Butterfly Parametric Curves

Overview

This butterfly design forms through polar parametric equations, where I used r equal to e raised to cos t, minus 2 times cos 4t, minus sin of t over 12 raised to the fifth power. The range of t spans from 0 to 12π, producing six full wings in the symmetry. Then I mapped the polar coordinates into Cartesian space using x equals sin t times r and y equals cos t times r. The sin and cos switch flips the usual role of radius and angle, which gives the figure its unique folded and flaring shape.

I generated 10000 steps for t to ensure the outline was smooth. I looped over all the points and plotted each small segment individually, using the twilight shifted colormap to match color to position along the path. The exponential and cosine combination controls the swell of the wings, while the sine to the fifth term adds ripples. I kept the line width thin at 0.8 so that the overlapping strokes would not blur the shape. This curve felt alive to me, like a breath moving outward then curling back inward.