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Geometric objects are created in the typical Python fashion, using the classes themselves as instance factories. A few of their intrinsic properties will be discussed in this sections, others in the following sections on operations and serializations.

Instances of Point , LineString , and LinearRing have as their most important attribute a finite sequence of coordinates that determines their interior, boundary, and exterior point sets. A line string can be determined by as few as 2 points, but contains an infinite number of points. Coordinate sequences are immutable. A third z coordinate value may be used when constructing instances, but has no effect on geometric analysis.

All operations are performed in the x-y plane. In all constructors, numeric values are converted to type float. In other words, Point 0, 0 and Point 0. Shapely does not check the topological simplicity or validity of instances when they are constructed as the cost is unwarranted in most cases.

Shapely is a planar geometry library and z , the height above or below the plane, is ignored in geometric analysis. There is a potential pitfall for users here: coordinate tuples that differ only in z are not distinguished from each other and their application can result in suprisingly invalid geometry objects. For example, LineString [ 0, 0, 0 , 0, 0, 1 ] does not return a vertical line of unit length, but an invalid line in the plane with zero length.

Similarly, Polygon [ 0, 0, 0 , 0, 0, 1 , 1, 1, 1 ] is not bounded by a closed ring and is invalid. Returns the area float of the object. Returns a minx, miny, maxx, maxy tuple float values that bounds the object.

Returns the length float of the object. This can be thought of as a measure of the robustness of a geometry, where larger values of minimum clearance indicate a more robust geometry. If no minimum clearance exists for a geometry, such as a point, this will return math. Returns a string specifying the Geometry Type of the object in accordance with 1.

Returns the minimum distance float to the other geometric object. Returns the Hausdorff distance float to the other geometric object.

The Hausdorff distance between two geometries is the furthest distance that a point on either geometry can be from the nearest point to it on the other geometry. Its x-y bounding box is a minx, miny, maxx, maxy tuple. Coordinate values are accessed via coords , x , y , and z properties.

The Point constructor also accepts another Point instance, thereby making a copy. The LineString constructor takes an ordered sequence of 2 or more x, y[, z] point tuples.

The constructed LineString object represents one or more connected linear splines between the points. Repeated points in the ordered sequence are allowed, but may incur performance penalties and should be avoided. A LineString may cross itself i. Source code , png , hires. Figure 1. A simple LineString on the left, a complex LineString on the right. The MultiPoint boundary of each is shown in black, the other points that describe the lines are shown in grey.

A LineString may also be constructed using a sequence of mixed Point instances or coordinate tuples. The individual coordinates are copied into the new object. The LinearRing constructor takes an ordered sequence of x, y[, z] point tuples.

The sequence may be explicitly closed by passing identical values in the first and last indices. Otherwise, the sequence will be implicitly closed by copying the first tuple to the last index.

As with a LineString , repeated points in the ordered sequence are allowed, but may incur performance penalties and should be avoided. A LinearRing may not cross itself, and may not touch itself at a single point. Figure 2. A valid LinearRing on the left, an invalid self-touching LinearRing on the right.

The points that describe the rings are shown in grey. Shapely will not prevent the creation of such rings, but exceptions will be raised when they are operated on.

As with LineString , a sequence of Point instances is not a valid constructor parameter. The Polygon constructor takes two positional parameters. The first is an ordered sequence of x, y[, z] point tuples and is treated exactly as in the LinearRing case.

Rings of a valid Polygon may not cross each other, but may touch at a single point only. Again, Shapely will not prevent the creation of invalid features, but exceptions will be raised when they are operated on. Figure 3. On the left, a valid Polygon with one interior ring that touches the exterior ring at one point, and on the right a Polygon that is invalid because its interior ring touches the exterior ring at more than one point.

Figure 4. On the left, a Polygon that is invalid because its exterior and interior rings touch along a line, and on the right, a Polygon that is invalid because its interior rings touch along a line. Rectangular polygons occur commonly, and can be conveniently constructed using the shapely. Makes a rectangular polygon from the provided bounding box values, with counter-clockwise order by default. To obtain a polygon with a known orientation, use shapely.

