Question

How to generate buffer around lines or any type of geometric shapes?
Not interested in available packages such as Shapely etc but wish to implement.

Answer 1

Edited:
This can be seen an indirect solution. If you can have your geometric shapes as image i.e., 2D matrix then you may implement (see below) simple functions such as dilation/erosion (fairly simple kernel jobs) etc to make surrounding areas around all the shapes (now pixels) on your image.

This is so straightforward than doing hard mathematics. The following shows an experiment.
The algorithm is simple:

1. shift-four-direction your matrix each time one pixel and repeat until you get desired
   depth for the buffer.
2. sum resulting matrices

That is it.

Apparently, for fancy output you may use contour or other function to generate lines from your output.

Update:
Note that as mentioned this idea is so simple and actually is based on matrix demonstration of polygon, line or whatever is in the question. That is the resolution is defined by the dimensions of the representing matrix. One advantage is however it works with any complexity of the input.

The corners are seen to be descretized form of circle.