Points and transformations

Points

Just like vectors, points defined in nalgebra are elements of $\mathbb{R}^n$ However, they both have different uses, depending on your application.

• In a very general context: vectors are what you choose when you need a bag of coordinates (relative to a suitable basis). Points will not be useful for you as they support much less operations than vectors. Vectors support most common operations like cross product, dot product, componentwise binary operations, etc. Simply search for the name of the operation you want on the root module documentation and it will likely be listed as a free function or a method. If you can't find what you need, feel free to open an issue or ask about it on the user forum.
• In a geometric context: a point is an element of an euclidean space, while a vector is an element of the underlying vector space. In other words, a point is a location in space while a vector is a translation, i.e., the displacement required to move from one point to another:

Intuition can often be safely used to anticipate which operations are or are not allowed between points, and what is the relation between a vector and a point. In particular:

• A point's coordinates vector can be accessed as a public field: p.coords.
• The sum of a point $a$ with a vector $v$ returns another point which is the translation of $a$ by the vector $v$.
• The difference of two point $b$ and $a$ returns the vector $\vec{ab}$ such that $b = a + \vec{ab}$.
• The sum, multiplication and division of two points are meaningless and not supported by nalgebra.
• For convenience, you may compute the center of two points using the free function na::center(a, b).
• The homogeneous coordinates of a point usually end with a 1 while those of a vector always ends with a 0.
• Points are affected by the translational component of transformations while vectors are only rotated and scaled:

let t = Isometry2::new(Vector2::new(1.0, 1.0), f32::consts::PI);

let p = Point2::new(1.0, 0.0); // Will be affected by the rotation and the translation.

let v = Vector2::x(); // Will *not* be affected by the translation.

assert_relative_eq!(t * p, Point2::new(-1.0 + 1.0, 1.0));

// ^^^^ │ ^^^^^^^^

// rotated │ translated

assert_relative_eq!(t * v, Vector2::new(-1.0, 0.0));

// ^^^^^

// rotated only

Points can be created with various constructors:

Constructor	Effect
::origin()	Builds the point with all its coordinates set to zero.
::from_coordinates(v)	Builds the point with coordinates vector equal to v.
::from_homogeneous(v)	Builds a point with the given homogeneous coordinates, i.e., with all its components divided by the last one (which is then removed).
::new(x, y, ...)	Builds the point with the given coordinates. Works only for points with a dimension known at compile-time and smaller than 6.
Bounded::max_value()	Builds the point with all its coordinates set to the maximal value of the underlying scalar type.
Bounded::min_value()	Builds the point with all its coordinates set to the minimal value of the underlying scalar type.

// Build using components directly.

let p0 = Point3::new(2.0, 3.0, 4.0);

// Build from a coordinates vector.

let coords = Vector3::new(2.0, 3.0, 4.0);

let p1 = Point3::from_coordinates(coords);

// Build by translating the origin.

let translation = Vector3::new(2.0, 3.0, 4.0);

let p2 = Point3::origin() + translation;

// Build from homogeneous coordinates. The last component of the

// vector will be removed and all other components divided by 10.0.

let homogeneous_coords = Vector4::new(20.0, 30.0, 40.0, 10.0);

let p3 = Point3::from_homogeneous(homogeneous_coords);

assert_eq!(p0, p1);

assert_eq!(p0, p2);

assert_eq!(p0, p3);

Transformations

Transformations are algebraic entities that act on points and vectors to change their coordinates. The following figure shows the whole spectrum of transformations supported by dedicated types on nalgebra (notations like Affine2/3 means either Affine2 or Affine3):

All the types shown in this diagram are actually type aliases for types generic wrt. the dimension and with a name ending with Base instead of a number, i.e., the base implementation of the translation is TranslationBase. The more general transformations Transform2/3, Projective2/3, and Affine2/3 are aliases for the parametrized type TransformBase<..., Category> where its last type parameter Category specifies which of the three variants is represented. Note that raw matrices can also be interpreted as general transformations that are not necessarily invertible. This may be useful in a generic context.

Transformations can be composed (by multiplication) even if they do not have the same type. The type of the composition result is the most general transformation of both. For example, multiplying a Projective3 (inversible transformation) with a Similarity3 yields a Projective3 because all similarities are inversible. An exception is the product of any pure rotation by a pure translation: none is more general than the other so the result is an Isometry2/3 which is the most specific transformation enough to represent this composition.

