Basis Functions

gsBasis

Abstract base type for all basis function types in TinyGismo.

Details

gsBasis is the parent type for all spline basis functions, including:

• BSplineBasis: Univariate B-spline bases
• TensorBSplineBasis: Tensor product B-spline bases
• NurbsBasis: Univariate NURBS bases
• TensorNurbsBasis: Tensor product NURBS bases

All basis types support common operations such as evaluation, differentiation, degree elevation, and refinement.

See Also

• BSplineBasis
• TensorBSplineBasis
• NurbsBasis
• TensorNurbsBasis

kontSpans(basis::gsBasis)

Returns a list of the Knot Spans, i.e. elements in reference space.

Common Methods

• centerPoint
• lowerCorner
• upperCorner

elementIndex(basis::gsBasis, u::Union{Vector{Float64}, Matrix{Float64}})

Get the index of the knot span element containing a parametric point.

Arguments

• basis: The basis
• u: A parametric point (vector) or multiple points (matrix, one per column)

Returns

The element index (1-indexed)

Details

Returns the index of the knot span containing the given parameter value(s).

elementInSupportOf(basis::gsBasis, j::Int)

Get the knot spans in the support of the j-th basis function.

Arguments

• basis: The basis
• j: The basis function index (1-indexed)

Returns

The element indices where the basis function has nonzero support

active!(basis::gsBasis, u::Union{Vector{Float64}, Matrix{Float64}}, out::gsMatrix{Int32})

Get the indices of active basis functions at parametric point(s) (in-place).

Arguments

• basis: The basis
• u: A parametric point (vector) or multiple points (matrix)
• out: Output gsMatrix{Int32} to store active basis indices (modified in-place, 1-indexed)

Details

Returns the indices of all basis functions that are nonzero at the given parametric point(s).

isActive(basis::gsBasis, i::Int, u::Vector{Float64})

Check if the i-th basis function is active (nonzero) at a parametric point.

Arguments

• basis: The basis
• i: The basis function index (1-indexed)
• u: The parametric point

Returns

Boolean indicating whether the basis function is active at the point

boundary(basis::gsBasis, s::Int)

Returns the indices of the basis functions that are nonzero at the domain boundary as single-column-matrix.

Arguments:

• basis: The basis
• s: The boxSide describing the side.

degreeElevate!(obj::Union{gsBasis, gsGeometry}, i::Int=1, dir::Int=-1)

Elevate the polynomial degree of a basis or geometry.

Arguments

• obj: A basis or geometry object (modified in-place)
• i: The amount to elevate degree by (default: 1)
• dir: The direction for tensor product bases (default: -1 for all directions)

Note

This operation modifies the object in-place.

degreeReduce!(obj::Union{gsBasis, gsGeometry}, i::Int=1, dir::Int=-1)

Reduce the polynomial degree of a basis or geometry.

Arguments

• obj: A basis or geometry object (modified in-place)
• i: The amount to reduce degree by (default: 1)
• dir: The direction for tensor product bases (default: -1 for all directions)

Note

This operation modifies the object in-place.

degreeIncrease!(obj::Union{gsBasis, gsGeometry}, i::Int=1, dir::Int=-1)

Increase the polynomial degree of a basis or geometry.

Arguments

• obj: A basis or geometry object (modified in-place)
• i: The amount to increase degree by (default: 1)
• dir: The direction for tensor product bases (default: -1 for all directions)

Note

This operation modifies the object in-place.

degreeDecrease!(obj::Union{gsBasis, gsGeometry}, i::Int=1, dir::Int=-1)

Decrease the polynomial degree of a basis or geometry.

Arguments

• obj: A basis or geometry object (modified in-place)
• i: The amount to decrease degree by (default: 1)
• dir: The direction for tensor product bases (default: -1 for all directions)

Note

This operation modifies the object in-place.

elevateContinuity!(basis::gsBasis, i::Int=1, dir::Int=-1)

Elevate the continuity of the basis.

Arguments

• basis: The basis (modified in-place)
• i: The number of continuity levels to increase (default: 1)
• dir: The direction for tensor product bases (default: -1 for all directions)

Details

Elevates continuity by increasing the polynomial degree and adjusting knot multiplicities.

Note

This operation modifies the basis in-place.

reduceContinuity!(basis::gsBasis, i::Int=1, dir::Int=-1)

Reduce the continuity of the basis.

Arguments

• basis: The basis (modified in-place)
• i: The number of continuity levels to decrease (default: 1)
• dir: The direction for tensor product bases (default: -1 for all directions)

Note

This operation modifies the basis in-place.

setDegree!(basis::gsBasis, i::Int)

Set the polynomial degree of the basis to a specific value.

