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Modify polygons so that they don't overlap and area stays the same

I have a set of polygons and they can overlap with each other, like this:
I want to modify them in such a way that they don't overlap and the resulting surface area stays the same. Something like this:
It is okay if the shape or the position changes. The main thing is that they should not overlap with each other and the area should not change much (I know the area changed a little in the second image but I drew it manually thus let's just assume that the areas did not change).
I am trying to do it programmatically with the help of Python. Basically I stored polygons in a PostGIS database and with the help of a script I want to retrieve them and modify them.
I am very new to GIS and thus this seems like a difficult task.
What is the correct way of doing it? Is there an algorithm that solves this kind of problems?

Take a look at ST_buffer and try passing a signed float as the second argument (degrees to reduce radius by)
SELECT buffer(the_geom,-0.01) as geom
Be careful with negative buffers as you could run into issues if the buffer size exceeds the radius, see here.

Here is what I did:
Iterated over all the polygons and found overlapping polygons. Then, I moved the polygon in different directions and found the best moving direction by calculating the minimum resulting overlapping area. Then I simply moved the polygon in that best direction until there is no overlapping area.

How to determine and extract surface points of 3D object?

I have a 3D object which is not hollow, so there are many 3D points. How would you determine which of these points of such an object (especially with a very curvaceous surface) are on the surface? I understand how to extract them, but I need either a function somewhat like libraryUNK.surfacePoint... Which I don't know.
Or better an understanding of what is considered to be a surface point, which I don't know either and couldn't (yet) develop (for myself) any proper definition.
I know I can do triangulation to get the surface. But I don't get what to do next, as I will be left now with a set of triangles, some of which are on the surface, some of which are not... but again I have no definition how to consider what's on surface and what is not...

It seems to me that you want to compute the convex hull of your 3D points cloud.
It's not an easy problem, but there's plently of solutions (and algorithms) to do that. One of the efficients one is called "convex hull". There's a ConvexHull function in scipy.spatial.
Here is the details with an example (2D, but it works in any dimension)
http://scipy.github.io/devdocs/generated/scipy.spatial.ConvexHull.html
This function use the QHull library
http://www.qhull.org/
There's plently of ressources on the QHull page. There's also a Wikipedia page (Again, this is not the only method to compute convex hulls, you may want to try others):
https://en.wikipedia.org/wiki/Quickhull
Edit: after re-reading the question, it seems your volume may not be convex. Unfortunately, if it isn't, there's no way to tell whether a point is inside the volume or in the surface without additional informations on the volume or on the points cloud.

Optimizing scipy.spatial.Delaunay.find_simplex

I have a set of points in a plane where each point has an associated altitude. I'm thinking of using the scipy.spatial library to compute the Delaunay triangulation of the point set and then use the result to interpolate for the points in between.
The library implements a nice function that, given a point, finds the triangle it lies in. This would be particularly useful when calculating the depth map from the mesh. I assume though (please do correct me if I'm wrong) that the search function searches from the same starting point every time it is called. Since the points I will be looking for will tend to lie either on the triangle the previous one lied on or on an adjacent one, I figure that's unneccessary, but can't seem to find a way to optimize the search, other than to implement it myself.
Is there a way to set the initial triangle for the search, or to optimize the depth map calculation otherwise?

You can try point in location test, especially Kirkpatrick algorithm/data structure. Basically you subdivide the mesh in both axis and re-triangulate it. A better and simpler solution is to give each triangle a color and draw a bitmap then check the color of the bitmap with the point.

