\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 2D and Surface Function Interpolation

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GradientFittingTraits Concept Reference
Concepts

## Definition

Sibson gradient fitting functions are parameterized by a traits class that defines the primitives used by the algorithm. The concept `GradientFittingTraits` defines this common set of requirements.

Has Models:
  `CGAL::Interpolation_gradient_fitting_traits_2<K>`

See Also
  `InterpolationTraits`
  `CGAL::Interpolation_traits_2<K>`
  Sibson gradient fitting functions
  `CGAL::sibson_c1_interpolation()`
  `CGAL::farin_c1_interpolation()`
  `CGAL::quadratic_interpolation()`

## Types
typedef unspecified_type | FT
| The number type must follow the model `FieldNumberType`.
typedef unspecified_type | Point_d
| The point type on which the function is defined and interpolated.
typedef unspecified_type | Vector_d
| The corresponding vector type.
typedef unspecified_type | Aff_transformation_d
| defines a matrix type. More...
typedef unspecified_type | Construct_vector_d
| A constructor object for `Vector_d`. More...
typedef unspecified_type | Construct_scaled_vector_d
| Constructor object for `Vector_d`. More...
typedef unspecified_type | Construct_null_matrix_d
| Constructor object for `Aff_transformation_d`. More...
typedef unspecified_type | Construct_scaling_matrix_d
| Constructor object for `Aff_transformation_d`. More...
typedef unspecified_type | Construct_sum_matrix_d
| Constructor object for `Aff_transformation_d`. More...
typedef unspecified_type | Construct_outer_product_d
| Constructor object for `Aff_transformation_d`. More...

## Creation
| GradientFittingTraits ()
| default constructor.

## Operations
The following functions that create instances of the above constructor object types must exist.
Construct_vector_d | construct_vector_d_object ()
Construct_scaled_vector_d | construct_scaled_vector_d_object ()
Construct_null_matrix_d | construct_null_matrix_d_object ()
Construct_sum_matrix_d | construct_sum_matrix_d_object ()
Construct_outer_product_d | construct_outer_product_d_object ()

## Member Typedef Documentation

typedef unspecified_type GradientFittingTraits::Aff_transformation_d

defines a matrix type.

Must provide the following member functions :

`Aff_transformation tr.inverse ()` which gives the inverse transformation, and

`Aff_transformation tr.transform( Vector v)` which returns the multiplication of `tr` with `v`.

typedef unspecified_type GradientFittingTraits::Construct_null_matrix_d

Constructor object for `Aff_transformation_d`.

Provides :

`Aff_transformation_d operator()()` which introduces an affine transformation whose matrix has only zero entries.

typedef unspecified_type GradientFittingTraits::Construct_outer_product_d

Constructor object for `Aff_transformation_d`.

Provides :

`Aff_transformation_d operator()(Vector v)` which returns the outer product, i.e. the quadratic matrix `v` \( ^t\)`v`.

typedef unspecified_type GradientFittingTraits::Construct_scaled_vector_d

Constructor object for `Vector_d`.

Provides :

`Vector_d operator() (Vector_d v,FT scale)` which produces the vector `v` scaled by a factor `scale`.

typedef unspecified_type GradientFittingTraits::Construct_scaling_matrix_d

Constructor object for `Aff_transformation_d`.

Provides :

`Aff_transformation_d operator()(FT scale)` which introduces a scaling by a scale factor `scale`.

typedef unspecified_type GradientFittingTraits::Construct_sum_matrix_d

Constructor object for `Aff_transformation_d`.

Provides :

`Aff_transformation_d operator()(Aff_transformation_d tr1, Aff_transformation_d tr2)` which returns the sum of the two matrices representing `tr1` and `tr2`.

typedef unspecified_type GradientFittingTraits::Construct_vector_d

A constructor object for `Vector_d`.

Provides :

`Vector_d operator() (Point_d a, Point_d b)` which produces the vector `b - a` and

`Vector_d operator() (Null_vector NULL_VECTOR)` which introduces the null vector.