- -
Source code for collections

-'''This module implements specialized container datatypes providing
-alternatives to Python's general purpose built-in containers, dict,
-list, set, and tuple.
-
-* namedtuple   factory function for creating tuple subclasses with named fields
-* deque        list-like container with fast appends and pops on either end
-* ChainMap     dict-like class for creating a single view of multiple mappings
-* Counter      dict subclass for counting hashable objects
-* OrderedDict  dict subclass that remembers the order entries were added
-* defaultdict  dict subclass that calls a factory function to supply missing values
-* UserDict     wrapper around dictionary objects for easier dict subclassing
-* UserList     wrapper around list objects for easier list subclassing
-* UserString   wrapper around string objects for easier string subclassing
-
-'''
-
-__all__ = ['deque', 'defaultdict', 'namedtuple', 'UserDict', 'UserList',
-            'UserString', 'Counter', 'OrderedDict', 'ChainMap']
-
-import _collections_abc
-from operator import itemgetter as _itemgetter, eq as _eq
-from keyword import iskeyword as _iskeyword
-import sys as _sys
-import heapq as _heapq
-from _weakref import proxy as _proxy
-from itertools import repeat as _repeat, chain as _chain, starmap as _starmap
-from reprlib import recursive_repr as _recursive_repr
-
-try:
-    from _collections import deque
-except ImportError:
-    pass
-else:
-    _collections_abc.MutableSequence.register(deque)
-
-try:
-    from _collections import defaultdict
-except ImportError:
-    pass
-
-
-def __getattr__(name):
-    # For backwards compatibility, continue to make the collections ABCs
-    # through Python 3.6 available through the collections module.
-    # Note, no new collections ABCs were added in Python 3.7
-    if name in _collections_abc.__all__:
-        obj = getattr(_collections_abc, name)
-        import warnings
-        warnings.warn("Using or importing the ABCs from 'collections' instead "
-                      "of from 'collections.abc' is deprecated since Python 3.3,"
-                      "and in 3.9 it will stop working",
-                      DeprecationWarning, stacklevel=2)
-        globals()[name] = obj
-        return obj
-    raise AttributeError(f'module {__name__!r} has no attribute {name!r}')
-
-################################################################################
-### OrderedDict
-################################################################################
-
-class _OrderedDictKeysView(_collections_abc.KeysView):
-
-    def __reversed__(self):
-        yield from reversed(self._mapping)
-
-class _OrderedDictItemsView(_collections_abc.ItemsView):
-
-    def __reversed__(self):
-        for key in reversed(self._mapping):
-            yield (key, self._mapping[key])
-
-class _OrderedDictValuesView(_collections_abc.ValuesView):
-
-    def __reversed__(self):
-        for key in reversed(self._mapping):
-            yield self._mapping[key]
-
-class _Link(object):
-    __slots__ = 'prev', 'next', 'key', '__weakref__'
-
-class OrderedDict(dict):
-    'Dictionary that remembers insertion order'
-    # An inherited dict maps keys to values.
-    # The inherited dict provides __getitem__, __len__, __contains__, and get.
-    # The remaining methods are order-aware.
-    # Big-O running times for all methods are the same as regular dictionaries.
-
-    # The internal self.__map dict maps keys to links in a doubly linked list.
-    # The circular doubly linked list starts and ends with a sentinel element.
-    # The sentinel element never gets deleted (this simplifies the algorithm).
-    # The sentinel is in self.__hardroot with a weakref proxy in self.__root.
-    # The prev links are weakref proxies (to prevent circular references).
-    # Individual links are kept alive by the hard reference in self.__map.
-    # Those hard references disappear when a key is deleted from an OrderedDict.
-
-    def __init__(*args, **kwds):
-        '''Initialize an ordered dictionary.  The signature is the same as
-        regular dictionaries.  Keyword argument order is preserved.
-        '''
-        if not args:
-            raise TypeError("descriptor '__init__' of 'OrderedDict' object "
-                            "needs an argument")
-        self, *args = args
-        if len(args) > 1:
-            raise TypeError('expected at most 1 arguments, got %d' % len(args))
-        try:
-            self.__root
-        except AttributeError:
-            self.__hardroot = _Link()
-            self.__root = root = _proxy(self.__hardroot)
-            root.prev = root.next = root
-            self.__map = {}
-        self.__update(*args, **kwds)
-
-    def __setitem__(self, key, value,
-                    dict_setitem=dict.__setitem__, proxy=_proxy, Link=_Link):
-        'od.__setitem__(i, y) <==> od[i]=y'
-        # Setting a new item creates a new link at the end of the linked list,
-        # and the inherited dictionary is updated with the new key/value pair.
-        if key not in self:
-            self.__map[key] = link = Link()
-            root = self.__root
-            last = root.prev
-            link.prev, link.next, link.key = last, root, key
-            last.next = link
-            root.prev = proxy(link)
-        dict_setitem(self, key, value)
-
-    def __delitem__(self, key, dict_delitem=dict.__delitem__):
-        'od.__delitem__(y) <==> del od[y]'
-        # Deleting an existing item uses self.__map to find the link which gets
-        # removed by updating the links in the predecessor and successor nodes.
-        dict_delitem(self, key)
-        link = self.__map.pop(key)
-        link_prev = link.prev
-        link_next = link.next
-        link_prev.next = link_next
-        link_next.prev = link_prev
-        link.prev = None
-        link.next = None
-
-    def __iter__(self):
-        'od.__iter__() <==> iter(od)'
-        # Traverse the linked list in order.
-        root = self.__root
-        curr = root.next
-        while curr is not root:
-            yield curr.key
-            curr = curr.next
-
-    def __reversed__(self):
-        'od.__reversed__() <==> reversed(od)'
-        # Traverse the linked list in reverse order.
-        root = self.__root
-        curr = root.prev
-        while curr is not root:
-            yield curr.key
-            curr = curr.prev
-
-    def clear(self):
-        'od.clear() -> None.  Remove all items from od.'
-        root = self.__root
-        root.prev = root.next = root
-        self.__map.clear()
-        dict.clear(self)
-
-    def popitem(self, last=True):
-        '''Remove and return a (key, value) pair from the dictionary.
-
-        Pairs are returned in LIFO order if last is true or FIFO order if false.
-        '''
-        if not self:
-            raise KeyError('dictionary is empty')
-        root = self.__root
-        if last:
-            link = root.prev
-            link_prev = link.prev
-            link_prev.next = root
-            root.prev = link_prev
-        else:
-            link = root.next
-            link_next = link.next
-            root.next = link_next
-            link_next.prev = root
-        key = link.key
-        del self.__map[key]
-        value = dict.pop(self, key)
