Geometry¶
This tutorial provides a quick tour of the functionality in compas.geometry
.
For a complete overview, visit the API Reference:
https://compas.dev/compas/latest/api/compas.geometry.html
Points and Vectors¶
In Python, the simplest way to represent a point or a vector is through a list of XYZ components. To retrieve or modify one of the components, simply access the corresponding index in the list
>>> point = [1, 1, 1]
>>> point[0]
1
>>> point[0] = 5
>>> point[0]
5
>>> point
[5, 1, 1]
To add two points, compute the length of a vector, … you can apply simple math to the items of these lists
>>> a = [1, 0, 0]
>>> b = [0, 1, 0]
>>> c = [a[i] + b[i] for i in range(3)]
>>> c
[1, 1, 0]
Most geometric operations can not be expressed so concisely as the addition of two points or vectors, and writing this out quickly becomes quite tedious.
Therefore, COMPAS provides many functions for points and vectors that simplify the use of basic operations.
>>> from compas.geometry import add_vectors, cross_vectors
>>> a = [1, 0, 0]
>>> b = [0, 1, 0]
>>> add_vectors(a, b)
[1, 1, 0]
>>> cross_vectors(a, b)
[0, 0, 1]
In addition to basic vector algebra functions, COMPAS provides Point
and Vector
classes
that can be used interchangeably with native Python types for geometrical calculations.
They provide access to XYZ coordinates through indexing as well as through x
, y
, and z
attributes,
support basic operations such as addition, subtraction, and multiplication,
and bind many of the basic geometry functions as methods.
>>> from compas.geometry import Point, Vector
>>> point = Point(1, 0, 0)
>>> point[0]
1
>>> point.x
1
>>> a = Point(1, 0, 0)
>>> b = Point(0, 1, 0)
>>> c = a + b
>>> c
Point(1.000, 1.000, 0.000)
>>> u = Vector(1, 0, 0)
>>> u * 3
Operators such as +
or *
involving COMPAS geometry objects always return a new COMPAS geometry object.
However, the result type is not always the same as the type of the inputs
>>> a = Point(0, 0, 0)
>>> b = Point(1, 1, 0)
>>> b  a
Vector(1.000, 1.000, 0.000)
Basic functions, on the other hand, always return native Python objects, regardless of the input
>>> x = Vector(1, 0, 0)
>>> y = Vector(0, 1, 0)
>>> cross_vectors(x, y)
[0.0, 0.0, 1.0]
Many of the basic functions are also available as object methods
>>> x.cross(y)
Vector(0.000, 0.000, 1.000)
>>> x.scale(3)
None
>>> x[0]
3.0
>>> x.scaled(3)
Vector(9.000, 0.000, 0.000)
>>> x[0]
3.0
>>> x.dot(y)
0.0
>>> z = Vector(1, 2, 3)
>>> z.unitize()
Vector(0.267, 0.535, 0.802)
>>> x.cross([0, 1, 0])
Vector(0.000, 0.000, 1.000)
>>> x.angle(y)
1.5707963267948966
For an overview of all functionality, see Points and Vectors in the API Reference.
Other Primitives¶
In addition to points and vectors, COMPAS provides Line
, Plane
,
Polyline
, Polygon
, Circle
, Ellipse
, Frame
, and Quaternion
.
All COMPAS primitives can be used interchangeably with native Python objects as input for geometry functions and object methods. The following representations of geometric objects are entirely equivalent.
Object 
Python 
COMPAS 

