# The Evolution of Plane Curves

This is the third post in the “Evolution Of…” series; the first and second posts can be found here and here.

This time around, we’ll explore some of the words that have come to be used for various plane curves. First of all, a disclaimer: often, the names of the curves existed many centuries before the development of modern algebra and the Cartesian coordinate system. As a consequence, the original names for the curves are more geometric in origin (imagine one of the ancient Greeks saying “umm, well, it looks like a flower… so let’s call it the flower curve“).

While reviewing the curves listed in Schwartzman’s book, I noticed that most of them can be classified into four major groups: the conics, the chrones, the trixes, and the oids.

1. Conics. You’ve probably heard of them—circle, ellipse, parabola, and hyperbola. The first one has its origins in the Latin word circus, which means “ring” or “hoop.” The other three are Greek, with their original meanings reflecting the Greeks’ use of conic sectionsEllipse comes from en (meaning “in”) and leipein (meaning “to leave out”). For the other two, note that –bola comes from ballein which means “to throw” or “to cast”. So hyperbola means “to cast over” and parabola means “to cast alongside”. (If you check out this image from Wikipedia, it may start to make more sense.)
2. Chrones. The two curves I have in mind here are brachistochrone and tautochrone. In Greek, chrone means “time”. The prefixes come from brakhus and tauto-, which mean “short” and “same”, respectively. So these curves’ names are really “short time” and “same time.” Naturally enough, the brachistochrone is the curve on which a ball will take the least amount of time to roll down, while the tautochrone is the curve on which the time to roll down is the same regardless of the ball’s starting point. Finding equations for these curves occupied the time of many scientists and mathematicians in the 17th century, including a controversy between the brothers Jakob and Johann Bernoulli. You could say they had a “chronic” case of sibling rivalry.
3. Trixes. I am a fan of the feminine suffix –trix because it also provides us with the modern word obstetrics (literally, “the woman who gets in the way”—i.e., a midwife). We don’t use this suffix very much anymore, though aviatrix comes to mind. The algebraic curves in this category are trisectrix (“cut into three”) and tractrix (“the one that pulls”), along with the parabola-related term directrix (“the one that directs”). Interestingly, the masculine form of tractrix gives us the English word tractor.
4. Oids. These were the most fun for me. The suffix is Greek, originating in oeides, which means “form” (though in modern English, “like” might be more appropriate). Here are a some examples: astroidcardioidcissoidcochleoidcycloidramphoid, strophoid. Here are their original Greek/Latin meanings: “star-like”, “heart-like”, “ivy-like”, “snail-like”, “circle-like”, “like a bird’s beak”, “(having the) form of turning”. I’ve provided images of each one below—see if you can match the name to the curve!

Don’t worry, there are plenty more word origins coming later! However, I’ll need a break to recharge my etymology batteries. Expect an “intermission” post in the next few weeks.