Insectoid Curve


by Zevan Rosser

The following short article will discuss the Insectoid Curve - a parametric curve inspired by the Scarabaeus and Cornoid curves.

The curve itself is a weighted average of varations on both the Cornoid and the Scarabaeus. The equation is paremetric, with $\theta$ as the parameter in a range from $0$ to $2\pi$.

$$ r = b \cos(2 \theta) - a \cos(\theta) \\ x_1 = 2|r \cos(\theta + \pi/2)| \\ y_1 = \text{atan}( r \sin(\theta 2)) \\ x_2 = c \cos(\theta) (1 - (2 \sin^2(\theta)) \\ y_2 = c \sin(\theta) (1 + (2 \cos^2(\theta)) \\ x = x_1 d + x_2 e \\ y = y_1 d + y_2 e $$

Values $a, b, c, d$ and $e$ all range from $0$ to $1$. In the below image these values were simply randomized:

Note each of the above plots is actually the result of $4$ randonly generated plots on top of one another and increasingly offset on the $y$ axis (slightly).

Interactive Version

Click the below plot - values $a$ through $e$ will be randomized.

Again, this is $4$ plots on top of one another. To render this plot with a single layer hold your shift key and click.

The Scarabaeus and Cornoid

The original equation for the Scarabaeus curve in polar coordinates is:

$$r = b \cos(2 \theta) - a \cos(\theta)$$
Produced by GNUPLOT 4.2 patchlevel 6 -6 -4 -2 0 2 4 6 -8 -6 -4 -2 0 2

... and the Cornoid in parametric form:

$$ x = a \cos (\theta) (1 - 2 \sin^2(\theta)) \\ y = a \sin (\theta) (1 + 2 \cos^2(\theta)) $$
Produced by GNUPLOT 4.2 patchlevel 6 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5

Quick Background

The original impetus for the creation of the Insectoid Curve was to create a curve that had features similar to an insect - specifically a beatle. After combining the Cornoid and Scarabaeus the result you see here came about rather quickly.