Open Space Toolkit ▸ Mathematics

Geometry, curve fitting, optimization.

Getting Started

Want to get started? This is the simplest and quickest way:

Nothing to download or install! This will automatically start a JupyterLab environment in your browser with Open Space Toolkit libraries and example notebooks ready to use.

Alternatives

Docker Images

Docker must be installed on your system.

iPython

The following command will start an iPython shell within a container where the OSTk components are already installed:

docker run -it openspacecollective/open-space-toolkit-mathematics-development python3.11 -m IPython

Once the shell is up and running, playing with it is easy:

import numpy
from ostk.mathematics.geometry import Angle
from ostk.mathematics.geometry.d3.transformation.rotation import Quaternion
from ostk.mathematics.geometry.d3.transformation.rotation import RotationVector

rv = RotationVector(numpy.array([[0.0], [0.0], [1.0]], dtype=float), Angle.degrees(15.0)) # Construct rotation vector

q_AB = Quaternion(1.0, 2.0, 3.0, 4.0, Quaternion.Format.XYZS).to_normalized() # Construct quaternion and normalize
q_BC = Quaternion.rotation_vector(rv) # Construct quaternion from rotation vector

q_AC = q_AB * q_BC # Multiply quaternions

Tip: Use tab for auto-completion!

JupyterLab

The following command will start a JupyterLab server within a container where the OSTk components are already installed:

docker run --publish=8888:8888 openspacecollective/open-space-toolkit-mathematics-jupyter

Once the container is running, access http://localhost:8888/lab and create a Python 3 Notebook.

Installation

C++

The binary packages are hosted using GitHub Releases:

• Runtime libraries: open-space-toolkit-mathematics-X.Y.Z-1.x86_64-runtime
• C++ headers: open-space-toolkit-mathematics-X.Y.Z-1.x86_64-devel
• Python bindings: open-space-toolkit-mathematics-X.Y.Z-1.x86_64-python

Debian / Ubuntu

After downloading the relevant .deb binary packages, install:

apt install open-space-toolkit-mathematics-*.deb

Python

Install from PyPI:

pip install open-space-toolkit-mathematics

Documentation

Documentation is available here:

• C++
• Python

Structure

The library exhibits the following structure:

├── Object
│   ├── Vector
│   ├── Matrix
│   └── Interval
├── Geometry
│   ├── 2D
|   │   ├── Object
│   │   │   ├── Point
│   │   │   ├── Point Set
│   │   │   ├── Line
│   │   │   ├── Line String
│   │   │   ├── Multi Line String
│   │   │   └── Polygon
│   │   ├── Intersection
│   │   └── Transformation
│   │       ├── Identity
│   │       ├── Translation
│   │       ├── Rotation
│   │       ├── Reflection
│   │       ├── Scaling
│   │       └── Shear
│   ├── 3D
|   │   ├── Object
│   │   │   ├── Point
│   │   │   ├── Point Set
│   │   │   ├── Line
│   │   │   ├── Ray
│   │   │   ├── Segment
│   │   │   ├── Line String
│   │   │   ├── Polygon
│   │   │   ├── Plane
│   │   │   ├── Cuboid
│   │   │   ├── Sphere
│   │   │   ├── Ellipsoid
│   │   │   ├── Cone
│   │   │   ├── Pyramid
│   │   │   └── Composite
│   │   ├── Intersection
│   │   └── Transformation
│   │       ├── Identity
│   │       ├── Translation
│   │       ├── Rotation
│   │       │   ├── Quaternion
│   │       │   ├── Euler Angle
│   │       │   ├── Rotation Vector
│   │       │   └── Rotation Matrix
│   │       ├── Reflection
│   │       ├── Scaling
│   │       └── Shear
├── Dynamics
│   ├── State
│   ├── Solver
│   │   ├── Runge–Kutta 4 (RK4)
│   │   ├── Dormand–Prince 5 (DP5)
│   │   └── Runge–Kutta–Fehlberg 78 (F78)
│   └── Systems
├── Curve Fitting
│   ├── Interpolator
│   │   ├── Linear
│   │   ├── Barycentric Rational
│   │   ├── Cubic Spline
│   │   └── Lagrange
│   └── Smoothing
├── Optimization
│   ├── Problem
│   └── Algorithms
│       ├── Gradient Descent
│       └── Evolutionary
│           ├── Genetic
│           ├── Differential Evolution
│           └── Swarm
└── Statistics

Tutorials

Tutorials are available here:

• C++
• Python

Features

Geometry Queries

• ○: query only
• ✔: query / intersection set
• : to be implemented
• -: undefined

3D

Intersect	Point	Point Set	Line	Ray	Segment	Line String	Polygon	Plane	Cuboid	Sphere	Ellipsoid	Cone	Pyramid	Composite
Point			○	○				✔	○	○	○
Point Set								✔	○	○	○
Line	○							✔	○	○	✔
Ray	○							✔		○	✔
Segment								✔		○	✔
Line String
Polygon
Plane	✔	✔	✔	✔	✔					○	○
Cuboid	○	○	○
Sphere	○	○	○	○	○			○					○
Ellipsoid	○	○	✔	✔	✔			○				✔	✔
Cone											✔
Pyramid										○	✔
Composite

Contain	Point	Point Set	Line	Ray	Segment	Line String	Polygon	Plane	Cuboid	Sphere	Ellipsoid	Cone	Pyramid	Composite
Point			-	-	-	-	-	-	-	-	-	-	-
Point Set			-	-	-	-	-	-	-	-	-	-	-
Line	✔						-	-	-	-	-	-	-
Ray	✔	✔	-				-	-	-	-	-	-	-
Segment	✔		-	-			-	-	-	-	-	-	-
Line String			-	-			-	-	-	-	-	-	-
Polygon			-	-				-	-	-	-	-	-
Plane	✔	✔	✔	✔	✔				-	-	-	-	-
Cuboid	✔	✔	-	-				-
Sphere	✔	✔	-	-	-	-	-	-
Ellipsoid	✔	✔	-	-	-	-	-	-
Cone						-	-	-
Pyramid	✔							-
Composite

Setup

Development Environment

Using Docker for development is recommended, to simplify the installation of the necessary build tools and dependencies. Instructions on how to install Docker are available here.

To start the development environment:

make start-development

This will:

1. Build the openspacecollective/open-space-toolkit-mathematics-development Docker image.
2. Create a development environment container with local source files and helper scripts mounted.
3. Start a bash shell from the ./build working directory.

If installing Docker is not an option, you can manually install the development tools (GCC, CMake) and all required dependencies, by following a procedure similar to the one described in the Development Dockerfile.

Build

From the ./build directory:

cmake ..
make

Tip: ostk-build simplifies building from within the development environment.

Test

To start a container to build and run the tests:

make test

Or to run them manually:

./bin/open-space-toolkit-mathematics.test

Tip: ostk-test simplifies running tests from within the development environment.

Dependencies

Name	Version	License	Link
Pybind11	2.12.0	BSD-3-Clause	github.com/pybind/pybind11
Eigen	3.3.7	MPL2	eigen.tuxfamily.org
Geometric Tools Engine	3.28	Boost Software License	geometrictools.com
Core	main	Apache License 2.0	github.com/open-space-collective/open-space-toolkit-core

Contribution

Contributions are more than welcome!

Please read our contributing guide to learn about our development process, how to propose fixes and improvements, and how to build and test the code.

Special Thanks

License

Apache License 2.0