Groove guides have sharp corners in the guide proper so are not low loss and high power carrying wave guides. Proposed has been a groove rectangular waveguide with rounded internal angles . However, the latter is not practical in having the infinite arms. Let us try the Conchoid of Nicomedes and Limacon of Pascal to see how can they be boundaries of static fields or high frequency waves. Also has been proposed the pentagon waveguide  for the high power transmission in high frequency. Yet we feel that we have very limited forms of waveguides, so we study the existing curves to see whether we can find some better guides such as rounded corner rectangular waveguide and others by starting from the boundary curves of ρ = a/ cos ϕ + l and ρ = a cos ϕ + l.
1. Lin, W., E. K.-N. Yung, and K.-M. Luk, "Groove rectangular waveguide with rounded internal angles," MWSYM, 1843-1844, 1996.
2. Lin, W., H. Zhou, Z. Yu, E. K.-N. Yung, and K.-M. Luk, "Theoretical and experimental study of a new family high-power waveguides," Journal of Electromagnetic waves and Applications, Vol. 13, No. 8, 1013-1020, 1999. doi:10.1163/156939399X01140
3. Vygodsky, M., Mathematical Handbook, 4th ed., Mir. Publishers, Moscus, 1984 or Bayer, CRC Stan. Math Table, 28th Ed.