You can make polygons by using a raise of points, so long as they are in either counterclockwise or clockwise order. There's some math in order to do hit detection with them, but I don't really know what off the top of my head. You could Google it, or I could try and find my to  2-D geometry library for BGT. (I can't really upload it right now) basically, there is an equation that will tell you which side of a line segment a point is, and whether not to line segments intersect. To tell if something why is inside a polygon, you need to be on the same side of each line segment, which is why they need to be    In order.

I've tested a tileset engine implementation with free-roaming sprites not strictly bound to tiles using circular collision detection, with each tile representing a 16 by 16 area. When a sprite walks into a tile it would register itself in a list at that particular grid point, other sprites in that sector would do the same. Each cycle, each individual sprite would then do a collision check only for those sprites in their immediate sectors to see if their two circle boundaries collided, so as to prevent having to comb through the entire sprite list for each and every sprite doing collision checks. So what you could do is have a collision grid represent an X by Y area, and then place your sprites over top, registering them in the various sectors they occupy. Each turn you'd check if their sector occupies the same space with another object and then do a collision check for them.

Here's the most up-to-date version of my BGT geometry library. Note that I combined it with Sam_Tuby's rotation package, which resulted in variable name collisions with pi, so if you try to use 2d or 3d and you get a compilation error that it can't find pi, just add

const double pi=3.14159265[...]

Somewhere in your code.
Readme2d.txt explains how to use 2d.bgt.
3d, phys, and object have not been documented. Phys has more 3d hit detection functions, and object has more factory functions and a minimalist object class. If you only need 2d, you can ignore these.

Eeeee! For real this time:
https://www.sendspace.com/file/q37jld
2d contains:
- shape interface
- geometry functions
- some basic shapes (rectangle, circle, polygon, right triangle)
3d contains:
- form interface
- basic forms (sphere, box, cylinder, cone, mesh)
phys contains:
- additional collision functions for forms
- super incomplete 3d movement functions
Phys includes 3d, and 3d includes 2d.
The collision functions in phys are pretty slow, especially when cylinders and cones are involved.
I could not figure out how to do 2d paths that aren't convex, since the examples I found used proprietary code to calculate path crossings for curves and lines. Otherwise there would probably be paths in there, too.
3d meshes use axis-aligned polygons to define their boundary. It's not relevant for what you're asking about, but I'm not sure if I can explain exactly how they work...