-        return key, value
-
-    def move_to_end(self, key, last=True):
-        '''Move an existing element to the end (or beginning if last is false).
-
-        Raise KeyError if the element does not exist.
-        '''
-        link = self.__map[key]
-        link_prev = link.prev
-        link_next = link.next
-        soft_link = link_next.prev
-        link_prev.next = link_next
-        link_next.prev = link_prev
-        root = self.__root
-        if last:
-            last = root.prev
-            link.prev = last
-            link.next = root
-            root.prev = soft_link
-            last.next = link
-        else:
-            first = root.next
-            link.prev = root
-            link.next = first
-            first.prev = soft_link
-            root.next = link
-
-    def __sizeof__(self):
-        sizeof = _sys.getsizeof
-        n = len(self) + 1                       # number of links including root
-        size = sizeof(self.__dict__)            # instance dictionary
-        size += sizeof(self.__map) * 2          # internal dict and inherited dict
-        size += sizeof(self.__hardroot) * n     # link objects
-        size += sizeof(self.__root) * n         # proxy objects
-        return size
-
-    update = __update = _collections_abc.MutableMapping.update
-
-    def keys(self):
-        "D.keys() -> a set-like object providing a view on D's keys"
-        return _OrderedDictKeysView(self)
-
-    def items(self):
-        "D.items() -> a set-like object providing a view on D's items"
-        return _OrderedDictItemsView(self)
-
-    def values(self):
-        "D.values() -> an object providing a view on D's values"
-        return _OrderedDictValuesView(self)
-
-    __ne__ = _collections_abc.MutableMapping.__ne__
-
-    __marker = object()
-
-    def pop(self, key, default=__marker):
-        '''od.pop(k[,d]) -> v, remove specified key and return the corresponding
-        value.  If key is not found, d is returned if given, otherwise KeyError
-        is raised.
-
-        '''
-        if key in self:
-            result = self[key]
-            del self[key]
-            return result
-        if default is self.__marker:
-            raise KeyError(key)
-        return default
-
-    def setdefault(self, key, default=None):
-        '''Insert key with a value of default if key is not in the dictionary.
-
-        Return the value for key if key is in the dictionary, else default.
-        '''
-        if key in self:
-            return self[key]
-        self[key] = default
-        return default
-
-    @_recursive_repr()
-    def __repr__(self):
-        'od.__repr__() <==> repr(od)'
-        if not self:
-            return '%s()' % (self.__class__.__name__,)
-        return '%s(%r)' % (self.__class__.__name__, list(self.items()))
-
-    def __reduce__(self):
-        'Return state information for pickling'
-        inst_dict = vars(self).copy()
-        for k in vars(OrderedDict()):
-            inst_dict.pop(k, None)
-        return self.__class__, (), inst_dict or None, None, iter(self.items())
-
-    def copy(self):
-        'od.copy() -> a shallow copy of od'
-        return self.__class__(self)
-
-    @classmethod
-    def fromkeys(cls, iterable, value=None):
-        '''Create a new ordered dictionary with keys from iterable and values set to value.
-        '''
-        self = cls()
-        for key in iterable:
-            self[key] = value
-        return self
-
-    def __eq__(self, other):
-        '''od.__eq__(y) <==> od==y.  Comparison to another OD is order-sensitive
-        while comparison to a regular mapping is order-insensitive.
-
-        '''
-        if isinstance(other, OrderedDict):
-            return dict.__eq__(self, other) and all(map(_eq, self, other))
-        return dict.__eq__(self, other)
-
-
-try:
-    from _collections import OrderedDict
-except ImportError:
-    # Leave the pure Python version in place.
-    pass
-
-
-################################################################################
-### namedtuple
-################################################################################
-
-_nt_itemgetters = {}
-
-def namedtuple(typename, field_names, *, rename=False, defaults=None, module=None):
-    """Returns a new subclass of tuple with named fields.
-
-    >>> Point = namedtuple('Point', ['x', 'y'])
-    >>> Point.__doc__                   # docstring for the new class
-    'Point(x, y)'
-    >>> p = Point(11, y=22)             # instantiate with positional args or keywords
-    >>> p[0] + p[1]                     # indexable like a plain tuple
-    33
-    >>> x, y = p                        # unpack like a regular tuple
-    >>> x, y
-    (11, 22)
-    >>> p.x + p.y                       # fields also accessible by name
-    33
-    >>> d = p._asdict()                 # convert to a dictionary
-    >>> d['x']
-    11
-    >>> Point(**d)                      # convert from a dictionary
-    Point(x=11, y=22)
-    >>> p._replace(x=100)               # _replace() is like str.replace() but targets named fields
-    Point(x=100, y=22)
-
-    """
-
-    # Validate the field names.  At the user's option, either generate an error
-    # message or automatically replace the field name with a valid name.
-    if isinstance(field_names, str):
-        field_names = field_names.replace(',', ' ').split()
-    field_names = list(map(str, field_names))
-    typename = _sys.intern(str(typename))
-
-    if rename:
-        seen = set()
-        for index, name in enumerate(field_names):
-            if (not name.isidentifier()
-                or _iskeyword(name)
-                or name.startswith('_')
-                or name in seen):
-                field_names[index] = f'_{index}'
-            seen.add(name)
-
-    for name in [typename] + field_names:
-        if type(name) is not str:
-            raise TypeError('Type names and field names must be strings')
-        if not name.isidentifier():
-            raise ValueError('Type names and field names must be valid '
-                             f'identifiers: {name!r}')
-        if _iskeyword(name):
-            raise ValueError('Type names and field names cannot be a '
-                             f'keyword: {name!r}')
-
-    seen = set()
-    for name in field_names:
-        if name.startswith('_') and not rename:
-            raise ValueError('Field names cannot start with an underscore: '
-                             f'{name!r}')
-        if name in seen:
-            raise ValueError(f'Encountered duplicate field name: {name!r}')
-        seen.add(name)
-
-    field_defaults = {}
-    if defaults is not None:
-        defaults = tuple(defaults)
-        if len(defaults) > len(field_names):
-            raise TypeError('Got more default values than field names')
-        field_defaults = dict(reversed(list(zip(reversed(field_names),
-                                                reversed(defaults)))))
-
-    # Variables used in the methods and docstrings
-    field_names = tuple(map(_sys.intern, field_names))
-    num_fields = len(field_names)
-    arg_list = repr(field_names).replace("'", "")[1:-1]
-    repr_fmt = '(' + ', '.join(f'{name}=%r' for name in field_names) + ')'
-    tuple_new = tuple.__new__
-    _len = len
-
