The Flight of Insects. Conchospirals
- Irina Boyadzhiev
- Parametric Curves
How do insects orientate themselves when they fly in the night? They try to keep a constant angle, say between the direction of the flight and the direction to some light source. They also keep a constant angle with the direction “down-up”. When the light source is very far away (like the moon), the light rays are practically parallel and the flight is on a straight line. When the light source is a light bulb, the rays are radial and the flight is on a conchospiral (see reference). Assuming the light source is at the origin of the coordinate -system, and above the insect when it starts flying, we can describe the initial position of the insect by the vertical distance to the source and angle . The trajectory is a conchospiral winding on a cone with a vertex at the light source and opening of . The trajectory cuts the light rays on the surface of the cone at a constant angle .
Reference: Boyadzhiev, Khristo N. "Spirals and Conchospirals in the Flight of Insects." The College Mathematics Journal 30:1 (1999): 23-31. or: http://xahlee.info/SpecialPlaneCurves_dir/EquiangularSpiral_dir/_p/Khristo_Boyadzhiev_CMJ-99.pdf