6 KiB
Fundamental Data Types
Most of the data that NumIdr operates on is stored in arrays, but there are a few other types and structures that are important to understand.
Arrays
What is an Array?
In most programming languages and libraries, the word "array" is used to mean a one-dimensional list of values that is contiguous in memory. A typical array of integers may be written in list form like this:
[1, 4, 10, 2, -5, 18]
In a typical one-dimensional array, elements are indexed by a single integer, starting at zero and increasing from left to right.
NumIdr, however, uses the word a bit more generally: a NumIdr array is a multi-dimensional structure that can be indexed by any number of integers. NumIdr arrays are written as nested lists:
[[4, -9, -2],
[5, -6, 1]]
Unlike in other languages, however, this is not a nested structure. The above is a single array, and it is always manipulated as one object.
Properties of Arrays
The Array datatype has the following parameters:
Array : (s : Vect rk Nat) -> (a : Type) -> Type
The first parameter is the shape, a list of numbers (the dimensions) where each dimension is the length of a particular axis of the array. The second parameter is the element type, the type of the values inside the array.
Let's return to the array example from earlier:
[[4, -9, -2],
[5, -6, 1]]
This is a rank-2 array, meaning that it has two axes. Rank-2 arrays are typically called matrices. To determine the dimensions of the array, we count the size of each nested list from the outside in, which in the case of matrices means the row axis comes before the column axis. This matrix has 2 rows and 3 columns, making its shape [2, 3]. Thus, a possible type for this array could be Array [2, 3] Int.
When determining the index of a value inside the array, the order of the indices is the same as the order of the dimensions, and each index number counts from zero. For example, the index [1, 0] indicates the second row and first column, which contains 5.
Note
The word "dimensions" is often ambiguously used to either refer to the rank of an array (as in "multi-dimensional array" in the previous section), or to the lengths of its axes. Conventionally, NumIdr reserves "dimension" for the second meaning, and uses "rank" for the first meaning.
This guide has ignored this convention until now to be more understandable to newcomers, but will follow it from this point onward.
Types of Arrays
Arrays are loosely divided into multiple subtypes mostly based on their rank. Each array subtype has an alias for convenience.
Scalars
A scalar is a rank-0 array, meaning that it is indexed by 0 integers. Its alias is Scalar:
Scalar : (a : Type) -> Type
Scalar = Array []
A scalar has exactly one index, the empty list []. This means that it is exactly the same as a single value and as such is largely pointless, but NumIdr still provides an alias for it just in case you need it.
Vectors
A vector is a rank-1 array:
Vector : (n : Nat) -> (a : Type) -> Type
Vector n = Array [n]
A vector's type signature and stored data is effectively identical to that of the standard library type Vect, whose elements are confusingly also called "vectors"; we often refer to these as "vects" to differentiate.
Indices are typically written as lists of integers, but for vectors it is occasionally acceptable to write the single index number without putting it inside a list. This is mostly the case for indexing, where each indexing function has an alternate definition specifically for vectors.
Matrices
As mentioned before, a matrix is a rank-2 array:
Matrix : (m, n : Nat) -> (a : Type) -> Type
Matrix m n = Array [m, n]
There is also an alias Matrix' for square matrices.
Matrix' : (n : Nat) -> (a : Type) -> Type
Matrix' n = Array [n, n]
As a linear algebra library, the majority of the operations in NumIdr revolve around matrices.
Homogeneous Matrices
NumIdr also provides aliases for homogeneous matrices:
HMatrix : (m, n : Nat) -> (a : Type) -> Type
HMatrix m n = Array [S m, S n]
HMatrix' : (n : Nat) -> (a : Type) -> Type
HMatrix' n = Array [S n, S n]
-- To use with homogeneous matrices
HVector : (n : Nat) -> (a : Type) -> Type
HVector n = Array [S n]
These are useful for clarity when working with both homogeneous and non-homogeneous matrices.
Other Datatypes
Transforms
A transform is a wrapper type for a matrix with certain properties that can be used to transform points in space.
Transform : (ty : TransType) -> (n : Nat) -> (a : Type) -> Type
The TransType parameter dictates what kind of transform it is. These eight options are currently available:
Linear Types:
- Trivial (always the identity transformation)
- Rotation
- Orthonormal (rotation + reflection)
- Linear
Affine Types:
- Translation
- Rigid (rotation + translation)
- Isometry (rotation + reflection + translation)
- Affine
The TransType value is obtained by prepending a capital T to these names. For example, an isometry may have the type Isometry 3 Double, which is an alias for Transform TIsometry 3 Double.
The Point Type
Transforms behave differently from regular matrices when applied to a vector. When a non-linear transform is used, the transform is first linearized, so that vectors only have linear transformations applied to them. This is not a bug.
In order to properly apply these transforms, the Point type must be used, which is a wrapper around the Vector type that supports these transforms. This separation between points and vectors is intended to make working with affine transformations more convenient, as it mirrors the separation between points and vectors in affine algebra.
Permutations
The type Permutation n represents a permutation of n elements. Permutations are mostly used internally for various algorithms, but they are also an input in various operations, such as those that permute the axes of an array.