 # Oloid surface

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2022-06-21 18:00:09

An oloid is a three-dimensional curved geometric object that was discovered by Paul Schatz in 1929. It’s a ruled surface, the convex hull of a skeletal frame made by placing two linked congruent circles in perpendicular planes, so that the center of each circle lies on the edge of the other circle. The following 2 circles generate an oloid where $$r$$ is the radius.

\begin{equation} k_a: \begin{cases} x^2+ \left( y-\frac{r}{2} \right)^2 = r^2 \newline z=0 \end{cases}\ \end{equation}

\begin{equation} k_b: \begin{cases} \left( y-\frac{r}{2} \right)^2 + z^2 = r^2 \newline x=0 \end{cases}\, \end{equation}

In this article we’ll parameterize this beautiful surface, and show that it’s surface is the same as the sphere ($$4 \ \pi \ r^2$$), apart of some other properties.

As mention above, the oloid is a ruled surface, and it’s formed by the segments AB, where A belongs to $$k_a$$ and B to $$k_b$$, respectively, along both circles.