-    # Create all the named tuple methods to be added to the class namespace
-
-    s = f'def __new__(_cls, {arg_list}): return _tuple_new(_cls, ({arg_list}))'
-    namespace = {'_tuple_new': tuple_new, '__name__': f'namedtuple_{typename}'}
-    # Note: exec() has the side-effect of interning the field names
-    exec(s, namespace)
-    __new__ = namespace['__new__']
-    __new__.__doc__ = f'Create new instance of {typename}({arg_list})'
-    if defaults is not None:
-        __new__.__defaults__ = defaults
-
-    @classmethod
-    def _make(cls, iterable):
-        result = tuple_new(cls, iterable)
-        if _len(result) != num_fields:
-            raise TypeError(f'Expected {num_fields} arguments, got {len(result)}')
-        return result
-
-    _make.__func__.__doc__ = (f'Make a new {typename} object from a sequence '
-                              'or iterable')
-
-    def _replace(_self, **kwds):
-        result = _self._make(map(kwds.pop, field_names, _self))
-        if kwds:
-            raise ValueError(f'Got unexpected field names: {list(kwds)!r}')
-        return result
-
-    _replace.__doc__ = (f'Return a new {typename} object replacing specified '
-                        'fields with new values')
-
-    def __repr__(self):
-        'Return a nicely formatted representation string'
-        return self.__class__.__name__ + repr_fmt % self
-
-    def _asdict(self):
-        'Return a new OrderedDict which maps field names to their values.'
-        return OrderedDict(zip(self._fields, self))
-
-    def __getnewargs__(self):
-        'Return self as a plain tuple.  Used by copy and pickle.'
-        return tuple(self)
-
-    # Modify function metadata to help with introspection and debugging
-
-    for method in (__new__, _make.__func__, _replace,
-                   __repr__, _asdict, __getnewargs__):
-        method.__qualname__ = f'{typename}.{method.__name__}'
-
-    # Build-up the class namespace dictionary
-    # and use type() to build the result class
-    class_namespace = {
-        '__doc__': f'{typename}({arg_list})',
-        '__slots__': (),
-        '_fields': field_names,
-        '_field_defaults': field_defaults,
-        # alternate spelling for backward compatibility
-        '_fields_defaults': field_defaults,
-        '__new__': __new__,
-        '_make': _make,
-        '_replace': _replace,
-        '__repr__': __repr__,
-        '_asdict': _asdict,
-        '__getnewargs__': __getnewargs__,
-    }
-    cache = _nt_itemgetters
-    for index, name in enumerate(field_names):
-        try:
-            itemgetter_object, doc = cache[index]
-        except KeyError:
-            itemgetter_object = _itemgetter(index)
-            doc = f'Alias for field number {index}'
-            cache[index] = itemgetter_object, doc
-        class_namespace[name] = property(itemgetter_object, doc=doc)
-
-    result = type(typename, (tuple,), class_namespace)
-
-    # For pickling to work, the __module__ variable needs to be set to the frame
-    # where the named tuple is created.  Bypass this step in environments where
-    # sys._getframe is not defined (Jython for example) or sys._getframe is not
-    # defined for arguments greater than 0 (IronPython), or where the user has
-    # specified a particular module.
-    if module is None:
-        try:
-            module = _sys._getframe(1).f_globals.get('__name__', '__main__')
-        except (AttributeError, ValueError):
-            pass
-    if module is not None:
-        result.__module__ = module
-
-    return result
-
-
-########################################################################
-###  Counter
-########################################################################
-
-def _count_elements(mapping, iterable):
-    'Tally elements from the iterable.'
-    mapping_get = mapping.get
-    for elem in iterable:
-        mapping[elem] = mapping_get(elem, 0) + 1
-
-try:                                    # Load C helper function if available
-    from _collections import _count_elements
-except ImportError:
-    pass
-
-class Counter(dict):
-    '''Dict subclass for counting hashable items.  Sometimes called a bag
-    or multiset.  Elements are stored as dictionary keys and their counts
-    are stored as dictionary values.
-
-    >>> c = Counter('abcdeabcdabcaba')  # count elements from a string
-
-    >>> c.most_common(3)                # three most common elements
-    [('a', 5), ('b', 4), ('c', 3)]
-    >>> sorted(c)                       # list all unique elements
-    ['a', 'b', 'c', 'd', 'e']
-    >>> ''.join(sorted(c.elements()))   # list elements with repetitions
-    'aaaaabbbbcccdde'
-    >>> sum(c.values())                 # total of all counts
-    15
-
-    >>> c['a']                          # count of letter 'a'
-    5
-    >>> for elem in 'shazam':           # update counts from an iterable
-    ...     c[elem] += 1                # by adding 1 to each element's count
-    >>> c['a']                          # now there are seven 'a'
-    7
-    >>> del c['b']                      # remove all 'b'
-    >>> c['b']                          # now there are zero 'b'
-    0
-
-    >>> d = Counter('simsalabim')       # make another counter
-    >>> c.update(d)                     # add in the second counter
-    >>> c['a']                          # now there are nine 'a'
-    9
-
-    >>> c.clear()                       # empty the counter
-    >>> c
-    Counter()
-
-    Note:  If a count is set to zero or reduced to zero, it will remain
-    in the counter until the entry is deleted or the counter is cleared:
-
-    >>> c = Counter('aaabbc')
-    >>> c['b'] -= 2                     # reduce the count of 'b' by two
-    >>> c.most_common()                 # 'b' is still in, but its count is zero
-    [('a', 3), ('c', 1), ('b', 0)]
-
-    '''
-    # References:
-    #   http://en.wikipedia.org/wiki/Multiset
-    #   http://www.gnu.org/software/smalltalk/manual-base/html_node/Bag.html
-    #   http://www.demo2s.com/Tutorial/Cpp/0380__set-multiset/Catalog0380__set-multiset.htm
-    #   http://code.activestate.com/recipes/259174/
-    #   Knuth, TAOCP Vol. II section 4.6.3
-
-    def __init__(*args, **kwds):
-        '''Create a new, empty Counter object.  And if given, count elements
-        from an input iterable.  Or, initialize the count from another mapping
-        of elements to their counts.
-
-        >>> c = Counter()                           # a new, empty counter
-        >>> c = Counter('gallahad')                 # a new counter from an iterable
-        >>> c = Counter({'a': 4, 'b': 2})           # a new counter from a mapping
-        >>> c = Counter(a=4, b=2)                   # a new counter from keyword args
-
-        '''
-        if not args:
-            raise TypeError("descriptor '__init__' of 'Counter' object "
-                            "needs an argument")
-        self, *args = args
-        if len(args) > 1:
-            raise TypeError('expected at most 1 arguments, got %d' % len(args))
-        super(Counter, self).__init__()
-        self.update(*args, **kwds)
-
-    def __missing__(self, key):
-        'The count of elements not in the Counter is zero.'
-        # Needed so that self[missing_item] does not raise KeyError
-        return 0
-
-    def most_common(self, n=None):