Converting a transformation to one that is more general is possible with the na::convert(...) function. The other way round is sometimes possible using na::try_convert(...) which returns a non-None value in case of success. For example, converting a Similarity2 to an Isometry2 will succeed only if the similarity scaling factor is one (which is checked at run-time by ::try_convert(...)). Note that if you do not want to consume your input value upon conversion, use na::convert_ref(...) and na::try_convert_ref(...) with references instead.

// Isometry -> Similarity conversion always succeeds.

let iso = Isometry2::new(Vector2::new(1.0f32, 2.0), na::zero());

let _: Similarity2<f32> = na::convert(iso);

// Similarity -> Isometry conversion fails if the scaling factor is not 1.0.

let sim_without_scaling = Similarity2::new(Vector2::new(1.0f32, 2.0), 3.14, 1.0);

let sim_with_scaling = Similarity2::new(Vector2::new(1.0f32, 2.0), 3.14, 2.0);

let iso_success: Option<Isometry2<f32>> = na::try_convert(sim_without_scaling);

let iso_fail: Option<Isometry2<f32>> = na::try_convert(sim_with_scaling);

assert!(iso_success.is_some());

assert!(iso_fail.is_none());

// Similarity -> Isometry conversion can be forced at your own risks.

let iso_forced: Isometry2<f32> = na::convert_unchecked(sim_with_scaling);

assert_eq!(iso_success.unwrap(), iso_forced);

Forcing the conversion of a transformation to a more specific one, without performing any invariants check at runtime, is possible using na::convert_unchecked(...) and na::convert_unchecked_ref(...). This is typically used when you know that the conversion is valid and want to avoid the run-time algebraic properties tests performed by ::try_convert(...).

Rotations

A rotation is an invertible origin-, distance- and orientation-preserving transformation. It is commonly known by algebraists as the $n$-dimensional Special Orthogonal group $\mathbf{SO}(n)$:

2D (resp. 3D) rotations are available as the Rotation2 and UnitComplex (resp. Rotation3 and UnitQuaternion) structures. Unit quaternions and rotation matrices have three construction methods in common:

Constructor	Effect
::from_axis_angle(ax, ang)	Builds a rotation of ang radians around the axis ax.
::from_scaled_axis(axang)	This is the same as ::from_axis_angle(...) with axang = ax * an.
::from_euler_angles(...)	Builds a rotation from its roll, pitch, and yaw components applied in that order.

In addition, ::from_matrix_unchecked(...) will initialize a rotation matrix from a raw Matrix2 or Matrix3. In that case, it is not checked whether or not it is actually a rotation. This should be used with great care.

Finally, an unit complex or an unit quaternion may be constructed from, or converted to, a Rotation2/3 by ::from_rotation_matrix(...) and ::to_rotation_matrix(...) respectively.

Direct isometries

Direct isometries are rotations followed by translations, i.e., they preserve orientations and distances, but not the origin. They are commonly known by algebraists as the $n$-dimensional Special Euclidean group $\mathbf{SE}(n)$. Note that isometries that are not direct because they include a reflection are not yet supported by nalgebra.

2D and 3D isometries are available as the Isometry2 and Isometry3 structures. They are internally represented as a translation vector and an unit complex number (for 2D) or an unit quaternion (for 3D). For isometries containing rotation matrices instead, use IsometryMatrix2 and IsometryMatrix3.

Constructor	Effect
::new(t, axang)	Builds an isometry that rotates by axang.norm() radians around the axis axang, and translates by the vector t. For 2D isometries, axang is just a scalar.
::identity()	The isometry with does nothing, i.e., a zero translation and a identity rotation.
::from_parts(t, rot)	Builds an isometry from a translation t and a rotation rot (which is a unit complex number, unit quaternion or rotation matrix, depending on the kind of isometry being built). This is equivalent to the product t * rot.
One::one()	Same as ::identity().

Direct isometries are commonly used to represent the position and orientation of a solid object or a camera. Convenient constructors are thus provided to center and orient an isometry as if it was the head of an observer:

Constructor	Effect
::look_at_rh(...)	A right-handed look-at view matrix that makes the $\mathbf{z}$ axis point toward the staring direction. This builds a transformation that maps a point from global coordinates into the local coordinates of a camera centered at point and looking at a second one.
::look_at_lh(...)	A left-handed look-at view matrix. This is the same as ::look_at_rh(...) except that it makes the $\mathbf{z}$ axis points toward the opposite of the staring direction.
::new_observer_frame(...)	Builds a transformation that maps a point from the local coordinates of an observer into global coordinates. This is the inverse transformation of ::look_at_rh(...).

