Geometries

Geometric objects represent spline curves, surfaces, and volumes. All geometry types inherit from gsGeometry and support evaluation, derivatives, and various geometric queries.

Base Types

TinyGismo.gsGeometry — Type

gsGeometry

Abstract base type for all geometry types in TinyGismo.

Details

gsGeometry is the parent type for all geometric objects, including:

BSpline: Univariate and multivariate B-spline curves/surfaces
TensorBSpline: Tensor product B-spline surfaces and volumes
Nurbs: Univariate and multivariate NURBS curves/surfaces
TensorNurbs: Tensor product NURBS surfaces and volumes

All geometry types support common operations such as evaluation, derivatives, closest point queries, and geometric refinement.

See Also

BSpline
TensorBSpline
Nurbs
TensorNurbs

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B-Spline Geometries

Univariate B-Splines

TinyGismo.BSpline — Type

BSpline(basis::BSplineBasis, coefs::Union{Vector{Float64}, Matrix{Float64}})

Construct a B-spline curve or surface from a basis and control points.

Arguments

basis: The B-spline basis
coefs: Control point coefficients (vector for curves, matrix for surfaces)

Returns

A BSpline object

Details

A B-spline is defined by a basis and control points. The basis determines the spline space, and the control points define the actual geometry.

Examples

basis = BSplineBasis(kv)
coefs = [1.0, 2.0, 3.0, 4.0]
spline = BSpline(basis, coefs)

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Properties and Queries

These query functions are also available for basis functions. See the Basis Functions section for detailed documentation.

TinyGismo.knots - Get the knot vector
TinyGismo.degree - Get the polynomial degree
TinyGismo.basis - Get the underlying basis
boundary - Extract a boundary curve/surface

TinyGismo.basis — Function

basis(spline::BSpline)

Get the underlying B-spline basis.

Arguments

spline: The B-spline

Returns

The BSplineBasis used by the spline

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TinyGismo.numCoefs — Function

numCoefs(spline::BSpline)

Get the number of control points.

Arguments

spline: The B-spline

Returns

The number of control points

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TinyGismo.coefAtCorner — Function

coefAtCorner(geometry::gsGeometry, c::Int)

Get the control point at a specific corner.

Arguments

geometry: The geometry
c: The corner index

Returns

The control point coordinates at that corner

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Refinement operations are shared with basis functions. See the Basis Functions - Refinement section for detailed documentation.

insertKnot! - Insert a knot into the geometry
uniformRefine! - Uniformly refine the mesh
uniformCoarsen! - Uniformly coarsen the mesh

Degree Operations

These degree operations are shared with basis functions. See the Basis Functions - Degree and Continuity Operations section for detailed documentation.

degreeElevate! - Elevate the polynomial degree
degreeReduce! - Reduce the polynomial degree
degreeIncrease! - Increase the polynomial degree
degreeDecrease! - Decrease the polynomial degree

Tensor Product B-Splines

TinyGismo.TensorBSpline — Type

TensorBSpline{1}(basis::TensorBSplineBasis{1}, coefs::Union{Vector{Float64}, Matrix{Float64}})
TensorBSpline{2}(basis::TensorBSplineBasis{2}, coefs::Matrix{Float64})
TensorBSpline{2}(kv1::KnotVector, kv2::KnotVector, coefs::Matrix{Float64})
TensorBSpline{2}(corner::Matrix{Float64}, kv1::KnotVector, kv2::KnotVector)
TensorBSpline{3}(basis::TensorBSplineBasis{3}, coefs::Matrix{Float64})
TensorBSpline{3}(kv1::KnotVector, kv2::KnotVector, kv3::KnotVector, coefs::Matrix{Float64})

Construct a tensor product B-spline surface or volume from a basis and control points.

Arguments

basis: The tensor product B-spline basis
coefs: Control point coefficients (matrix)
kv1, kv2, [kv3]: Knot vectors for each parametric direction
corner: Corner points (for constructing bilinear patches)

Returns

A TensorBSpline object (surface or volume)

Details

Tensor product B-splines are formed by taking the tensor product of univariate B-spline bases. They are parameterized by 2 or more parameters and are commonly used for representing surfaces and volumes.

All methods from gsGeometry are available for TensorBSpline objects.

Examples

kv1 = KnotVector([0.0, 0.0, 0.0, 1.0, 1.0, 1.0])
kv2 = KnotVector([0.0, 0.0, 0.0, 1.0, 1.0, 1.0])
coefs = [0.0 1.0 2.0; 0.0 0.0 0.0; 0.0 1.0 2.0; 0.0 0.0 0.0; ...]
basis = TensorBSplineBasis{2}(kv1, kv2)
surface = TensorBSpline{2}(basis, coefs)

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Properties and Queries

These query functions are also available for basis functions. See the Basis Functions section for detailed documentation.

