oval_relatives

A few years back I experimented with shapes that are almost round, but not quite yet. Slight Imperfections that become irritations are very interesting for me. Imagine a round mirror, or a picture frame shaped like the object [1] above — would you immediately see that it isn’t round? Most people would subconsciously expect it to be round or oval and find it to slightly deviate from the norm.

1. the basic shape

The basic shape - the edge of the red area - is described by

$$ \mathbf{v}(\phi) = \begin{pmatrix}x(\phi) \\ y(\phi) \end{pmatrix} = \overbrace{(1 + f(\phi))}^{\text{radius}} \times \underbrace{\begin{pmatrix} \sin(\phi) \\ \cos(\phi) \end{pmatrix}}_{\text{unit circle}}. $$

with $\phi \in [0,2\pi]$. If $f(\phi) = const$ i.e. $f’(\phi) = 0$ the shape $\mathbf{v}(\phi)$ becomes a perfect circle. If we now make the deviation of the radius a (small) function over $\phi$, we can simply create a shape like the one above. In my case I wanted to the deviation $f(\phi)$ to be a simple Fourier series of only few components:

$$ f(\phi) = \sum_{i=1}^{N} a_i \sin(i\phi) + b_i \cos(i\phi) $$ Alternatively we can describe $f$ as a sum of only sine-functions, each with its respective phase-offset $\theta$: $$ f(\phi) = \sum_{i=1}^{N} a_i \sin(i\phi + \theta_i) $$

2. offset curves

Graphic designers already know “offset curves”, known in math as a parallel curve. If we draw parallel curves outside or inside our curve $\mathbf{v}$ we can make beautiful parallel curves $\mathbf{v}_1, \mathbf{v}_2, …$. The outer dark curve in [1] is such a parallel curve derived from the edge of the red shape.

To make things short, given the two functions $x(\phi)$ and $y(\phi)$ describing the $x$ and $y$ coordinates of a curve $\mathbf{v}(\phi)$ we can define the parallel curve $\mathbf{v}_d(\phi)$ with offset $o$ as

$$ \mathbf{v}_d(\phi) = \mathbf{v}(\phi) + d \times \overbrace{\begin{pmatrix} \frac{ y’(\phi)}{\sqrt{x’(\phi)^2+y’(\phi)^2}} \\ \frac{ y’(\phi)}{\sqrt{x’(\phi)^2+y’(\phi)^2}} \end{pmatrix}}^{\text{orthonormal vector}}. $$

In words: We follow the original curve $\mathbf{v}$ and at each point we go into the direction orthonormal - i.e. orthogonal/perpendicular and normalized to unit length of $1$ - to the tangent of the curve at that point.

todo:

• animation: wabbelgifs, increasing length of the series.