-        '''List the n most common elements and their counts from the most
-        common to the least.  If n is None, then list all element counts.
-
-        >>> Counter('abcdeabcdabcaba').most_common(3)
-        [('a', 5), ('b', 4), ('c', 3)]
-
-        '''
-        # Emulate Bag.sortedByCount from Smalltalk
-        if n is None:
-            return sorted(self.items(), key=_itemgetter(1), reverse=True)
-        return _heapq.nlargest(n, self.items(), key=_itemgetter(1))
-
-    def elements(self):
-        '''Iterator over elements repeating each as many times as its count.
-
-        >>> c = Counter('ABCABC')
-        >>> sorted(c.elements())
-        ['A', 'A', 'B', 'B', 'C', 'C']
-
-        # Knuth's example for prime factors of 1836:  2**2 * 3**3 * 17**1
-        >>> prime_factors = Counter({2: 2, 3: 3, 17: 1})
-        >>> product = 1
-        >>> for factor in prime_factors.elements():     # loop over factors
-        ...     product *= factor                       # and multiply them
-        >>> product
-        1836
-
-        Note, if an element's count has been set to zero or is a negative
-        number, elements() will ignore it.
-
-        '''
-        # Emulate Bag.do from Smalltalk and Multiset.begin from C++.
-        return _chain.from_iterable(_starmap(_repeat, self.items()))
-
-    # Override dict methods where necessary
-
-    @classmethod
-    def fromkeys(cls, iterable, v=None):
-        # There is no equivalent method for counters because setting v=1
-        # means that no element can have a count greater than one.
-        raise NotImplementedError(
-            'Counter.fromkeys() is undefined.  Use Counter(iterable) instead.')
-
-    def update(*args, **kwds):
-        '''Like dict.update() but add counts instead of replacing them.
-
-        Source can be an iterable, a dictionary, or another Counter instance.
-
-        >>> c = Counter('which')
-        >>> c.update('witch')           # add elements from another iterable
-        >>> d = Counter('watch')
-        >>> c.update(d)                 # add elements from another counter
-        >>> c['h']                      # four 'h' in which, witch, and watch
-        4
-
-        '''
-        # The regular dict.update() operation makes no sense here because the
-        # replace behavior results in the some of original untouched counts
-        # being mixed-in with all of the other counts for a mismash that
-        # doesn't have a straight-forward interpretation in most counting
-        # contexts.  Instead, we implement straight-addition.  Both the inputs
-        # and outputs are allowed to contain zero and negative counts.
-
-        if not args:
-            raise TypeError("descriptor 'update' of 'Counter' object "
-                            "needs an argument")
-        self, *args = args
-        if len(args) > 1:
-            raise TypeError('expected at most 1 arguments, got %d' % len(args))
-        iterable = args[0] if args else None
-        if iterable is not None:
-            if isinstance(iterable, _collections_abc.Mapping):
-                if self:
-                    self_get = self.get
-                    for elem, count in iterable.items():
-                        self[elem] = count + self_get(elem, 0)
-                else:
-                    super(Counter, self).update(iterable) # fast path when counter is empty
-            else:
-                _count_elements(self, iterable)
-        if kwds:
-            self.update(kwds)
-
-    def subtract(*args, **kwds):
-        '''Like dict.update() but subtracts counts instead of replacing them.
-        Counts can be reduced below zero.  Both the inputs and outputs are
-        allowed to contain zero and negative counts.
-
-        Source can be an iterable, a dictionary, or another Counter instance.
-
-        >>> c = Counter('which')
-        >>> c.subtract('witch')             # subtract elements from another iterable
-        >>> c.subtract(Counter('watch'))    # subtract elements from another counter
-        >>> c['h']                          # 2 in which, minus 1 in witch, minus 1 in watch
-        0
-        >>> c['w']                          # 1 in which, minus 1 in witch, minus 1 in watch
-        -1
-
-        '''
-        if not args:
-            raise TypeError("descriptor 'subtract' of 'Counter' object "
-                            "needs an argument")
-        self, *args = args
-        if len(args) > 1:
-            raise TypeError('expected at most 1 arguments, got %d' % len(args))
-        iterable = args[0] if args else None
-        if iterable is not None:
-            self_get = self.get
-            if isinstance(iterable, _collections_abc.Mapping):
-                for elem, count in iterable.items():
-                    self[elem] = self_get(elem, 0) - count
-            else:
-                for elem in iterable:
-                    self[elem] = self_get(elem, 0) - 1
-        if kwds:
-            self.subtract(kwds)
-
-    def copy(self):
-        'Return a shallow copy.'
-        return self.__class__(self)
-
-    def __reduce__(self):
-        return self.__class__, (dict(self),)
-
-    def __delitem__(self, elem):
-        'Like dict.__delitem__() but does not raise KeyError for missing values.'
-        if elem in self:
-            super().__delitem__(elem)
-
-    def __repr__(self):
-        if not self:
-            return '%s()' % self.__class__.__name__
-        try:
-            items = ', '.join(map('%r: %r'.__mod__, self.most_common()))
-            return '%s({%s})' % (self.__class__.__name__, items)
-        except TypeError:
-            # handle case where values are not orderable
-            return '{0}({1!r})'.format(self.__class__.__name__, dict(self))
-
-    # Multiset-style mathematical operations discussed in:
-    #       Knuth TAOCP Volume II section 4.6.3 exercise 19
-    #       and at http://en.wikipedia.org/wiki/Multiset
-    #
-    # Outputs guaranteed to only include positive counts.
-    #
-    # To strip negative and zero counts, add-in an empty counter:
-    #       c += Counter()
-
-    def __add__(self, other):
-        '''Add counts from two counters.
-
-        >>> Counter('abbb') + Counter('bcc')
-        Counter({'b': 4, 'c': 2, 'a': 1})
-
-        '''
-        if not isinstance(other, Counter):
-            return NotImplemented
-        result = Counter()
-        for elem, count in self.items():
-            newcount = count + other[elem]
-            if newcount > 0:
-                result[elem] = newcount
-        for elem, count in other.items():
-            if elem not in self and count > 0:
-                result[elem] = count
-        return result
-
-    def __sub__(self, other):
-        ''' Subtract count, but keep only results with positive counts.
-
-        >>> Counter('abbbc') - Counter('bccd')
-        Counter({'b': 2, 'a': 1})
-
-        '''
-        if not isinstance(other, Counter):
-            return NotImplemented
-        result = Counter()
-        for elem, count in self.items():
-            newcount = count - other[elem]
-            if newcount > 0:
-                result[elem] = newcount
-        for elem, count in other.items():
-            if elem not in self and count < 0:
-                result[elem] = 0 - count
-        return result
