Specification
The aim of the Geometry Library is to clean up and unify KiCad's 2D
geometry engine and provide a solid base
for tools demanding more complex geometry operations, such as a
Push&Shove
router.
Rationale
Currently every object on a PCB/Schematic has its own notion of geometric shape.
Different subsystems deal with basic 2D geometry operations in their own
way.
There are at least 5 of such places:
 drawing
 plotting
 hit testing
 DRC/polygon processing
 LIB_ITEM derived classes in eeschema.
Adding another pad type could serve as an example of how much existing
code would have to be modified to full support the new pads in all
subsystems.
Need for extra geometry processing functions:
 Push & Shove requires spatial indexing and convex hull generation.
 Convex partitions may be required for rendering polygons (unless
some
sophisticated polygon cache is implemented inside the GAL). DRC may also
require need them to fully check polygontoanythingelse clearance (including
graphical objects).  DRC needs many more checks to be implemented  for example
mask/silkscreen clearance, which require anytoany shape distance check.  Smart selection (i.e. guessing what the user wanted to select in
case of
clicking over multiple objects, depending on the neighbourhood) requires fast conversion
of complex shapes to to polygons, clipping and polygon area & coverage calculation.
In case of the DRC, geometry functions are very modelspecific, so
they
cannot be easily reused on different types of objects.
This results in artificial limitations, such as inability to stitch
polygons (because DRC redoes them instead of just checking the
clearances) or place arcs on copper layers (e.g. teardrops or
mitered traces, a musthave for high speed designs).
Numerical consistency between subsystems:
Any numerical differences between different geometry implementations
will very likely result with DRC dropping lots of clearance errors
caused by P&S and P&S unnecessarily adjusting objects that have
passed
the DRC.
Current geometry operations take a lot of code to maintain.
... and sometimes they are not very well designed (like
polygon/math_for_graphics.cpp).
In many places, intersection and distance calculations internally use
floating point
(basing all calculations on y=mx + b line equations),
which results with rounding errors for more vertical lines.
Functional requirements
The Geometry library incorporates the 3 parts:
 basic classes for 2D vector, axisaligned bounding box, line segment and angle.
 collection of basic 2D shapes used in schematics and PCB geometry
 shape groups (a set of shapes treated as a single solid shape) and indexes.
Features required for all shapes:
 Basic geometry operations: mirror/rotate/translate
 Axisaligned bounding box generation. Each shape must provide a BoundingBox() method.
 Potential intersection test (using bounding boxes).
 Convex hull generation. For shapes that incorporate curves, the convex hull can be approximated by first converting the shape to a polygon.
 Intersection test, returning list of intersecting points.
 Minimum distance between shapes, returing the segment connecting the closest points in both shapes.
 Point inside test.
 Area calculation.
 Conversion to a polygon (closed shapes or shapes with thickness) or a line chain, with userdefined approximation accuracy.
 Serialization/deserialization in sexpression format.
Segments/line chains:
 Perpendicular projection of a point on a line/segment.
 Segment collinearity check.
 Line chain simplification (remove duplicate vertices and collinear segments).
Polygons:
 a cleaned up version of CPolyLine class, following coding standards
 boolean operations
 chamfered/filleted/offset polygon generation
 hatching
Containers / Indexes:
 add/remove, owning (container) / nonowning(index)
 collision and minimum distance queries (with respect to a VECTOR2,
BOX2 or any
other SHAPE)  interface compatible with STL (iterators)
Basic classes

template VECTOR2
Numerically stable (explicit specialization for T == int to avoid overflow on multiplication) 2D vector class.
Provides standard vector arithmentic, product, and distance operators. Replacement of all wxPoints in the code that is not related to the GUI. 
template class BOX2
Bounding box class, based on EDA_RECT. Intended to replace EDA_RECT, GAL's BOUNDING_BOX and wxRects in the parts of code that are not UIspecific. 
class ANGLE, implementing a restricted set of angles. Provides
accurate point rotation
and vector>magnitude operators (in certain cases, without floating point trigonometry and rounding errors
associated with it). For example an ANGLE45 class could be derived
from ANGLE and provide floatless (n * 45 degree) directions for all other geometry operations. 
template class SEG
Class defining a line segment. Provides segment/line/point intersection, pointtosegment and segmenttosegment distance operators.
Supported shapes
Shapes mandatory for the P&S router:
 class SHAPE : abstract interface for all SHAPE_ classes.
 class SHAPE_RECT: axisaligned rectangle
 class SHAPE_CIRCLE: full circle
 class SHAPE_LINE_CHAIN (_POLYLINE): string of line segments (zerowidth polyline). Polygon outline is of type SHAPE_LINE_CHAIN.
More complex shapes (for DRC):
 class SHAPE_ORIENTED_RECT: rectangle with rotation
 class SHAPE_POLYGON: polygon with holes. Outline/hole edges are stored as SHAPE_LINE_CHAINs
 class SHAPE_ARC: circular arc (zerowidth)
 class SHAPE_BEZIER: quadratic bezier curve (zerowidth)
Shape groups and indexes
 class SHAPE_GROUP: group of shapes, behaving as a single solid body. Each member shape can have an individual offset (a vector) with respect to the group origin.
 class SHAPE_INDEX: class for spatial indexing of shapes. Internally implements an R/R*/Bounding Volume/Bounding Interval Tree. Provides fast distance/collision query method.
Implementation
To consider:
 whether to design and write something from scratch (lots of
GPL/LGPL/BSD/MIT code available),  or rely on a library (consider builing and integration issues).
boost::geometry seems to be worth checking out,
clipper already does all what we need regarding polygons.  first clients/test targets: new graphics subsystem and C** P&S implementation.
Targets:
 P&S will largely depend on spatial indexing and collision detection.
 Even partially done geometry library can be used to quickly develop
a number
of new DRC tests: componenttocomponent, solder/pastemask sliver, and
silkscreenoverpads test.  As soon as polygon support is available, the DRC could be improved
with a proper
solution to the polygon stitching issue. With the new geometry code,
DRC would not need to modify zones but only check their clearances, and zone
refilling would be performed when the user requests it.