Class LinearAlgebra

Namespace

MathAssertions

Assembly

MathAssertions.dll

Linear-algebra invariants over Matrix4x4 and Vector3: matrix symmetry, orthogonality, identity, determinant, trace, invertibility, plus vector-pair orthogonality and parallelism and the AreLinearlyIndependent(ReadOnlySpan<Vector3>, double) triple-product check for sets of up to three vectors in R^3.

public static class LinearAlgebra

Inheritance

object

LinearAlgebra

Inherited Members

object.Equals(object)

object.Equals(object, object)

object.GetHashCode()

object.GetType()

object.MemberwiseClone()

object.ReferenceEquals(object, object)

object.ToString()

Remarks

0.1.0 Cluster 5. Sits alongside MathTolerance, Sequences, and Statistics in the same package; the MathAssertions.TUnit adapter delegates to these helpers from its [GenerateAssertion] extensions.

Methods

AngleBetween(Vector3, Vector3)

Returns the unsigned angle, in radians on [0, pi], between two vectors, computed as atan2(|u x v|, u . v). This form stays numerically accurate across the whole range, unlike acos((u . v) / (|u| |v|)), which loses precision near 0 and pi where the cosine flattens. A zero vector yields an angle of 0 by convention (the angle is otherwise undefined).

public static double AngleBetween(Vector3 u, Vector3 v)

Parameters

u Vector3

First vector.

v Vector3

Second vector.

Returns

double

The angle between the vectors in radians, on [0, pi].

AreLinearlyIndependent(ReadOnlySpan<Vector3>, double)

Returns true when the supplied vectors are linearly independent in R^3. Up to three vectors can be linearly independent; spans of four or more vectors in R^3 are always dependent and return false. The empty span is vacuously independent and returns true.

public static bool AreLinearlyIndependent(ReadOnlySpan<Vector3> vectors, double tolerance)

Parameters

vectors ReadOnlySpan<Vector3>

Vectors to test.

tolerance double

Threshold for the length and triple-product checks. Must be non-negative and not NaN.

Returns

bool

true if the vectors are linearly independent.

Remarks

Length-1: the single vector is independent iff its length exceeds tolerance. Length-2: equivalent to AreParallel(Vector3, Vector3, double) returning false. Length-3: triple-product test |v1 . (v2 x v3)| > tolerance. The absolute scalar triple product is the volume of the parallelepiped spanned by the three vectors and is non-zero iff the three are linearly independent.

The length-1 case compares vector length directly against tolerance rather than length-squared against tolerance-squared, so the verdict is well-defined for extreme tolerance magnitudes (where squaring would underflow or overflow).

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

AreOrthogonal(Vector3, Vector3, double)

Returns true when the two vectors are orthogonal: their dot product is within tolerance of zero.

public static bool AreOrthogonal(Vector3 u, Vector3 v, double tolerance)

Parameters

u Vector3

First vector.

v Vector3

Second vector.

tolerance double

Maximum allowed absolute deviation of the dot product from zero. Must be non-negative and not NaN.

Returns

bool

true if u . v ~ 0.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

AreParallel(Vector3, Vector3, double)

Returns true when the two vectors are parallel (or anti-parallel): the sine of the angle between their directions, |u x v| / (|u| |v|), is within tolerance of zero. The measure is scale-invariant, so parallelism depends only on direction, not on the vectors' magnitudes. A zero (or shorter-than-tolerance) vector is treated as parallel to every other vector, both because 0 x v = 0 and to avoid dividing by a zero length.

public static bool AreParallel(Vector3 u, Vector3 v, double tolerance)

Parameters

u Vector3

First vector.

v Vector3

Second vector.

tolerance double

Maximum allowed sine of the angle between the directions. Must be non-negative and not NaN.

Returns

bool

true if the directions are parallel within tolerance.