Returns a properly oriented copy of the given polygon. The signed area of the result will have the given sign. A sign of 1. Heterogeneous collections of geometric objects may result from some Shapely operations. For example, two LineStrings may intersect along a line and at a point. To represent these kind of results, Shapely provides frozenset -like, immutable collections of geometric objects. The collections may be homogeneous MultiPoint etc. Figure 5. Members of a GeometryCollection are accessed via the geoms property or via the iterator protocol using in or list.

The MultiPoint constructor takes a sequence of x, y[, z ] point tuples. Members of a multi-point collection are accessed via the geoms property or via the iterator protocol using in or list. The constructor also accepts another MultiPoint instance or an unordered sequence of Point instances, thereby making copies.

Figure 6. The points defining the objects are shown in gray, the boundaries of the objects in black. Its members are instances of LineString and are accessed via the geoms property or via the iterator protocol using in or list.

The constructor also accepts another instance of MultiLineString or an unordered sequence of LineString instances, thereby making copies. The MultiPolygon constructor takes a sequence of exterior ring and hole list tuples: [ a1, …, aM , [ b1, …, bN , …] , …].

More clearly, the constructor also accepts an unordered sequence of Polygon instances, thereby making copies. Figure 7. On the left, a valid MultiPolygon with 2 members, and on the right, a MultiPolygon that is invalid because its members touch at an infinite number of points along a line.

Its members are instances of Polygon and are accessed via the geoms property. Empty features can be created by calling the various constructors with no arguments. Almost no operations are supported by empty features. The list of coordinates that describe a geometry are represented as the CoordinateSequence object. These sequences should not be initialised directly, but can be accessed from an existing geometry as the Geometry. Coordinate sequences can be indexed, sliced and iterated over as if they were a list of coordinate tuples.

Multipart geometries do not have a coordinate sequence. Instead the coordinate sequences are stored on their component geometries. It can be useful to specify position along linear features such as LineStrings and MultiLineStrings with a 1-dimensional referencing system.

Shapely supports linear referencing based on length or distance, evaluating the distance along a geometric object to the projection of a given point, or the point at a given distance along the object. Returns the distance along this geometric object to a point nearest the other object. If the normalized arg is True , return the distance normalized to the length of the object. The project method is the inverse of interpolate.

Objects of the types explained in Geometric Objects provide standard 1 predicates as attributes for unary predicates and methods for binary predicates. Whether unary or binary, all return True or False. Standard unary predicates are implemented as read-only property attributes.

An example will be shown for each. Returns True if the feature has not only x and y , but also z coordinates for 3D or so-called, 2.

Returns True if coordinates are in counter-clockwise order bounding a region with positive signed area. This method applies to LinearRing objects only. Returns True if the feature is a closed and simple LineString. This property is applicable to LineString and LinearRing instances, but meaningless for others.

Returns True if the feature does not cross itself. The simplicity test is meaningful only for LineStrings and LinearRings. The validity test is meaningful only for Polygons and MultiPolygons. True is always returned for other types of geometries. A valid Polygon may not possess any overlapping exterior or interior rings.

A valid MultiPolygon may not collect any overlapping polygons. Operations on invalid features may fail. The two points above are close enough that the polygons resulting from the buffer operations explained in a following section overlap.

Standard binary predicates are implemented as methods. These predicates evaluate topological, set-theoretic relationships. In a few cases the results may not be what one might expect starting from different assumptions. All take another geometric object as argument and return True or False. Returns True if the two objects are of the same geometric type, and the coordinates of the two objects match precisely.

Returns True if the set-theoretic boundary , interior , and exterior of the object coincide with those of the other. The coordinates passed to the object constructors are of these sets, and determine them, but are not the entirety of the sets.

Equivalent lines, for example, can be constructed differently. Returns True if the object is approximately equal to the other at all points to specified decimal place precision. Returns True if no points of other lie in the exterior of the object and at least one point of the interior of other lies in the interior of object. This predicate applies to all types, and is inverse to within.

The expression a. Binary predicates can be used directly as predicates for filter or itertools. Returns True if every point of other is a point on the interior or boundary of object. This is similar to object. Returns True if every point of object is a point on the interior or boundary of other. This is equivalent to other. Returns True if the interior of the object intersects the interior of the other but does not contain it, and the dimension of the intersection is less than the dimension of the one or the other.