Arguments

• basis: The basis (modified in-place)
• i: The target polynomial degree

Note

This operation modifies the basis in-place.

setDegreePreservingMultiplicity!(basis::gsBasis, i::Int)

Set the polynomial degree while preserving knot multiplicities.

Arguments

• basis: The basis (modified in-place)
• i: The target polynomial degree

Note

This operation modifies the basis in-place.

reverse!(basis::gsBasis)

Reverse the basis (flip the parametric direction).

Arguments

• basis: The basis (modified in-place)

Note

This operation modifies the basis in-place.

eval!(obj::Union{gsBasis, gsGeometry}, u::Vector{Float64}, out::gsMatrix{Float64})

Evaluate basis functions or geometry at parametric point(s) (in-place).

Arguments

• obj: A basis or geometry object
• u: Parametric point(s) (vector or matrix where columns are points)
• out: Output gsMatrix{Float64} to store values (modified in-place)

Details

For basis objects, evaluates all basis functions that are nonzero at the given point(s). For geometry objects, evaluates the geometry at the given parametric point(s).

_eval(obj::Union{gsBasis, gsGeometry}, u::Vector{Float64})

Evaluate basis functions or geometry at parametric point(s).

Arguments

• obj: A basis or geometry object
• u: Parametric point(s)

Returns

For basis objects, returns a matrix of basis function values. For geometry objects, returns a matrix of evaluated points.

evalSingle!(basis::gsBasis, i::Int, u::Vector{Float64}, out::gsMatrix{Float64})

Evaluate a single basis function at parametric point(s) (in-place).

Arguments

• basis: The basis
• i: The basis function index (1-indexed)
• u: Parametric point(s)
• out: Output gsMatrix{Float64} (modified in-place)

evalSingle(basis::gsBasis, i::Int, u::Vector{Float64})

Evaluate a single basis function at parametric point(s).

Arguments

• basis: The basis
• i: The basis function index (1-indexed)
• u: Parametric point(s)

Returns

The basis function values

evalFunc!(basis::gsBasis, u::Vector{Float64}, coefs::Union{Vector{Float64}, Matrix{Float64}}, out::gsMatrix{Float64})

Evaluate a function represented in the basis (in-place).

Arguments

• basis: The basis
• u: Parametric evaluation points
• coefs: Control point coefficients (vector or matrix)
• out: Output gsMatrix{Float64} to store function values (modified in-place)

Details

Evaluates the function defined by ∑(coefsi × basisi) at the given points.

evalFunc(basis::gsBasis, u::Vector{Float64}, coefs::Union{Vector{Float64}, Matrix{Float64}})

Evaluate a function represented in the basis.

Arguments

• basis: The basis
• u: Parametric evaluation points
• coefs: Control point coefficients (vector or matrix)

Returns

A matrix of function values

deriv!(obj::Union{gsBasis, gsGeometry}, u::Vector{Float64}, out::gsMatrix{Float64})

Evaluate the first derivative (in-place).

Arguments

• obj: A basis or geometry object
• u: Parametric evaluation points
• out: Output gsMatrix{Float64} to store derivatives (modified in-place)

Details

For basis objects, evaluates the derivatives of basis functions. For geometry objects, evaluates the derivative (tangent) of the geometry.

deriv(obj::Union{gsBasis, gsGeometry}, u::Union{Vector{Float64}, Matrix{Float64}})

Evaluate the first derivative.

Arguments

• obj: A basis or geometry object
• u: Parametric evaluation points

Returns

For basis objects, returns a matrix of basis function derivatives. For geometry objects, returns a matrix of derivative (tangent) vectors.

derivSingle!(basis::gsBasis, i::Int, u::Vector{Float64}, out::gsMatrix{Float64})

Evaluate the first derivative of a single basis function (in-place).

Arguments

• basis: The basis
• i: The basis function index (1-indexed)
• u: Parametric evaluation points
• out: Output gsMatrix{Float64} (modified in-place)

derivSingle(basis::gsBasis, i::Int, u::Vector{Float64})

Evaluate the first derivative of a single basis function.

Arguments

• basis: The basis
• i: The basis function index (1-indexed)
• u: Parametric evaluation points

Returns

The derivative values

derivFunc(basis::gsBasis, u::Vector{Float64}, coefs::Union{Vector{Float64}, Matrix{Float64}})

Evaluate the first derivative of a function represented in the basis.

Arguments

• basis: The basis
• u: Parametric evaluation points
• coefs: Control point coefficients (vector or matrix)

Returns

A matrix of derivative values

deriv2!(obj::Union{gsBasis, gsGeometry}, u::Vector{Float64}, out::gsMatrix{Float64})

Evaluate the second derivative (in-place).

