Ambiguous Objects

Michael Tritle

10/20/19

MATH 401 – 3D Printing

Assignment 3: Ambiguous Curves

There’s common saying: “A magician never reveals his secret.” For me personally, this is understandable since most reveals have an element of deception to them. For mathematicians, it’s important that all secrets are revealed, so that we achieve consistency and mutual understanding. In figures 1 and 2, we see two different shapes at the top.

These images are actually of the same object. The noticeable difference comes from the viewing orientation. The shapes we see are generated by portions of these curves:

Figure 1:

Figure 2:

The first set of curves generate a pair of semi-circles, and the second set of curves generate two parabolas.

If we were to view this object from no specific position, the surface of the object looks like this:

This is generated by the space parameterizations

Where 

For more details on how to mathematically derive these equations, please visit here.

Creating a 3D printable object can be successfully done using software like Mathematica.

The following code creates a graphical representation of the 3D object seen in figures 1 and 2. 

ParametricPlot3D[

 20 { {t, alpha ((-Sqrt[1 - (-Abs[t] + 1)^2] + 1) + (-t^2 + 1)), 

     u*(r + alpha ((-t^2 + 1) - (-Sqrt[1 - (-Abs[t] + 1)^2] + 

             1)))}, {t, 

     alpha ((Sqrt[1 - (-Abs[t] + 1)^2] - 1) + (t^2 - 1)), 

     u*(r + alpha ((t^2 - 1) - (Sqrt[1 - (-Abs[t] + 1)^2] - 1)))}} /. 

  r -> 2, {u, 0, 1}, {t, -1, 1}, PlotStyle -> Thickness[2], 

 Mesh -> False, PlotPoints -> 100, AspectRatio -> Equal, 

 ViewPoint -> {0, Infinity, Infinity}]

A major difference here is that were are adding points below the curve. We are achieving this by adding an extra parameter  in our earlier parameterization. The result looks like this:

.