-
-    def __or__(self, other):
-        '''Union is the maximum of value in either of the input counters.
-
-        >>> Counter('abbb') | Counter('bcc')
-        Counter({'b': 3, 'c': 2, 'a': 1})
-
-        '''
-        if not isinstance(other, Counter):
-            return NotImplemented
-        result = Counter()
-        for elem, count in self.items():
-            other_count = other[elem]
-            newcount = other_count if count < other_count else count
-            if newcount > 0:
-                result[elem] = newcount
-        for elem, count in other.items():
-            if elem not in self and count > 0:
-                result[elem] = count
-        return result
-
-    def __and__(self, other):
-        ''' Intersection is the minimum of corresponding counts.
-
-        >>> Counter('abbb') & Counter('bcc')
-        Counter({'b': 1})
-
-        '''
-        if not isinstance(other, Counter):
-            return NotImplemented
-        result = Counter()
-        for elem, count in self.items():
-            other_count = other[elem]
-            newcount = count if count < other_count else other_count
-            if newcount > 0:
-                result[elem] = newcount
-        return result
-
-    def __pos__(self):
-        'Adds an empty counter, effectively stripping negative and zero counts'
-        result = Counter()
-        for elem, count in self.items():
-            if count > 0:
-                result[elem] = count
-        return result
-
-    def __neg__(self):
-        '''Subtracts from an empty counter.  Strips positive and zero counts,
-        and flips the sign on negative counts.
-
-        '''
-        result = Counter()
-        for elem, count in self.items():
-            if count < 0:
-                result[elem] = 0 - count
-        return result
-
-    def _keep_positive(self):
-        '''Internal method to strip elements with a negative or zero count'''
-        nonpositive = [elem for elem, count in self.items() if not count > 0]
-        for elem in nonpositive:
-            del self[elem]
-        return self
-
-    def __iadd__(self, other):
-        '''Inplace add from another counter, keeping only positive counts.
-
-        >>> c = Counter('abbb')
-        >>> c += Counter('bcc')
-        >>> c
-        Counter({'b': 4, 'c': 2, 'a': 1})
-
-        '''
-        for elem, count in other.items():
-            self[elem] += count
-        return self._keep_positive()
-
-    def __isub__(self, other):
-        '''Inplace subtract counter, but keep only results with positive counts.
-
-        >>> c = Counter('abbbc')
-        >>> c -= Counter('bccd')
-        >>> c
-        Counter({'b': 2, 'a': 1})
-
-        '''
-        for elem, count in other.items():
-            self[elem] -= count
-        return self._keep_positive()
-
-    def __ior__(self, other):
-        '''Inplace union is the maximum of value from either counter.
-
-        >>> c = Counter('abbb')
-        >>> c |= Counter('bcc')
-        >>> c
-        Counter({'b': 3, 'c': 2, 'a': 1})
-
-        '''
-        for elem, other_count in other.items():
-            count = self[elem]
-            if other_count > count:
-                self[elem] = other_count
-        return self._keep_positive()
-
-    def __iand__(self, other):
-        '''Inplace intersection is the minimum of corresponding counts.
-
-        >>> c = Counter('abbb')
-        >>> c &= Counter('bcc')
-        >>> c
-        Counter({'b': 1})
-
-        '''
-        for elem, count in self.items():
-            other_count = other[elem]
-            if other_count < count:
-                self[elem] = other_count
-        return self._keep_positive()
-
-
-########################################################################
-###  ChainMap
-########################################################################
-
-class ChainMap(_collections_abc.MutableMapping):
-    ''' A ChainMap groups multiple dicts (or other mappings) together
-    to create a single, updateable view.
-
-    The underlying mappings are stored in a list.  That list is public and can
-    be accessed or updated using the *maps* attribute.  There is no other
-    state.
-
-    Lookups search the underlying mappings successively until a key is found.
-    In contrast, writes, updates, and deletions only operate on the first
-    mapping.
-
-    '''
-
-    def __init__(self, *maps):
-        '''Initialize a ChainMap by setting *maps* to the given mappings.
-        If no mappings are provided, a single empty dictionary is used.
-
-        '''
-        self.maps = list(maps) or [{}]          # always at least one map
-
-    def __missing__(self, key):
-        raise KeyError(key)
-
-    def __getitem__(self, key):
-        for mapping in self.maps:
-            try:
-                return mapping[key]             # can't use 'key in mapping' with defaultdict
-            except KeyError:
-                pass
-        return self.__missing__(key)            # support subclasses that define __missing__
-
-    def get(self, key, default=None):
-        return self[key] if key in self else default
-
-    def __len__(self):
-        return len(set().union(*self.maps))     # reuses stored hash values if possible
-
-    def __iter__(self):
-        d = {}
-        for mapping in reversed(self.maps):
-            d.update(mapping)                   # reuses stored hash values if possible
-        return iter(d)
-
-    def __contains__(self, key):
-        return any(key in m for m in self.maps)
-
-    def __bool__(self):
-        return any(self.maps)
-
-    @_recursive_repr()
-    def __repr__(self):
-        return '{0.__class__.__name__}({1})'.format(
-            self, ', '.join(map(repr, self.maps)))
-
-    @classmethod
-    def fromkeys(cls, iterable, *args):
-        'Create a ChainMap with a single dict created from the iterable.'
-        return cls(dict.fromkeys(iterable, *args))
-
-    def copy(self):
-        'New ChainMap or subclass with a new copy of maps[0] and refs to maps[1:]'
-        return self.__class__(self.maps[0].copy(), *self.maps[1:])
-
-    __copy__ = copy
-
-    def new_child(self, m=None):                # like Django's Context.push()
-        '''New ChainMap with a new map followed by all previous maps.
-        If no map is provided, an empty dict is used.
-        '''
-        if m is None:
-            m = {}
-        return self.__class__(m, *self.maps)
-
-    @property
-    def parents(self):                          # like Django's Context.pop()
-        'New ChainMap from maps[1:].'
-        return self.__class__(*self.maps[1:])
-
-    def __setitem__(self, key, value):
-        self.maps[0][key] = value
-
-    def __delitem__(self, key):
-        try:
-            del self.maps[0][key]
-        except KeyError:
-            raise KeyError('Key not found in the first mapping: {!r}'.format(key))
-
-    def popitem(self):
-        'Remove and return an item pair from maps[0]. Raise KeyError is maps[0] is empty.'
-        try:
-            return self.maps[0].popitem()
-        except KeyError:
-            raise KeyError('No keys found in the first mapping.')
-
-    def pop(self, key, *args):
-        'Remove *key* from maps[0] and return its value. Raise KeyError if *key* not in maps[0].'
-        try:
