---
title: "Compute an inertia tensor in C#"
url: https://recursiveai.org/onemath/cookbook/inertia-tensors/
description: "Compute the mass, center of mass, inertia tensor and principal axes of a rigid body made of boxes, spheres and cylinders in C#, the same bits on every machine. Runnable code, with OneMath."
version: "1.0.0"
---

# Compute an inertia tensor in C#

Build each part from its shape and mass, combine them, and diagonalize the result. The program below takes a body made of a 2 kg box and a 1 kg sphere and prints its mass, center of mass, inertia tensor and principal axes.

```shell
dotnet new console -o Inertia && cd Inertia
dotnet add package RecursiveAI.OneMath --version 1.0.0
```

Put this in `Program.cs` and run `dotnet run`:

```csharp
using RecursiveAI.OneMath;
using RecursiveAI.OneMath.Physics;

// A body made of two parts: a 2 kg box and a 1 kg sphere offset along x.
var box = new MassPropertiesD(2, new Vector3D(0, 0, 0), InertiaD.SolidBox(2, new Vector3D(1, 0.5, 0.25)));
var ball = new MassPropertiesD(1, new Vector3D(1.5, 0.5, 0), InertiaD.SolidSphere(1, 0.4));

MassPropertiesD body = InertiaD.Combine([box, ball]);
Console.WriteLine($"mass {body.Mass}, center {body.Center}");
Console.WriteLine($"inertia about the center:\n{body.Inertia}");

PrincipalAxesD axes = InertiaD.PrincipalAxes(body.Inertia);
Console.WriteLine($"principal moments {axes.Moments}");
Console.WriteLine($"orientation {axes.Orientation}");
```

```text
mass 3, center (0.5, 0.16666666666666666, 0)
inertia about the center:
[0.43900000000000006, -0.5, 0; -0.5, 2.2723333333333335, 0; 0, 0, 2.564]
principal moments (0.3115029928215277, 2.399830340511806, 2.564)
orientation (0, 0, 0.12451268641694911, 0.9922180158217418)
```

## The pieces

- `InertiaD.SolidBox(mass, halfExtents)`, `SolidSphere(mass, radius)` and `SolidCylinder` (axis along Y, from its mass, radius and half height) give the tensor of a solid about its own center, axes along the shape. `InertiaF` is the float form.
- `MassPropertiesD(mass, center, inertia)` is one part; `InertiaD.Combine(parts)` returns the total mass, the center of mass and the tensor about that center (parallel-axis shifts included). `ShiftToPoint`, `ShiftToCenter` and `Rotate` are there for doing it by hand.
- `InertiaD.PrincipalAxes(tensor)` diagonalizes it: `Moments` are the principal moments in ascending order, `Rotation` the axes as a matrix and `Orientation` the same as a quaternion, with a right-handed frame and fixed signs.

The decomposition is cyclic Jacobi with a written-down order of operations, so the principal frame comes out the same, bit for bit, on every platform. NaN, infinity, a negative mass or a non-symmetric tensor throw an exception naming the field. Details in [inertia and eigen decomposition](https://recursiveai.org/onemath/docs/physics/).

## Limits

- **Three solid shapes are built in:** box, sphere and cylinder. For any other shape, compute its tensor yourself and pass it to `MassPropertiesD`.
- **Solids of uniform density.** Hollow shapes and shells aren’t provided.