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Ruling of a Plastic Heart


In mathematics, a surface is ruled if at every point on the surface, there is a distinct straight line on the surface. To better illustrate this concept, I give a picture of a ruled surface provided by user Ag2gaeh from the Wikipedia page regarding this topic. 
Point P lies on the surface, which has been constructed using only straight lines between the two arbitrary black curves. It is also possible to have a surface which has two distinct lines that lie on it at every point. These surfaces are doubly ruled. A famous example of a doubly ruled surface is the hyperboloid of one sheet. Below is an image of the hyperboloid of one sheet defined by two sets of distinct lines, rendered in OpenScad using code written by Dr. Kate Ponto.
Since a hyperboloid of one sheet can be constructed using only straight lines, this allows for their application in architecture and engineering since the doubly ruled structure provides exceptional support. Many people are able to recognize this structural motif in the cooling towers of nuclear reactors. In an article written by Duke Energy's Nuclear Information Center regarding cooling towers for nuclear reactors, they supply an image of a hyperbolic cooling tower under construction. The two sets of lines can be seen at the bottom of the image. 

In a final aside regarding ruled surfaces, is that the only surface in which three distinct lines sit on every point is the plane!

To create my own ruled surface, I chose the shape of a heart. Since OpenScad allows for the definition of an arbitrary function, I was able to write a parametric equation which defined the heart shape. Below I give the equation for the heart, along with how OpenScad renders it, defined on the interval 0-360 degrees with a step size of 3 degrees per iteration. 

\[r(t) = (x(t), y(t))  \\  x(t) = 16 \sin(t) ^3 \\ y(t)=13 \cos(t) - 5 \cos(2t) - 2 \cos(3t) - \cos(4t) \]

The equation above can be modified to accept a phase argument which will shift the order in which points appear to create the heart, but interestingly does not affect the orientation of the heart. The modified expression is below. 
\[x(t) = 16 \sin(t+phase) ^3 \\ y(t)=13 \cos(t+phase) - 5 \cos(2(t+phase)) - 2 \cos(3(t+phase)) - \cos(4(t+phase)) \]

The single ruled surface between these two curves is shown below. The above heart was affected by a phase shift of 55 degrees. 
Interestingly, I was able to wrap a second pair of lines between the surface, but I do not believe it qualifies as a doubly ruled surface, since it appeared there was a sharp intersection, rather than a gradual one. Even if it could qualify as a doubly ruled surface, it would probably be a similar case for the hyperboloid of one sheet qualifying as a doubly ruled surface, rather than a new discovery. Currently, I do not have the skill to prove or disprove this claim, and choose instead to show a rough rendering. To make the intersection more apparent, I shift the heart affected by the phase shift, up by 50 units from the bottom heart. 
As I stated above, a ruled surface can exist between any two arbitrary curves. In the above examples, the surface exists between two surfaces that only differ by a phase shift, but are identical in shape. To demonstrate the robustness of the singly ruled surface, I change the shape of the above heart by only shifting the phase of \(y(t)\). The modified equation is shown below.

\[x(t) = 16 \sin(t) ^3 \\ y(t)=13 \cos(t+phase) - 5 \cos(2(t+phase)) - 2 \cos(3(t+phase)) - \cos(4(t+phase)) \]
This causes the above heart to become slightly lopsided but the two hearts are still connectable by single lines!
For the print required for this project, I decided to print both the straight heart, and well as the lopsided heart. Both span 60mm (~2.5in) in the x-direction, and 50mm (~2.2in) in the y-direction, and have a height of 20mm (~0.8in). 












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