-            return self.maps[0].pop(key, *args)
-        except KeyError:
-            raise KeyError('Key not found in the first mapping: {!r}'.format(key))
-
-    def clear(self):
-        'Clear maps[0], leaving maps[1:] intact.'
-        self.maps[0].clear()
-
-
-################################################################################
-### UserDict
-################################################################################
-
-class UserDict(_collections_abc.MutableMapping):
-
-    # Start by filling-out the abstract methods
-    def __init__(*args, **kwargs):
-        if not args:
-            raise TypeError("descriptor '__init__' of 'UserDict' object "
-                            "needs an argument")
-        self, *args = args
-        if len(args) > 1:
-            raise TypeError('expected at most 1 arguments, got %d' % len(args))
-        if args:
-            dict = args[0]
-        elif 'dict' in kwargs:
-            dict = kwargs.pop('dict')
-            import warnings
-            warnings.warn("Passing 'dict' as keyword argument is deprecated",
-                          DeprecationWarning, stacklevel=2)
-        else:
-            dict = None
-        self.data = {}
-        if dict is not None:
-            self.update(dict)
-        if len(kwargs):
-            self.update(kwargs)
-    def __len__(self): return len(self.data)
-    def __getitem__(self, key):
-        if key in self.data:
-            return self.data[key]
-        if hasattr(self.__class__, "__missing__"):
-            return self.__class__.__missing__(self, key)
-        raise KeyError(key)
-    def __setitem__(self, key, item): self.data[key] = item
-    def __delitem__(self, key): del self.data[key]
-    def __iter__(self):
-        return iter(self.data)
-
-    # Modify __contains__ to work correctly when __missing__ is present
-    def __contains__(self, key):
-        return key in self.data
-
-    # Now, add the methods in dicts but not in MutableMapping
-    def __repr__(self): return repr(self.data)
-    def __copy__(self):
-        inst = self.__class__.__new__(self.__class__)
-        inst.__dict__.update(self.__dict__)
-        # Create a copy and avoid triggering descriptors
-        inst.__dict__["data"] = self.__dict__["data"].copy()
-        return inst
-
-    def copy(self):
-        if self.__class__ is UserDict:
-            return UserDict(self.data.copy())
-        import copy
-        data = self.data
-        try:
-            self.data = {}
-            c = copy.copy(self)
-        finally:
-            self.data = data
-        c.update(self)
-        return c
-
-    @classmethod
-    def fromkeys(cls, iterable, value=None):
-        d = cls()
-        for key in iterable:
-            d[key] = value
-        return d
-
-
-
-################################################################################
-### UserList
-################################################################################
-
-class UserList(_collections_abc.MutableSequence):
-    """A more or less complete user-defined wrapper around list objects."""
-    def __init__(self, initlist=None):
-        self.data = []
-        if initlist is not None:
-            # XXX should this accept an arbitrary sequence?
-            if type(initlist) == type(self.data):
-                self.data[:] = initlist
-            elif isinstance(initlist, UserList):
-                self.data[:] = initlist.data[:]
-            else:
-                self.data = list(initlist)
-    def __repr__(self): return repr(self.data)
-    def __lt__(self, other): return self.data <  self.__cast(other)
-    def __le__(self, other): return self.data <= self.__cast(other)
-    def __eq__(self, other): return self.data == self.__cast(other)
-    def __gt__(self, other): return self.data >  self.__cast(other)
-    def __ge__(self, other): return self.data >= self.__cast(other)
-    def __cast(self, other):
-        return other.data if isinstance(other, UserList) else other
-    def __contains__(self, item): return item in self.data
-    def __len__(self): return len(self.data)
-    def __getitem__(self, i):
-        if isinstance(i, slice):
-            return self.__class__(self.data[i])
-        else:
-            return self.data[i]
-    def __setitem__(self, i, item): self.data[i] = item
-    def __delitem__(self, i): del self.data[i]
-    def __add__(self, other):
-        if isinstance(other, UserList):
-            return self.__class__(self.data + other.data)
-        elif isinstance(other, type(self.data)):
-            return self.__class__(self.data + other)
-        return self.__class__(self.data + list(other))
-    def __radd__(self, other):
-        if isinstance(other, UserList):
-            return self.__class__(other.data + self.data)
-        elif isinstance(other, type(self.data)):
-            return self.__class__(other + self.data)
-        return self.__class__(list(other) + self.data)
-    def __iadd__(self, other):
-        if isinstance(other, UserList):
-            self.data += other.data
-        elif isinstance(other, type(self.data)):
-            self.data += other
-        else:
-            self.data += list(other)
-        return self
-    def __mul__(self, n):
-        return self.__class__(self.data*n)
-    __rmul__ = __mul__
-    def __imul__(self, n):
-        self.data *= n
-        return self
-    def __copy__(self):
-        inst = self.__class__.__new__(self.__class__)
-        inst.__dict__.update(self.__dict__)
-        # Create a copy and avoid triggering descriptors
-        inst.__dict__["data"] = self.__dict__["data"][:]
-        return inst
-    def append(self, item): self.data.append(item)
-    def insert(self, i, item): self.data.insert(i, item)
-    def pop(self, i=-1): return self.data.pop(i)
-    def remove(self, item): self.data.remove(item)
-    def clear(self): self.data.clear()
-    def copy(self): return self.__class__(self)
-    def count(self, item): return self.data.count(item)
-    def index(self, item, *args): return self.data.index(item, *args)
-    def reverse(self): self.data.reverse()
-    def sort(self, *args, **kwds): self.data.sort(*args, **kwds)
-    def extend(self, other):
-        if isinstance(other, UserList):
-            self.data.extend(other.data)
-        else:
-            self.data.extend(other)
-
-
-
-################################################################################
-### UserString
-################################################################################
-
-class UserString(_collections_abc.Sequence):
-    def __init__(self, seq):
-        if isinstance(seq, str):
-            self.data = seq
-        elif isinstance(seq, UserString):
-            self.data = seq.data[:]
-        else:
-            self.data = str(seq)
-    def __str__(self): return str(self.data)
-    def __repr__(self): return repr(self.data)
-    def __int__(self): return int(self.data)
-    def __float__(self): return float(self.data)
-    def __complex__(self): return complex(self.data)
-    def __hash__(self): return hash(self.data)
-    def __getnewargs__(self):
-        return (self.data[:],)
-
-    def __eq__(self, string):
-        if isinstance(string, UserString):
-            return self.data == string.data
-        return self.data == string
