Of course I've read the wiki
The rapier you're thinking of is a fashionable weapon of later design. Latter rapiers sometimes didn't even have a sharpened edge, being exclusive a thrusting weapon.
Here's an appropriate quote:
The rapier began to develop around 1500 as the Spanish espada ropera, or "dress sword".[citation needed] The espada ropera was a cut-and-thrust civilian weapon for self-defense and the duel, while earlier weapons were equally at home on the battlefield. Its development began at a time period when the need for a lighter and faster sword became mandatory thanks to the introduction of firearm use in warfare .[4] Throughout the 16th century, a variety of new, single-handed civilian weapons were being developed, including the German Rapier, another cut-and-thrust weapon used for sportive fencing, as described in Joachim Meyer's Fechtbuch of 1570. 1570 is also the year in which the Italian swordmaster Signior Rocco Bonetti first settled in England advocating the use of the rapier for thrusting as opposed to cutting or slashing when engaged in a duel. Nevertheless, the English word "rapier" generally refers to a primarily thrusting weapon, developed by the year 1600 as a result of the geometrical theories of such masters as Camillo Agrippa, Ridolfo Capoferro and Vincentio Saviolo.
The weapon rapidly evolved thanks to rapid firearms and, I assume, metallurgy advances. Things like the Epee came, which is what you're probably thinking of.
Also, what Mephansteras said