TinyGismo.knots - Get the knot vector
TinyGismo.degree - Get the polynomial degree
TinyGismo.basis - Get the underlying basis
TinyGismo.boundary - Extract a boundary curve/surface

Refinement operations are shared with basis functions. See the Basis Functions - Refinement section for detailed documentation.

insertKnot! - Insert a knot into the geometry
uniformRefine! - Uniformly refine the mesh
uniformCoarsen! - Uniformly coarsen the mesh

Degree Operations

These degree operations are shared with basis functions. See the Basis Functions - Degree and Continuity Operations section for detailed documentation.

degreeElevate! - Elevate the polynomial degree
degreeReduce! - Reduce the polynomial degree
degreeIncrease! - Increase the polynomial degree
degreeDecrease! - Decrease the polynomial degree

NURBS Geometries

Univariate NURBS

TinyGismo.Nurbs — Type

Nurbs(basis::NurbsBasis, coefs::Union{Vector{Float64}, Matrix{Float64}})

Construct a NURBS curve from a NURBS basis and control points.

Arguments

basis: The NURBS basis
coefs: Control point coefficients (vector for curves, matrix for surfaces)

Returns

A Nurbs object

Details

A NURBS (Non-Uniform Rational B-Spline) curve or surface is defined by a NURBS basis (which includes weights) and control points. The rational nature allows exact representation of conic sections and provides local control through weights.

All methods from gsGeometry are available for Nurbs objects.

Examples

kv = KnotVector([0.0, 0.0, 0.0, 1.0, 1.0, 1.0])
weights = [1.0, sqrt(2)/2, 1.0]
basis = NurbsBasis(kv, weights)
coefs = [0.0, 1.0, 1.0, 1.0, 0.0, 1.0]
curve = Nurbs(basis, coefs)

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Properties and Queries

These query functions are also available for basis functions. See the Basis Functions section for detailed documentation.

TinyGismo.knots - Get the knot vector
TinyGismo.degree - Get the polynomial degree
TinyGismo.basis - Get the underlying basis
TinyGismo.boundary - Extract a boundary curve/surface
TinyGismo.weights - Get the control point weights

TinyGismo.weight — Function

weight(nurbs::Nurbs, i::Int)

Get the weight of a specific control point.

Arguments

nurbs: The NURBS curve
i: The control point index (1-indexed)

Returns

The weight value at control point i

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Refinement operations are shared with basis functions. See the Basis Functions - Refinement section for detailed documentation.

insertKnot! - Insert a knot into the geometry
uniformRefine! - Uniformly refine the mesh
uniformCoarsen! - Uniformly coarsen the mesh

Degree Operations

These degree operations are shared with basis functions. See the Basis Functions - Degree and Continuity Operations section for detailed documentation.

degreeElevate! - Elevate the polynomial degree
degreeReduce! - Reduce the polynomial degree
degreeIncrease! - Increase the polynomial degree
degreeDecrease! - Decrease the polynomial degree

Tensor Product NURBS

TinyGismo.TensorNurbs — Type

TensorNurbs{2}(basis::TensorNurbsBasis{2}, coefs::Matrix{Float64})
TensorNurbs{2}(kv1::KnotVector, kv2::KnotVector, coefs::Matrix{Float64})
TensorNurbs{3}(basis::TensorNurbsBasis{3}, coefs::Matrix{Float64})
TensorNurbs{3}(kv1::KnotVector, kv2::KnotVector, kv3::KnotVector, coefs::Matrix{Float64})

Construct a tensor product NURBS surface or volume from a basis and control points.

Arguments

basis: The tensor product NURBS basis
coefs: Control point coefficients (matrix)
kv1, kv2, [kv3]: Knot vectors for each parametric direction

Returns

A TensorNurbs object (surface or volume)

Details

Tensor product NURBS surfaces and volumes are formed by taking the tensor product of univariate NURBS bases. They combine the flexibility of tensor products with the rational properties of NURBS, enabling exact representation of complex shapes including quadric surfaces.

All methods from gsGeometry are available for TensorNurbs objects.

Examples

kv1 = KnotVector([0.0, 0.0, 0.0, 1.0, 1.0, 1.0])
kv2 = KnotVector([0.0, 0.0, 0.0, 1.0, 1.0, 1.0])
weights = [1.0 sqrt(2)/2 1.0; 1.0 sqrt(2)/2 1.0; 1.0 sqrt(2)/2 1.0]
basis = TensorNurbsBasis{2}(kv1, kv2, weights)
coefs = [0.0 1.0 1.0; 0.0 0.0 1.0; ...]
surface = TensorNurbs{2}(basis, coefs)

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Properties and Queries

These query functions are also available for basis functions. See the Basis Functions section for detailed documentation.