Arguments

• obj: A basis or geometry object
• u: Parametric evaluation points
• out: Output gsMatrix{Float64} to store second derivatives (modified in-place)

Details

For basis objects, evaluates the second derivatives of basis functions. For geometry objects, evaluates the second derivative (curvature information) of the geometry.

deriv2(obj::Union{gsBasis, gsGeometry}, u::Union{Vector{Float64}, Matrix{Float64}})

Evaluate the second derivative.

Arguments

• obj: A basis or geometry object
• u: Parametric evaluation points

Returns

A matrix of second derivative vectors

deriv2Single!(basis::gsBasis, i::Int, u::Vector{Float64}, out::gsMatrix{Float64})

Evaluate the second derivative of a single basis function (in-place).

Arguments

• basis: The basis
• i: The basis function index (1-indexed)
• u: Parametric evaluation points
• out: Output gsMatrix{Float64} (modified in-place)

deriv2Single(basis::gsBasis, i::Int, u::Vector{Float64})

Evaluate the second derivative of a single basis function.

Arguments

• basis: The basis
• i: The basis function index (1-indexed)
• u: Parametric evaluation points

Returns

The second derivative values

deriv2Func(basis::gsBasis, u::Vector{Float64}, coefs::Union{Vector{Float64}, Matrix{Float64}})

Evaluate the second derivative of a function represented in the basis.

Arguments

• basis: The basis
• u: Parametric evaluation points
• coefs: Control point coefficients (vector or matrix)

Returns

A matrix of second derivative values

uniformRefine!(basis::gsBasis, numKnots::Int=1, mul::Int=1)

Uniformly refine the basis by inserting knots.

Arguments

• basis: The basis (modified in-place)
• numKnots: Number of knots to insert in each span (default: 1)
• mul: Multiplicity of each inserted knot (default: 1)

Note

This operation modifies the basis in-place.

uniformCoarsen!(basis::gsBasis, numKnots::Int=1)

Uniformly coarsen the basis by removing knots.

Arguments

• basis: The basis (modified in-place)
• numKnots: Number of knots to remove from each span (default: 1)

Note

This operation modifies the basis in-place.

uniformRefine_withCoefs!(basis::gsBasis, coefs::Matrix{Float64}, numKnots::Int=1, mul::Int=1)

Uniformly refine the basis and update control point coefficients accordingly.

Arguments

• basis: The basis (modified in-place)
• coefs: Control point coefficients (modified in-place)
• numKnots: Number of knots to insert in each span (default: 1)
• mul: Multiplicity of each inserted knot (default: 1)

Details

This function refines the basis and automatically computes the new control points such that the refined basis represents the same geometry/function.

Note

Both the basis and coefficients are modified in-place.

BSplineBasis(knotVector::gsKnotVector)

Construct a univariate B-spline basis from a knot vector.

Arguments

• knotVector: The knot vector defining the basis

Returns

A B-spline basis object

Examples

kv = gsKnotVector(...)
basis = BSplineBasis(kv)

TensorBSplineBasis{2}(knotVector1::gsKnotVector, knotVector2::gsKnotVector)
TensorBSplineBasis{3}(knotVector1::gsKnotVector, knotVector2::gsKnotVector, knotVector3::gsKnotVector)

Construct a tensor product B-spline basis from knot vectors.

Arguments

• knotVector1, knotVector2, [knotVector3]: Knot vectors for each parametric direction

Returns

A tensor product B-spline basis object (2D or 3D)

Examples

kv1 = gsKnotVector(...)
kv2 = gsKnotVector(...)
basis = TensorBSplineBasis(kv1, kv2)

knots(basis::BSplineBasis, i::Int=1)

Get the knot vector for a given parametric direction.

Arguments

• basis: The B-spline basis
• i: The parametric direction (default: 1)

Returns

The knot vector for direction i

knot(basis::BSplineBasis, i::Int)

Get the i-th knot value.

Arguments

• basis: The B-spline basis
• i: The knot index

Returns

The knot value at index i

Base.size(basis::BSplineBasis)

Get the number of basis functions in the B-spline basis.

Alias for the C++ size() method.

Base.size(basis::TensorBSplineBasis)

Get the number of basis functions in the tensor product B-spline basis.

Alias for the C++ size() method.

Base.size(basis::NurbsBasis)

Get the number of basis functions in the NURBS basis.

Alias for the C++ size() method.

Base.size(basis::TensorNurbsBasis)

Get the number of basis functions in the tensor product NURBS basis.

Alias for the C++ size() method.

numElements(basis::BSplineBasis, side::Int=0)

Get the number of elements (knot spans) in the basis.

