The key search term here is convolution, as essentially your tool $A$ is convolved
with the shape $B$ whose boundary is the curve.
The first paper below provides more than you need (in that your moving shape
is rigid):
In this paper we suggest an algebraic algorithm to compute the exact general sweep boundary
of a 2D curved object which moves in its own $xy$ plane along a
parametric curve trajectory while
changing its shape parametrically.
In this case the general sweep boundary is composed of algebraic
curve segments.
The second paper below is more specific to your question (despite its
non-descriptive title).
Kim, Myung-Soo, Jae-Woo Ahn, and Soon-Bum Lim. "An algebraic algorithm to compute the exact general sweep boundary of a 2D curved object." Information Processing Letters 47.5 (1993): 221-229.
Bajaj, Chanderjit, and M-S. Kim. "Generation of configuration space obstacles: The case of a moving sphere." IEEE Journal on Robotics and Automation 4.1 (1988): 94-99.