# 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

point = [0, 0, 0]

point = Point(0, 0, 0)

vector

vector = [0, 0, 1]

vector = Vector(0, 0, 1)

line

line = [0, 0, 0], [1, 0, 0]

line = Line(point, point)

plane

plane = [0, 0, 0], [0, 0, 1]

plane = Plane(point, vector)

circle

circle = ([0, 0, 0], [0, 0, 1]), 1.0

circle = Circle(plane, radius)

polyline

polyline = [0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 0, 0]

polyline = Polyline(points)

polygon

polygon = [0, 0, 0], [1, 0, 0], [1, 1, 0]

polygon = Polygon(points)

frame

frame = [0, 0, 0], [1, 0, 0], [0, 1, 0]

frame = Frame(point, xaxis, xypoint)

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.

## 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 axis-aligned 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 serialized to JSON format, and deserialized from such a representation to reconstruct an equivalent object without loss of information. The serialization mechanism relies on the underlying data model.