Arguments

• basis: The B-spline basis
• side: The side/direction to query (default: 0 for all sides)

Returns

The number of elements in the specified direction

numTotalElements(basis::TensorBSplineBasis)

Get the total number of elements in a tensor product basis.

Arguments

• basis: The tensor product B-spline basis

Returns

The total number of elements across all parametric directions

degree(obj::Union{gsBasis, KnotVector, gsGeometry}, i::Int)

Get the polynomial degree of the basis, knot vector, or geometry.

For univariate bases, returns a single degree. For tensor product bases, returns the degree in a specified direction.

Arguments

• obj: The B-spline basis, knot vector, or geometry object
• i: The parametric direction (for tensor product bases, optional)

Returns

The polynomial degree

order(basis::BSplineBasis)

Get the order of the B-spline basis.

The order equals the polynomial degree plus one.

Arguments

• basis: The B-spline basis

Returns

The order of the basis (degree + 1)

component(basis::TensorBSplineBasis, i::Int)

Get the univariate B-spline basis component for a parametric direction.

Arguments

• basis: The tensor product B-spline basis
• i: The parametric direction (1-indexed)

Returns

The univariate B-spline basis for direction i

numActive(basis::TensorBSplineBasis, u::Vector{Float64})

Get the number of active basis functions at a given parametric point.

For tensor product bases, returns a vector with the number of active basis functions in each direction.

Arguments

• basis: The tensor product B-spline basis
• u: The parametric point

Returns

The number of active basis functions at point u

numActive!(basis::TensorBSplineBasis, u::Vector{Float64}, out::Vector{Int})

Get the number of active basis functions at a given parametric point (in-place).

Stores the result in the output vector out.

Arguments

• basis: The tensor product B-spline basis
• u: The parametric point
• out: Output vector to store the number of active basis functions in each direction (modified in-place)

insertKnot!(basis::BSplineBasis, knot::Float64, mult::Int=1)

Insert a knot into the basis.

This operation refines the basis by inserting a knot value. The multiplicity determines how many times the knot is inserted.

Arguments

• basis: The B-spline basis (modified in-place)
• knot: The knot value to insert
• mult: The multiplicity of insertion (default: 1)

Note

This operation modifies the basis in-place.

removeKnot!(basis::BSplineBasis, knot::Float64, mult::Int=1)

Remove a knot from the basis.

Arguments

• basis: The B-spline basis (modified in-place)
• knot: The knot value to remove
• mult: The multiplicity of removal (default: 1)

Note

This operation modifies the basis in-place.

insertKnots!(basis::BSplineBasis, knots::Vector{Float64})

Insert multiple knots into the basis.

Arguments

• basis: The B-spline basis (modified in-place)
• knots: A vector of knot values to insert

Note

This operation modifies the basis in-place.

NurbsBasis(knotVector::KnotVector)
NurbsBasis(knotVector::KnotVector, weights::Vector{Float64})
NurbsBasis(basis::BSplineBasis, weights::Vector{Float64})

Construct a univariate NURBS (Non-Uniform Rational B-Spline) basis from a knot vector and optional weights.

Arguments

• knotVector: The knot vector defining the basis
• weights: Control point weights (optional, defaults to uniform weights)
• basis: Existing B-spline basis to convert to NURBS

Returns

A NURBS basis object

Details

NURBS bases generalize B-spline bases by associating weights with control points, allowing for more flexible shape control and exact representation of conic sections.

Examples

kv = gsKnotVector(...)
basis = NurbsBasis(kv)

# With weights
w = [1.0, 1.0, sqrt(2)/2, 1.0, 1.0]
basis = NurbsBasis(kv, w)

TensorNurbsBasis{2}(knotVector1::KnotVector, knotVector2::KnotVector, weights::Matrix{Float64})
TensorNurbsBasis{3}(knotVector1::KnotVector, knotVector2::KnotVector, knotVector3::KnotVector, weights::Matrix{Float64})

Construct a tensor product NURBS basis from knot vectors and weights.

Arguments

• knotVector1, knotVector2, [knotVector3]: Knot vectors for each parametric direction
• weights: Control point weights arranged in a matrix

Returns

A tensor product NURBS basis object (2D or 3D)

Details

Tensor product NURBS surfaces and volumes are formed by taking the tensor product of univariate NURBS bases, with weights controlling the influence of each control point.

Examples

kv1 = gsKnotVector(...)
kv2 = gsKnotVector(...)
w = [1.0 1.0 1.0; 1.0 sqrt(2)/2 1.0; 1.0 1.0 1.0]
basis = TensorNurbsBasis(kv1, kv2, w)

weights(nurbs::Nurbs)

Get all control point weights.

Arguments

• nurbs: The NURBS curve

Returns

A vector of all weights