L-MATH is a library for performing simple linear algebra. Vector and matrix classes are available, as are simple linear interpolation functions, spline-based interpolations (Catmull-Rom and B-Spline methods), and various operations related to creating rotation matrices.

L-MATH also contains various spline implementations, including those implemented using matrices:

  • Hermite curves
  • Cubic Bezi?r curves
  • Uniform, non-rational B-Splines
  • Catmull-Rom splines

There are also general splines that are calculated using recurrence formulae (which is generally a more stable approach than using matrices), such as:

  • General Bezi?r curves of arbitrary degree
  • Non-Rational B-Splines, including uniform and non-uniform parametrisations.

In addition, L-MATH contains various functions for creating random data, including:

  • uniformly distributed random numbers and vectors
  • normal distribution of random numbers and vectors
  • Perlin noise.


Vectors can be constructed using the VECTOR and TO-VECTOR functions. VECTOR accepts a list of elements, like so:

(lm:vector 1 2 3) => #<L-MATH:VECTOR 1.000 2.000 3.000 >

The VECTOR's dimension is defined by the number of elements in the VECTOR function's lambda list.

(lm:dimension (lm:vector 1 2 3 4)) => 4

TO-VECTOR is intended to transform other types into the VECTOR type. At the moment it supports transforming lists and, trivially, other vector objects:

(lm:to-vector (list 1 2 3)) => #<L-MATH:VECTOR 1.000 2.000 3.000 >

Importantly, TO-VECTOR allows the vector's length to be modified:

(lm:to-vector (list 1 2 3) :dimension 2) => #<L-MATH:VECTOR 1.000 2.000 >
(lm:to-vector (lm:vector 1 2) :dimension 3) => #<L-MATH:VECTOR 1.000 2.000 0.000 >

Vectors can typically be represented as lists. For instance:

(lm:dot-product (lm:vector 1 0 1) (list 0 1 0)) => 0

Various operations (listed below) are available for VECTOR objects. Many of these functions will also accept lists as VECTOR representations.

(lm:dimension VECTOR)

Returns the VECTOR's dimension.

(lm:length VECTOR)

Synonym for DIMENSION.

(lm:norm VECTOR)

Returns the VECTOR's length.

(lm:vector= LHS RHS)

Returns T iff the two vectors are equal. Internally, the VECTOR class stores the data as an array of double-floats. Because of rounding errors it is not advisable to compare floating point values exactly. VECTOR= uses the special variable equivalence-tolerance to define the tolerance within which two vectors are considered equal. equivalence-tolerance defaults to 0.0001, which should be reasonable for most applications. For example:

(lm:vector= (lm:vector 1 2 3) (lm:vector 1 2 3.1)) => NIL
(lm:vector= (lm:vector 1 2 3) (lm:vector 1 2 3.00001)) => T
(lm:elt VECTOR index)

Returns the element at the given index. This is also a SETFable place. VECTORs are zero based.

(lm:x VECTOR) (lm:y VECTOR) (lm:z VECTOR) (lm:w VECTOR)

Returns the elements at indices 0, 1, 2 and 3 respectively. These are all SETFable places.

(lm:dot-product VECTOR)

Returns the VECTOR's dot product.

(lm:cross-product LHS RHS)

Calculates the cross product between two 3-vectors.

(lm:angle-between FROM-VECTOR TO-VECTOR)

Returns the angle, in radians, needed to align the FROM-VECTOR with the TO-VECTOR. The angle is signed, and is a left-handed rotation. Example:

(lm:to-degrees (lm:angle-between (lm:vector 1 0) (lm:vector 0 1))) => 90.0d0
(lm:to-degrees (lm:angle-between (lm:vector 1 0) (lm:vector 0 -1))) => -90.0d0
(lm:euclidean-distance LHS RHS)

Calculates the Euclidean distance between two vectors or two numbers.


Matrices can be constructed using (lm:MAKE-MATRIX row col &key initial-elements)

(lm:make-matrix 2 3 :initial-elements '(1 0 0
                            0 1 0))
1.000 0.000 0.000 
0.000 1.000 0.000 >

If :initial-elements isn't specified, the matrix elements are initialised to zero.

(lm:matrix= LHS RHS)

Ensures that two matrices are numerically equivalent. All the real-valued components must be within equivalence-tolerance of each other.

(lm:matrix-rows MATRIX) (lm:matrix-cols MATRIX)

Returns the number of rows and columns in the matrix.

(lm:matrix-elt MATRIX row col)

Returns the element at the given row and column. MATRIX objects are zero based. This is a SETFable place.

(lm:make-identity SIZE)

Returns a SIZE?SIZE identity matrix.

(lm:roll-matrix SIZE ANGLE)

Returns SIZE?SIZE matrix which will rotate a post multiplied vector around the z-axis. It is a left-handed rotation. The ANGLE is given in radians. SIZE should be either 3 or 4.

(lm:yaw-matrix SIZE ANGLE)

Returns SIZE?SIZE matrix which will rotate a post multiplied vector around the y-axis. It is a left-handed rotation. The ANGLE is given in radians. SIZE should be either 3 or 4.

(lm:pitch-matrix SIZE ANGLE)

Returns SIZE?SIZE matrix which will rotate a post multiplied vector around the x-axis. It is a left-handed rotation. The ANGLE is given in radians. SIZE should be either 3 or 4.