TinyGismo.knots - Get the knot vector
TinyGismo.degree - Get the polynomial degree
TinyGismo.basis - Get the underlying basis
TinyGismo.boundary - Extract a boundary curve/surface
TinyGismo.weights - Get the control point weights

Refinement operations are shared with basis functions. See the Basis Functions - Refinement section for detailed documentation.

insertKnot! - Insert a knot into the geometry
uniformRefine! - Uniformly refine the mesh
uniformCoarsen! - Uniformly coarsen the mesh

Degree Operations

These degree operations are shared with basis functions. See the Basis Functions - Degree and Continuity Operations section for detailed documentation.

degreeElevate! - Elevate the polynomial degree
degreeReduce! - Reduce the polynomial degree
degreeIncrease! - Increase the polynomial degree
degreeDecrease! - Decrease the polynomial degree

Generic Geometry Operations

These methods work on all geometry types (B-spline and NURBS).

Control Points

TinyGismo.coefsSize — Function

coefsSize(geometry::gsGeometry)

Get the size (number of control points).

Arguments

geometry: The geometry

Returns

The number of control points

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TinyGismo.coefs — Function

coefs(geometry::gsGeometry)

Get the control points (coefficients) of the geometry.

Arguments

geometry: The geometry

Returns

A matrix of control point coordinates

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Evaluation

These evaluation methods are shared with basis functions. See the Basis Functions - Evaluation section for detailed documentation.

eval! - Evaluate at parametric points (in-place)
_eval - Evaluate at parametric points

Derivatives

These derivative methods are shared with basis functions. See the Basis Functions - Derivatives section for detailed documentation.

deriv! - First derivative (in-place)
deriv - First derivative
deriv2! - Second derivative (in-place)
deriv2 - Second derivative

Geometric Queries

TinyGismo.closestPointTo — Function

closestPointTo(geometry::gsGeometry, pt::Vector{Float64}, result::gsVector{Float64})

Find the closest point on the geometry to a given point.

Arguments

geometry: The geometry
pt: The query point
result: Output gsVector{Float64} to store the closest parametric point (modified in-place)

Details

Computes the closest point on the geometry to the given point and returns its parametric coordinates in result.

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TinyGismo.jacobian! — Function

jacobian!(geometry::gsGeometry, u::Union{Vector{Float64}, Matrix{Float64}}, out::gsMatrix{Float64})

Evaluate the Jacobian matrix of the geometry (in-place).

Arguments

geometry: The geometry
u: Parametric evaluation points
out: Output gsMatrix{Float64} to store Jacobian (modified in-place)

Details

The Jacobian is the matrix of all first-order partial derivatives of the geometry mapping.

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TinyGismo.jacobian — Function

jacobian(geometry::gsGeometry, u::Union{Vector{Float64}, Matrix{Float64}})

Evaluate the Jacobian matrix of the geometry.

Arguments

geometry: The geometry
u: Parametric evaluation points

Returns

The Jacobian matrix

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TinyGismo.hessian! — Function

hessian!(geometry::gsGeometry, u::Union{Vector{Float64}, Matrix{Float64}}, out::gsMatrix{Float64}, coord::Int)

Evaluate the Hessian matrix of a coordinate function (in-place).

Arguments

geometry: The geometry
u: Parametric evaluation points
out: Output gsMatrix{Float64} to store Hessian (modified in-place)
coord: The coordinate component (1-indexed)

Details

The Hessian is the matrix of second-order partial derivatives for the specified coordinate.

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TinyGismo.hessian — Function

hessian(geometry::gsGeometry, u::Union{Vector{Float64}, Matrix{Float64}}, coord::Int)

Evaluate the Hessian matrix of a coordinate function.

Arguments

geometry: The geometry
u: Parametric evaluation points
coord: The coordinate component (1-indexed)

Returns

The Hessian matrix

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Dimension Queries

TinyGismo.targetDim — Function

targetDim(geometry::gsGeometry)

Get the dimension of the ambient space (target dimension).

Arguments

geometry: The geometry

Returns

The target dimension (physical space dimension)

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TinyGismo.coefDim — Function

coefDim(geometry::gsGeometry)

Get the dimension of the coefficient space.

Arguments

geometry: The geometry

Returns

The coefficient dimension

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TinyGismo.geoDim — Function

geoDim(geometry::gsGeometry)

Get the geometric dimension (same as target dimension).

Arguments

geometry: The geometry

Returns

The geometric dimension

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TinyGismo.parDim — Function

parDim(geometry::gsGeometry)

Get the parametric dimension (number of parameters).

Arguments

geometry: The geometry

Returns

The parametric dimension (1 for curves, 2 for surfaces, etc.)

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