-    def __lt__(self, string):
-        if isinstance(string, UserString):
-            return self.data < string.data
-        return self.data < string
-    def __le__(self, string):
-        if isinstance(string, UserString):
-            return self.data <= string.data
-        return self.data <= string
-    def __gt__(self, string):
-        if isinstance(string, UserString):
-            return self.data > string.data
-        return self.data > string
-    def __ge__(self, string):
-        if isinstance(string, UserString):
-            return self.data >= string.data
-        return self.data >= string
-
-    def __contains__(self, char):
-        if isinstance(char, UserString):
-            char = char.data
-        return char in self.data
-
-    def __len__(self): return len(self.data)
-    def __getitem__(self, index): return self.__class__(self.data[index])
-    def __add__(self, other):
-        if isinstance(other, UserString):
-            return self.__class__(self.data + other.data)
-        elif isinstance(other, str):
-            return self.__class__(self.data + other)
-        return self.__class__(self.data + str(other))
-    def __radd__(self, other):
-        if isinstance(other, str):
-            return self.__class__(other + self.data)
-        return self.__class__(str(other) + self.data)
-    def __mul__(self, n):
-        return self.__class__(self.data*n)
-    __rmul__ = __mul__
-    def __mod__(self, args):
-        return self.__class__(self.data % args)
-    def __rmod__(self, format):
-        return self.__class__(format % args)
-
-    # the following methods are defined in alphabetical order:
-    def capitalize(self): return self.__class__(self.data.capitalize())
-    def casefold(self):
-        return self.__class__(self.data.casefold())
-    def center(self, width, *args):
-        return self.__class__(self.data.center(width, *args))
-    def count(self, sub, start=0, end=_sys.maxsize):
-        if isinstance(sub, UserString):
-            sub = sub.data
-        return self.data.count(sub, start, end)
-    def encode(self, encoding=None, errors=None): # XXX improve this?
-        if encoding:
-            if errors:
-                return self.__class__(self.data.encode(encoding, errors))
-            return self.__class__(self.data.encode(encoding))
-        return self.__class__(self.data.encode())
-    def endswith(self, suffix, start=0, end=_sys.maxsize):
-        return self.data.endswith(suffix, start, end)
-    def expandtabs(self, tabsize=8):
-        return self.__class__(self.data.expandtabs(tabsize))
-    def find(self, sub, start=0, end=_sys.maxsize):
-        if isinstance(sub, UserString):
-            sub = sub.data
-        return self.data.find(sub, start, end)
-    def format(self, *args, **kwds):
-        return self.data.format(*args, **kwds)
-    def format_map(self, mapping):
-        return self.data.format_map(mapping)
-    def index(self, sub, start=0, end=_sys.maxsize):
-        return self.data.index(sub, start, end)
-    def isalpha(self): return self.data.isalpha()
-    def isalnum(self): return self.data.isalnum()
-    def isascii(self): return self.data.isascii()
-    def isdecimal(self): return self.data.isdecimal()
-    def isdigit(self): return self.data.isdigit()
-    def isidentifier(self): return self.data.isidentifier()
-    def islower(self): return self.data.islower()
-    def isnumeric(self): return self.data.isnumeric()
-    def isprintable(self): return self.data.isprintable()
-    def isspace(self): return self.data.isspace()
-    def istitle(self): return self.data.istitle()
-    def isupper(self): return self.data.isupper()
-    def join(self, seq): return self.data.join(seq)
-    def ljust(self, width, *args):
-        return self.__class__(self.data.ljust(width, *args))
-    def lower(self): return self.__class__(self.data.lower())
-    def lstrip(self, chars=None): return self.__class__(self.data.lstrip(chars))
-    maketrans = str.maketrans
-    def partition(self, sep):
-        return self.data.partition(sep)
-    def replace(self, old, new, maxsplit=-1):
-        if isinstance(old, UserString):
-            old = old.data
-        if isinstance(new, UserString):
-            new = new.data
-        return self.__class__(self.data.replace(old, new, maxsplit))
-    def rfind(self, sub, start=0, end=_sys.maxsize):
-        if isinstance(sub, UserString):
-            sub = sub.data
-        return self.data.rfind(sub, start, end)
-    def rindex(self, sub, start=0, end=_sys.maxsize):
-        return self.data.rindex(sub, start, end)
-    def rjust(self, width, *args):
-        return self.__class__(self.data.rjust(width, *args))
-    def rpartition(self, sep):
-        return self.data.rpartition(sep)
-    def rstrip(self, chars=None):
-        return self.__class__(self.data.rstrip(chars))
-    def split(self, sep=None, maxsplit=-1):
-        return self.data.split(sep, maxsplit)
-    def rsplit(self, sep=None, maxsplit=-1):
-        return self.data.rsplit(sep, maxsplit)
-    def splitlines(self, keepends=False): return self.data.splitlines(keepends)
-    def startswith(self, prefix, start=0, end=_sys.maxsize):
-        return self.data.startswith(prefix, start, end)
-    def strip(self, chars=None): return self.__class__(self.data.strip(chars))
-    def swapcase(self): return self.__class__(self.data.swapcase())
-    def title(self): return self.__class__(self.data.title())
-    def translate(self, *args):
-        return self.__class__(self.data.translate(*args))
-    def upper(self): return self.__class__(self.data.upper())
-    def zfill(self, width): return self.__class__(self.data.zfill(width))
-
- -
Operator	dunder method
`*`	__mul__ , __rmul__
`*=`	__imul__
`/`	__truediv__
`/=`	__itruediv__
`**`	__pow__
`**=`	__ipow__
`+`	__add__, __radd__
`+=`	__iadd__
`-`	__sub__, __rsub__
`-=`	__isub__
Operands		`*`
left	right	type	result
Pose	Pose	Pose	composition [1]
Pose	scalar	matrix	elementwise product
scalar	Pose	matrix	elementwise product
Pose	N-vector	N-vector	vector transform [2]
Pose	NxM matrix	NxM matrix	vector transform [2] [3]
Operands		`/`
left	right	type	result
Pose	Pose	Pose	matrix * inverse #1
Pose	scalar	matrix	elementwise product
scalar	Pose	matrix	elementwise product
Operands		`**`
left	right	type	result
Pose	int >= 0	Pose	exponentiation [1]
Pose	int <=0	Pose	exponentiation [1] then inverse
Operands		`+`
left	right	type	result
Pose	Pose	matrix	elementwise sum
Pose	scalar	matrix	add scalar to all elements
scalar	Pose	matrix	add scalarto all elements
Method	Operation
`clear`	Clear all elements, object now has zero length
`append`	Append a single element
`del`
`enumerate`	Iterate over the elments
`extend`	Append a list of same type pose objects
`insert`	Insert an element
`len`	Return the number of elements
`map`	Map a function of each element
`pop`	Remove first element and return it
`slice`	Index from a slice object
`zip`	Iterate over the elments
operands		results
len(X)	len(Y)	len(Z)	results
1	1	1	Z = X op Y
1	M	M	Z[i] = X op Y[i]
M	1	M	Z[i] = X[i] op Y
M	M	M	Z[i] = X[i] op Y[i]
Spatial object	equivalent class	numpy.ndarray shape
2D rotation SO(2)	SO2	(2,2)
2D pose SE(2)	SE2	(3,3)
3D rotation SO(3)	SO3	(3,3)
3D poseSE3 SE(3)	SE3	(3,3)
3D rotation	UnitQuaternion	(4,)
n/a	Quaternion	(4,)