(lm:set-rotation-naming-convention CONVENTION)

Rebinds YAW-MATRIX, PITCH-MATRIX, and ROLL-MATRIX to rotate around different axes. The available conventions are provided in *lm:*rotation-naming-conventions**

(lm:create-rotation-matrix VIEW RIGHT UP &optional (SIZE 3))

Creates a rotation matrix from three vectors. VIEW is the direction that the resulting vector should be pointing along, UP is the direction upwards. RIGHT is the vector orthogonal to this. Will return a left-handed rotation matrix. SIZE is the size of the matrix, and should be either 3 or 4.

(lm:create-rotation-from-view VIEW WORLD-UP &optional (SIZE (length SIZE)))

Given a direction to look in (VIEW), and the direction that is 'upwards' in a given coordinate system, this function creates a rotation matrix to translate into that coordinate system. This rotation is left-handed. SIZE should be either 3 or 4.

(lm:create-rotation-from-view-to-view FROM-VIEW TO-VIEW WORLD-UP)

Creates a rotation matrix that will rotate the vector FROM-VIEW on to the vector TO-VIEW, using WORLD-UP as the coordinate system's 'upward' direction. This is a left-handed rotation. Example:

(let ((rotation (lm:create-rotation-from-view-to-view (lm:vector 1 0 0) 
                              (lm:vector 0 1 0) 
                              (lm:vector 0 0 1))))
       (lm:* rotation (lm:vector 1 0 0)))
=> #<L-MATH:VECTOR 0.000 1.000 0.000 >


(lm:linear-interpolation START END T-VAL)

Given two vectors (START and END), and a real valued parameter (T-VAL), this returns a vector between START and END. When T-VAL is zero, this returns START. When T-VAL is 1, this returns END. Values between 0 and 1 return vectors between START and END; values below zero return vectors "before" START; values above 1 return vectors "after" END. The value 0.5 returns the vector exactly between START and END. Example:

(lm:linear-interpolation (lm:vector -1 0) (lm:vector 1 0) 0.5)
=> #<L-MATH:VECTOR 0.000 0.000 >
(lm:linear-interpolation (lm:vector 0 0 0) (lm:vector 10 10 10) 2)
=> #<L-MATH:VECTOR 20.000 20.000 20.000 >

This method also accepts arbitrary numbers, and will interpolate between them:

(lm:linear-interpolation 0 100 0.5) => 50.0
(lm:between START END)

Returns the vector exactly between the VECTORs START and END.

General Operations

(lm:equivalent LHS RHS)

Returns t iff the two objects are numerically equivalent. Numbers are tested using =. Real-valued objects (REAL types, VECTORs and MATRIXs) are compared to each other using a tolerance equivalence-tolerance. VECTORs and MATRIX objects are compared using VECTOR= and MATRIX=.

(lm:copy OBJECT)

Returns a copy of the given VECTOR, MATRIX or list.

(lm:negate OBJECT) (lm:negate! OBJECT)

Returns the arithmetic inverse of the given object. NEGATE! does so destructively. Example:

(lm:negate (list 1 -2 3)) => (-1 2 -3)
(lm:to-radians ANGLE) (lm:to-degrees ANGLE)

Converts from radians to degrees, and vice versa.

(lm:test-dimensions LHS RHS)

Ensures that the two items have the same dimensions. The items may be lists, vectors or matrices in most sensible combinations. This function is useful when implementing your own operations between vectors and matrices to ensure that their dimensions agree. If they do not, a DIMENSION-ERROR condition is signalled.

Arithmetic operations

All the general arithmetic operations are defined:

* (lm:+ LHS RHS)
* (lm:- LHS RHS)
* (lm:- OBJECT)
* (lm:* LHS RHS)
* (lm:/ LHS RHS)


L-MATH-ERROR: A general condition from which all error conditions for the package inherit.

DIMENSION-ERROR: This is signalled when an operation is requested on objects whose dimensions are inappropriate.

ZERO-NORM-ERROR: This is signalled on operations which do not make sense for vectors with zero norm.

OPERATION-NOT-SUPPORTED: This is signalled when an arithmetic operation is requested on two objects for which the operation is not supported. This should usually not occur, and probably should be considered a bug if it does.

General Comments

Both VECTOR and MATRIX classes have load forms (MAKE-LOAD-FORM). Internally, the data is stored as arrays of DOUBLE-FLOAT values.

For those operations which deal with rotations, note that rotation matrices should be post-multiplied by the vectors. The coordinate system is left-handed, as are the rotations.

Supported Compilers

L-MATH is known to work on SBCL 1.0.29 and 1.0.55. While it should work on other compilers, this so far has not been tested. Please feel free to send in reports of which compilers you've successfully run this with, or to file bug reports where L-MATH is having problems.

Getting and Installing

L-MATH is available from its home page at Common Lisp.net, or from its github repository. L-MATH can be installed using ASDF-INSTALL and QUICKLISP:

(require 'asdf-install)
(asdf-install:install 'l-math)

(quicklisp:quickload 'l-math)

Reporting Bugs

Bugs can be reported to https://github.com/TheRiver/L-MATH.


See the file LICENSE for the licensing details. In brief, L-MATH is licensed under the GPL, with additional permissions giving link exceptions (aka the Classpath exception). Importantly for a Common Lisp library, this exception allows you to use this GPLed library in your application regardless of the licenses of the compiler and the other libraries you are using (as long, of course, as you satisfy those licenses).

Note that this does not remove the obligation that the rest of the GPL places on you, such as supplying the source code of this library.

Rudolph Neeser <rudy.neeser@gmail.com>
GPLv3 with Classpath Exception
linear algebra