs.1 and 2 are obtained. These figures are supplied by Mr. Joshua Gottdenker and Mr. Svein Daniel Solvenus. I appreciate them. Thank you.
If we use the following equation instead of Eq.(6), Figs 3 and 4 are obtained. These are also supplied by Mr. Joshua and Mr. Daniel.