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Do Over: Ruled Surfaces

Why to choose this project to repeat

For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected.

What to be improved

In this do over project, I would like to improve in two aspects.
Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangle. Then the basic shape of the ruled surface is hyperbolic paraboloid. This kind of shapes is also used as roofs in architecture, as shown in the following picture.
Secondly, the step to move the straight line is adjusted larger, so that the gaps between lines are clear enough to show the formation of the ruled surface.

How to implement improvements

The chosen ruled surface is defined by a central line and two border lines. The central line is a line segment connecting two points, \((-10, 0 ,10)\) and \((10, 0, 10)\). The two border lines are symmetric respect to the \(x-\) axis and \(y-\) axis. The first border line is defined by the following equations: \[\left\{\begin{split} \begin{aligned} x&=t, -10 \le t \le 10\\ y&=-10 \\ z&=10(1+t/10), -10 \le t \le 0 \\ z&=10(1-t/10), 0 \le t \le 10 \\ \end{aligned} \end{split}\right.\] The second border line is defined by the following equations: \[\left\{\begin{split} \begin{aligned} x&=t, -10 \le t \le 10\\ y&=10 \\ z&=10(1+t/10), -10 \le t \le 0 \\ z&=10(1-t/10), 0 \le t \le 10 \\ \end{aligned} \end{split}\right.\] In the following picture, the central line is marked in red, and the border lines are marked in blue.
The 3D model of the ruled surface is designed by using OpenSCAD.

Why to use this example

This example of the ruled surface was chosen for the following reasons:
  1. In geometry, a hyperbolic is a doubly-curved surface that resembles the shape of a saddle. It is also a ruled surface, that is, every point on its surface lies on straight lines across the surface.
  2. The fact that hyperbolic paraboloids are ruled surfaces means that they are easy to construct using a series of straight structural materials. As a consequence, they are commonly used to construct thin ‘shell’ roofs. These ruled surfaces can be formed using timber or steel sections, etc. Being both lightweight and efficient, the form was used to minimize materials and increase structural performance. This kind of shapes can also create impressive and complex designs.
In the do over project, I have investigated a different example of rules surface. The straight lines in the 3D print model can be clearly observed.

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