Pipeline to create Voronoi Meshes

I would like to implement a Maya plugin (this question is independent from Maya) to create 3D Voronoi patterns, Something like
I just know that I have to start from point sampling (I implemented the adaptive poisson sampling algorithm described in this paper).
I thought that, from those points, I should create the 3D wire of the mesh applying Voronoi but the result was something different from what I expected.
Here are a few example of what I get handling the result i get from scipy.spatial.Voronoi like this (as suggested here):
vor = Voronoi(points)
for vpair in vor.ridge_vertices:
for i in range(len(vpair) - 1):
if all(x >= 0 for x in vpair):
v0 = vor.vertices[vpair[i]]
v1 = vor.vertices[vpair[i+1]]
create_line(v0.tolist(), v1.tolist())
The grey vertices are the sampled points (the original shape was a simple sphere):
Here is a more complex shape (an arm)
I am missing something? Can anyone suggest the proper pipeline and algorithms I have to implement to create such patterns?

I saw your question since you posted it but didn’t have a real answer for you, however as I see you still didn’t get any response I’ll at least write down some ideas from me. Unfortunately it’s still not a full solution for your problem.
For me it seems you’re mixing few separate problems in this question so it would help to break it down to few pieces:
Voronoi diagram:
The diagram is by definition infinite, so when you draw it directly you should expect a similar mess you’ve got on your second image, so this seems fine. I don’t know how the SciPy does that, but the implementation I’ve used flagged some edge ends as ‘infinite’ and provided me the edges direction, so I could clip it at some distance by myself. You’ll need to check the exact data you get from SciPy.
In the 3D world you’ll almost always want to remove such infinite areas to get any meaningful rendering, or at least remove the area that contains your camera.
Points generation:
The Poisson disc is fine as some sample data or for early R&D but it’s also the most boring one :). You’ll need more ways to generate input points.
I tried to imagine the input needed for your ball-like example and I came up with something like this:
Create two spheres of points, with the same center but different radius.
When you create a Voronoi diagram out of it and remove infinite areas you should end up with something like a football ball.
If you created both spheres randomly you’ll get very irregular boundaries of the ‘ball’, but if you scale the points of one sphere, to use for the 2nd one you should get a regular mesh, similar to ball. You can also use similar points, but add some random offset to control the level of surface irregularity.
Get your computed diagram and for each edge create few points along this edge - this will give you small areas building up the edges of bigger areas. Play with random offsets again. Try to ignore edges, that doesn't touch any infinite region to get result similar to your image.
Get the points from both stages and compute the diagram once more.
Mesh generation:
Up to now it didn’t look like your target images. In fact it may be really hard to do it with production quality (for a Maya plugin) but I see some tricks that may help.
What I would try first would be to get all my edges and extrude some circle along them. You may modulate circle size to make it slightly bigger at the ends. Then do Boolean ‘OR’ between all those meshes and some Mesh Smooth at the end.
This way may give you similar results but you’ll need to be careful at mesh intersections, they can get ugly and need some special treatment.

Calculating the fraction of each cell in a grid overlapped by a 2D object

I have an arbitrary rectangular Cartesian grid divided into potentially 10^6 or so rectangular cells. (Arbitrary means that the $x$ grid is along points $x_1,...x_n$ and the same goes for the $y$ grid.) I would like to draw an arbitrary object on top of it (say a rotated rectangle, or a circle), and efficiently calculate what fraction of each cell is overlapped by the object: if the cell is entirely inside the bounds of the object, 1.0; if the cell is entirely outside, 0.0; if half of the cell is covered by the object, 0.5. If you displayed this as an image and scaled it where 1 is black and 0 is white, the result would look like an antialiased drawing of the black object.
My application for this question is in Python, and it seems like this capability might be provided by some existing graphics library. Is there a Python module that will test for the fractional intersection of a rectangle and an arbitrary object? Is there a Python library that can at least efficiently test if a point is inside an arbitrary object like a rotated rectangle?

You could use PyCairo, which has fast native routines to do its drawing. It's antialiased by default.
Implementing the drawing algorithms in Python would be very slow.

To find the area of a trapezoid resulting from a polygon-square intersection, you can follow the process described by Sean Barrett at https://nothings.org/gamedev/rasterize/
The shapely Python library can find the area of a trapezoid and perform point-in-object tests. However, for best performance this sounds like something that you'd want to write in C/C++ and provide numpy bindings.