bl_info = {
    "name": "Mercator Project",
    "author": "batFINGER",
    "version": (1, 0),
    "blender": (2, 79, 0),
    "location": "View3D > Mesh > UV UnWrap > Mercator Project",
    "description": "UV Mercator Projection",
    "warning": "",
    "wiki_url": "",
    "category": "UV",
    }
import bpy
import bmesh
from bpy.props import FloatProperty
from math import sin, log, radians, degrees, inf
from mathutils import Vector, Matrix

class Spherical:

    def __init__(self, vert, north=(0, 0, 1), long0=(0, -1)):
        self.vert = vert
        self.north, self.long0 = Vector(north), Vector(long0)        
        v = vert.co
        R = v.length
        lat = radians(90) - self.north.angle(v)
        is_pole = v.xy.length < 0.000001
        sign = 1 if v.x > 0 else -1
        long = 0 if is_pole else sign * self.long0.angle(v.xy)

        self.R = R
        self.lat = lat
        self.long = long
        self.west = long if long < 0 else long - radians(360)
        self.east = long if long >= 0 else long + radians(360)
        
    def __repr__(self):
        return "%.3f, %.3f, %.3f" % (self.R, degrees(self.lat), degrees(self.long))
                
class MercatorUV:
    def x(self, long):
        ''' mercator  longitude mapping '''
        return self.R * long
    
    def y(self, lat):
        ''' mercator latitude mapping '''
        s = sin(lat)
        return self.R * log((1 + s) / (1 - s)) / 2
    
    def uv(self, face, uv_layer):
        ''' Map a UV face '''
        # see if face is east / west
        orient = "long"
        c = face.calc_center_median()
        if c.y > 0: # on the "dark side"
            orient = "west" if c.x < 0 else "east"

        for l in face.loops:            
            luv = l[uv_layer]
            # apply the location of the vertex as a UV
            p = self.pts[l.vert]
            lat, long = p.lat, getattr(p, orient)
            luv.uv = self.scale * (self.translate + Vector([self.x(long), p.y]))
       
    def calc_uv(self):
        bm = self.bm
               
        uv_layer = bm.loops.layers.uv.verify()
        bm.faces.layers.tex.verify()  # currently blender needs both layers.
        bm.select_mode = {'FACE', 'VERT'}
        bm.select_flush_mode()
        bm.select_flush(True)
        # adjust UVs
        for f in bm.faces:
            if not f.select:
                continue
            self.uv(f, uv_layer)
        pass
        
    def __init__(self, me, bm, minlat, maxlat):
        def pt(v):
            s = Spherical(v)
            v.select = minlat <= s.lat <= maxlat
            setattr(s, "select", v.select)
            return s

        self.minlat, self.maxlat = minlat, maxlat
        # add a new uv map
        uv = me.uv_textures.new("Mercator")
        me.uv_textures.active = uv
        self.bm = bm
        # spherical coords for verts    
        self.pts = {v: pt(v) for v in self.bm.verts}
        # radius average of calc'd s.R
        self.R = sum(s.R for s in self.pts.values()) / len(self.pts)
        for s in self.pts.values():
            if s.select:
                # set y on a per verf basis , not per face.verts
                setattr(s, "y", self.y(s.lat))
        # scale UV to [0, 1] make scale matrix
        scale_x = 1 / (self.R * radians(360))        
        scale_y = 1 / (self.y(maxlat) - self.y(minlat))
        self.scale = Matrix([[scale_x, 0], [0, scale_y]])
        # and transform vector
        self.translate = Vector((self.R * radians(180), -self.y(minlat)))
        
class UV_OT_MercatorProject(bpy.types.Operator):
    """Create a Mercator Projection UV Map"""
    bl_idname = "uv.mercator_project"
    bl_label = "Mercator Project"
    bl_options = {'REGISTER', 'UNDO'}
    minlat = FloatProperty(default=radians(-82),
                           name="Minimum Latitude",
                           description="Minimum Latitude to project.", 
                           min=radians(-86),
                           max=radians(86),
                           precision=2,
                           unit='ROTATION')
    maxlat = FloatProperty(default=radians(82),
                           name="Minimum Latitude",
                           description="Maximum Latitude to project.", 
                           min=radians(-86),
                           max=radians(86),    
                           precision=2,
                           unit='ROTATION')
    @classmethod
    def poll(cls, context):
        return (context.mode == 'EDIT_MESH')

    def execute(self, context):
        obj = context.edit_object
        me = obj.data
        bm = bmesh.from_edit_mesh(me)
        merc = MercatorUV(me, bm, self.minlat, self.maxlat)
        merc.calc_uv()
        bmesh.update_edit_mesh(me)
        return {'FINISHED'}

def unwrapmenu(self, context):
    ''' menu item '''
    self.layout.operator("uv.mercator_project")

def register():
    # add to edit mesh > UV menu
    bpy.types.VIEW3D_MT_uv_map.append(unwrapmenu)
    bpy.utils.register_class(UV_OT_MercatorProject)

def unregister():
    bpy.types.VIEW3D_MT_uv_map.remove(unwrapmenu)
    bpy.utils.unregister_class(UV_OT_MercatorProject)
    
if __name__ == "__main__":
    register()