point 


vector 


line 


plane 


circle 


polyline 


polygon 


frame 


In addition to the default instantiation mechanism, which is always based on the default representation of geometric entities, many primitives provide “alternative constructors”.
>>> a = Vector(1, 0, 0)
>>> b = Vector.from_start_end([1, 0, 0], [2, 0, 0])
>>> a == b
True
>>> a = Plane([0, 0, 0], [0, 0, 1])
>>> b = Plane.from_three_points([0, 0, 0], [1, 0, 0], [0, 1, 0])
>>> a == b
True
>>> a = Frame([0, 0, 0], [3, 0, 0], [0, 2, 0])
>>> b = Frame.from_points([0, 0, 0], [5, 0, 0], [1, 2, 0])
>>> a == b
True
Primitives also provide easy access to many of the geometric properties of the represented objects.
>>> line = Line([0, 0, 0], [2, 0, 0])
>>> line.start
Point(0.000, 0.000, 0.000)
>>> line.start.x
0.0
>>> line.end
Point(2.000, 0.000, 0.000)
>>> line.vector
Vector(2.000, 0.000, 0.000)
>>> line.vector[0]
2.0
>>> line.direction
Vector(1.000, 0.000, 0.000)
>>> line.midpoint
Point(1.000, 0.000, 0.000)
>>> line.length
2.0
Frame
and Quaternion
are special primitives that play an important role
in transformations (see Transformations).
A frame defines a local coordinate system and quaternions provide an alternative formulation for rotations.
Predicates¶
Intersections¶
To compute intersections between primitives and/or shapes, use the intersection functions.
>>> line = [1, 1, 0], [1, 1, 1]
>>> plane = [0, 0, 0], [0, 0, 1]
>>> intersection_line_plane(line, plane)
[1.0, 1.0, 0.0]
>>> line = [1, 1, 0], [1, 1, 1]
>>> plane = Plane.worldXY()
>>> intersection_line_plane(line, plane)
[1.0, 1.0, 0.0]
>>> line = Line([1, 1, 0], [1, 1, 1])
>>> plane = Plane.worldXY()
>>> line.intersection(plane)
Point(1.000, 1.000, 0.000)
Transformations¶
All transformations of geometric objects are based on Transformation
,
which defines a general projective or affine transformation in eucledian space,
represented by a 4x4 transformation matrix.
The default transformation is an identity
>>> from compas.geometry import Transformation
>>> X = Transformation()
>>> a = Point(1, 0, 0)
>>> b = a.transformed(X)
>>> a == b
True
The base transformation object provides alternative constructors to create transformations between different coordinate systems represented by frames
>>> X = Transformation.from_frame(frame)
>>> X = Transformation.from_frame_to_frame(frame1, frame2)
>>> X = Transformation.from_change_of_basis(frame1, frame2)
Translation
, Rotation
, Scale
, Shear
, and Projection
define specific transformations
>>> import math
>>> from compas.geometry import Rotation
>>> R = Rotation.from_axis_and_angle([0, 0, 1], math.radians(90))
All primitives support transformations through the methods Primitive.transform()
and Primitive.transformed()
.
The former modifies the object in place, whereas the latter returns a new object
>>> point = Point(1, 0, 0)
>>> point.transformed(R)
Point(0.000, 1.000, 0.000)
>>> point.transform(R)
None
>>> point.y
1.0
All transformation objects support matrix multiplication with the *
operator.
Remember that the multiplication order of transformation matrices is important!
>>> T = Translation.from_vector([1, 1, 0])
>>> R = Rotation.from_axis_and_angle([0, 0, 1], math.radians(90))
>>> point = Point(1, 0, 0)
>>> point.transformed(T * R)
Point(1.000, 2.000, 0.000)
>>> point.transformed(R * T)
Point(1.000, 2.000, 0.000)
Note that points and vectors behave different in transformations. Applying the same transformation above to a vector instead of a point, we get a different result, because the translation component is ignored
>>> vector = Vector(1, 0, 0)
>>> vector.transformed(R * T)
Vector(0.000, 1.000, 0.000)
>>> vector.transformed(T * R)
Vector(0.000, 1.000, 0.000)
Note that geometries are not implicitly linked
>>> a = Point(0, 0, 0)
>>> b = Point(1, 0, 0)
>>> ab = Line(a, b)
>>> R = Rotation.from_axis_and_angle([0, 0, 1], math.radians(90))
>>> ab.transform(R)
>>> ab.end
Point(0.000, 1.000, 0.000)
>>> ab.end == b
False
Shapes¶
“Shapes” (or “Solids”) extend the primitives with volumetric geometries.
Box
, Capsule
, Cone
, Cylinder
, Polyhedron
, Sphere
, and Torus
are available.
As usual, there is a default “constructor” and several “alternative constructors”. The default constructor, corresponds to the canonical representation of the geometrical objects.
After construction, all shapes are axisaligned and centered at the origin. To move shapes to different locations in 3D space, change their orientations, or modify their geometry, use transformations
>>> b1 = Box(Frame.worldXY(), 5, 1, 3)
>>> b2 = Box.from_width_height_depth(5, 1, 3)
>>> b1 == b2
True
Boolean Operations¶
3D boolean operations are not supported in COMPAS by default,
but are available through the plugin mechanism if compas_cgal
is installed
Algorithms¶
For a complete overview of available algorithms, see compas.geometry
.
Serialization¶
All geometry objects (primitives, shapes, transformations) can be serialised to JSON format, and unserialised from such a representation to reconstruct an equivalent object without loss of information. The serialisation mechanism relies on the underlying data model.