While the right-handed look-at and the observer-frame isometries are closely related (they are inverses of each other), they have very different semantic meanings. The former will typically be used in the context of rendering a scene. Indeed, we want to bring the objects into the local space of the camera for rendering. The latter will typically be used to orient a solid, a robot, a head, etc. because, in that case, we want to be able to localize our object into the global scene.

Similarities

A similarity is an uniform scaling, followed by a rotation, followed by translation. Thus, it preserves angle sizes and maps parallel lines to parallel lines. Orientation is lost if the scaling factor is negative. They are usually known by algebraists as the $n$-dimensional Similarity group $\mathbf{S}(n)$.

Similarities share the same constructors as the isometry described in the previous section. Though most of them take one additional parameter: the uniform scaling factor.

Homogeneous coordinates

We are used to work with cartesian coordinates. However, it has some limitations when it comes to represent transformations in matrix form. For example, the translational part of an isometry cannot be expressed in cartesian coordinates in a compact way (using a matrix). That's why homogeneous coordinates exist: rotations, scaling, and translations can all be combined in a single matrix of higher dimension than the euclidean space we are working with.

Definition

Any affine transformation have equivalent representations as raw matrices. In particular, any 2D affine transformation is representable as a Matrix3, and any 3D affine transformation is representable as a Matrix4. Each such raw matrix is known as the homogeneous coordinates of the corresponding transformation. The following example shows the homogeneous coordinate matrix of a 2D similarity. Note that this process implicitly computes the rotation matrix representation (which components are numbered from $r_{11}$ to $r_{22}$) of the unit complex number that represents the rotational part of the underlying isometry:

Note that if $H$ is an homogeneous coordinate matrix of some transformation and $\lambda \neq 0$ any real number, $H$ and $\lambda H$ are considered equivalent. Therefore, any transformation has an infinite number of homogeneous coordinates (just multiply every single entry of the raw matrix by a non-zero real number). However, we usually only care about the representation with a 1 on the bottom-right entry as it simplifies computations. Use the .to_homogeneous() method to compute the homogeneous coordinates of any vector, point, or transformation.

Interactions with points

It is natural to wonder how transformation expressed in homogeneous coordinates can be applied to vectors and points. For example, multiplying a Vector2 by a Rotation2 will yield a rotated Vector2. On the other hand, multiplying a Vector2 by a Matrix3, obtained by converting a 2D rotation to homogeneous coordinates, will not even compile! The only solution is to somehow convert the 2D vector to a Vector3 and perform the multiplication. The same reasoning applies to points: we would need to convert the Point2 to a Vector3. Those conversions can be made by computing the homogeneous coordinates of vectors and points themselves with .to_homogeneous().

Computing a vector's homogeneous coordinates will append a 0 to it while computing a point's homogeneous coordinates will append a 1:

This subtle difference reflects the fact emphasized at the beginning of this chapter: transformations do not have the same effect on points as on vectors. The 0 appended to vectors will cancel the translational component of any transformation; the 1 appended to points will let it be translated normally.

Points and vectors in homogeneous coordinates can be transformed back using ::from_homogeneous(...). For vectors, the last coordinate entry will be removed. For points, the last coordinate will also be removed and every other coordinate will be divided by it:

The following example shows the application of a transformation using two methods: with dedicated transformation types, and with homogeneous coordinates. While the end-results are the same, using dedicated types is much more concise and efficient.

// With dedicated transform types.

let iso = Isometry2::new(Vector2::new(1.0, 1.0), f32::consts::PI);

let pt = Point2::new(1.0, 0.0);

let vec = Vector2::x();

let transformed_pt = iso * pt;

let transformed_vec = iso * vec;

assert_relative_eq!(transformed_pt, Point2::new(0.0, 1.0));

assert_relative_eq!(transformed_vec, Vector2::new(-1.0, 0.0));

// With homogeneous coordinates (plain matrices).

let iso = Isometry2::new(Vector2::new(1.0, 1.0), f32::consts::PI);

let pt = Point2::new(1.0, 0.0);

let vec = Vector2::x();

// Compute using homogeneous coordinates.

let hom_iso = iso.to_homogeneous();

let hom_pt = pt.to_homogeneous();

let hom_vec = vec.to_homogeneous();

let hom_transformed_pt = hom_iso * hom_pt;

let hom_transformed_vec = hom_iso * hom_vec;

// Convert back to the cartesian coordinates.

let transformed_pt = Point2::from_homogeneous(hom_transformed_pt).unwrap();

let transformed_vec = Vector2::from_homogeneous(hom_transformed_vec).unwrap();

assert_relative_eq!(transformed_pt, Point2::new(0.0, 1.0));

assert_relative_eq!(transformed_vec, Vector2::new(-1.0, 0.0));