Remarks

The tolerance is an angular measure (the sine of the maximum permitted deviation from exactly parallel), not an absolute cross-product magnitude. For small angles sin(theta) ~ theta, so a tolerance of 1e-3 admits directions within roughly a milliradian of parallel regardless of how long the vectors are.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

HasDeterminantApproximately(Matrix4x4, double, double)

Returns true when the matrix determinant is within tolerance of expected. Delegates the computation to GetDeterminant().

public static bool HasDeterminantApproximately(Matrix4x4 m, double expected, double tolerance)

Parameters

m Matrix4x4

Matrix to test.

expected double

Expected determinant value.

tolerance double

Maximum allowed absolute difference. Must be non-negative and not NaN.

Returns

bool

true if the determinant is approximately expected.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

HasTraceApproximately(Matrix4x4, double, double)

Returns true when the matrix trace (sum of diagonal elements) is within tolerance of expected.

public static bool HasTraceApproximately(Matrix4x4 m, double expected, double tolerance)

Parameters

m Matrix4x4

Matrix to test.

expected double

Expected trace value.

tolerance double

Maximum allowed absolute difference. Must be non-negative and not NaN.

Returns

bool

true if the trace is approximately expected.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

IsIdentity(Matrix4x4, double)

Returns true when the matrix equals the Identity matrix element-wise within tolerance. Convenience wrapper over IsApproximatelyEqual(Matrix4x4, Matrix4x4, double) with the identity as the second operand.

public static bool IsIdentity(Matrix4x4 m, double tolerance)

Parameters

m Matrix4x4

Matrix to test.

tolerance double

Maximum allowed absolute difference per element. Must be non-negative and not NaN.

Returns

bool

true if the matrix is approximately the identity.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

IsInvertible(Matrix4x4, double)

Returns true when the absolute value of the determinant exceeds tolerance. The threshold expresses "the matrix is far enough from singular to invert numerically"; choose a tolerance that reflects the expected condition number of the inputs.

public static bool IsInvertible(Matrix4x4 m, double tolerance)

Parameters

m Matrix4x4

Matrix to test.

tolerance double

Minimum acceptable absolute determinant for the matrix to be considered invertible. Must be non-negative and not NaN.

Returns

bool

true if |det(M)| > tolerance.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

IsOrthogonal(Matrix4x4, double)

Returns true when the matrix is orthogonal: M * M^T = I within tolerance. Orthogonal matrices preserve angles and lengths and have determinant +/- 1.

public static bool IsOrthogonal(Matrix4x4 m, double tolerance)

Parameters

m Matrix4x4

Matrix to test.

tolerance double

Maximum allowed absolute difference between M * M^T and the identity per element. Must be non-negative and not NaN.

Returns

bool

true if the matrix is orthogonal within tolerance.

Remarks

Translation matrices are not orthogonal because the translation column makes the product M * M^T deviate from identity in the off-diagonals. A pure rotation is orthogonal; rotations composed with uniform scaling are not (the diagonal of M * M^T picks up the squared scale).

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

IsRotation(Matrix4x4, double)

Returns true when the matrix is a proper rotation: orthogonal (M * M^T = I, so it preserves lengths and angles) and has determinant +1 within tolerance. The determinant condition rules out reflections (an improper orthogonal matrix has determinant -1). A matrix carrying a non-zero translation is not orthogonal and so is not a rotation under this definition.

public static bool IsRotation(Matrix4x4 m, double tolerance)

Parameters

m Matrix4x4

Matrix to test.

tolerance double

Maximum allowed absolute deviation, applied to both the orthogonality check (per element of M * M^T - I) and the determinant (from +1). Must be non-negative and not NaN.

Returns

bool

true if the matrix is a proper rotation within tolerance.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.

IsSymmetric(Matrix4x4, double)

Returns true when the matrix is symmetric: every off-diagonal element equals its transpose partner within tolerance, that is m[i, j] = m[j, i] for all i != j.

public static bool IsSymmetric(Matrix4x4 m, double tolerance)

Parameters

m Matrix4x4

Matrix to test.

tolerance double

Maximum allowed absolute difference between mirrored off-diagonal elements. Must be non-negative and not NaN.

Returns

bool

true if the matrix is symmetric within tolerance.

Exceptions

ArgumentOutOfRangeException

tolerance is NaN or negative.