	- s
	- `spatialmath`	-
	- `spatialmath.base.quaternions`	-
	- `spatialmath.base.transforms2d`	-
	- `spatialmath.base.transforms3d`	-
	- `spatialmath.base.transformsNd`	-
	- `spatialmath.base.vectors`	-
	- `spatialmath.geom3d`	-
	- `spatialmath.pose2d`	-
	- `spatialmath.pose3d`	-
	- `spatialmath.quaternion`	-
len(left)	len(right)	len	operation
1	1	1	`prod = left + right`
1	M	M	`prod[i] = left + right[i]`
N	1	M	`prod[i] = left[i] + right`
M	M	M	`prod[i] = left[i] + right[i]`
len(left)	len(right)	len	operation
1	1	1	`ret = left == right`
1	M	M	`ret[i] = left == right[i]`
N	1	M	`ret[i] = left[i] == right`
M	M	M	`ret[i] = left[i] == right[i]`
len(left)	len(right)	len	operation
1	1	1	`prod = left * right`
1	M	M	`prod[i] = left * right[i]`
N	1	M	`prod[i] = left[i] * right`
M	M	M	`prod[i] = left[i] * right[i]`
Multiplicands		Product
len(left)	right.shape	shape	operation
1	(N,)	(N,)	vector transformation
M	(N,)	(N,M)	vector transformations
1	(N,M)	(N,M)	column transformation
len(left)	len(right)	len	operation
1	1	1	`ret = left != right`
1	M	M	`ret[i] = left != right[i]`
N	1	M	`ret[i] = left[i] != right`
M	M	M	`ret[i] = left[i] != right[i]`
len(left)	len(right)	len	operation
1	1	1	`prod = left - right`
1	M	M	`prod[i] = left - right[i]`
N	1	M	`prod[i] = left[i] - right`
M	M	M	`prod[i] = left[i] right[i]`
len(left)	len(right)	len	operation
1	1	1	`prod = left * right.inv()`
1	M	M	`prod[i] = left * right[i].inv()`
N	1	M	`prod[i] = left[i] * right.inv()`
M	M	M	`prod[i] = left[i] * right[i].inv()`
len(X)	len(s)	len(result)	Result
1	1	1	Y = interp(identity, X, s)
M	1	M	Y[i] = interp(T0, X[i], s)
1	M	M	Y[i] = interp(T0, X, s[i])
Input	Output
Pose	Shape	Structure
SO2	(2,2)	skew-symmetric
SE2	(3,3)	augmented skew-symmetric
SO3	(3,3)	skew-symmetric
SE3	(4,4)	augmented skew-symmetric
Source code for collections

All modules for which code is available

Source code for spatialmath.base.quaternions

Source code for spatialmath.base.transforms2d

Source code for spatialmath.base.transforms3d

Source code for spatialmath.base.transformsNd

Source code for spatialmath.base.vectors

Source code for spatialmath.geom3d

Source code for spatialmath.pose2d

Source code for spatialmath.pose3d

Source code for spatialmath.quaternion

Source code for spatialmath.super_pose

Index

_

A

C

D

E

H

I

L

M

N

O

P

Q

R

S

T

U

V

W

X

Spatial Maths for Python¶

Indices¶

Introduction¶

High-level classes¶

Operators for pose objects¶

SO(n) and SE(n)¶

Unit quaternions and quaternions¶

List capability¶

Vectorization¶

Low-level spatial math¶

Graphics¶

Symbolic support¶

MATLAB compatability¶

spatialmath¶

Python Module Index

Search

Classes and functions¶

Pose classes¶

Pose in 2D¶

Pose in 3D¶

Geometry¶

Geometry in 3D¶

Functions (base)¶

Transforms in 2D¶

Transforms in 3D¶

Transforms in ND¶

Vectors¶

Quaternions¶

Support¶

Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.
