PHP: is there a way to convert a private ECDSA key to its corresponding public key?
Yeah, you do "need some code which multiplies some coordinates on the elliptic curve" :). If you don't have extensive cryptography experience I would recommend you use library rather than trying to implement details yourself.

Googling I've found Bitcoin-lib-php. Looking at  the code it seems it seems this function does exactly what you are looking for.