Precision Class Reference

The Precision package offers a set of functions defining precision criteria for use in conventional situations when comparing two numbers. Generalities It is not advisable to use floating number equality. Instead, the difference between numbers must be compared with a given precision, i.e. : double x1, x2 ; x1 = ... x2 = ... If ( x1 == x2 ) ... should not be used and must be written as indicated below: double x1, x2 ; double Precision = ... x1 = ... x2 = ... If ( Abs ( x1 - x2 ) < Precision ) ... Likewise, when ordering floating numbers, you must take the following into account : double x1, x2 ; double Precision = ... x1 = ... ! a large number x2 = ... ! another large number If ( x1 < x2 - Precision ) ... is incorrect when x1 and x2 are large numbers ; it is better to write : double x1, x2 ; double Precision = ... x1 = ... ! a large number x2 = ... ! another large number If ( x2 - x1 > Precision ) ... Precision in Cas.Cade Generally speaking, the precision criterion is not implicit in Cas.Cade. Low-level geometric algorithms accept precision criteria as arguments. As a rule, they should not refer directly to the precision criteria provided by the Precision package. On the other hand, high-level modeling algorithms have to provide the low-level geometric algorithms that they call, with a precision criteria. One way of doing this is to use the above precision criteria. Alternatively, the high-level algorithms can have their own system for precision management. For example, the Topology Data Structure stores precision criteria for each elementary shape (as a vertex, an edge or a face). When a new topological object is constructed, the precision criteria are taken from those provided by the Precision package, and stored in the related data structure. Later, a topological algorithm which analyses these objects will work with the values stored in the data structure. Also, if this algorithm is to build a new topological object, from these precision criteria, it will compute a new precision criterion for the new topological object, and write it into the data structure of the new topological object. The different precision criteria offered by the Precision package, cover the most common requirements of geometric algorithms, such as intersections, approximations, and so on. The choice of precision depends on the algorithm and on the geometric space. The geometric space may be : More...

#include <Precision.hxx>

Static Public Member Functions
static constexpr double	Angular ()
	Returns the recommended precision value when checking the equality of two angles (given in radians). double Angle1 = ... , Angle2 = ... ; If ( std::abs( Angle2 - Angle1 ) < Precision::Angular() ) ... The tolerance of angular equality may be used to check the parallelism of two vectors : gp_Vec V1, V2 ; V1 = ... V2 = ... If ( V1.IsParallel (V2, Precision::Angular() ) ) ... The tolerance of angular equality is equal to 1.e-12. Note : The tolerance of angular equality can be used when working with scalar products or cross products since sines and angles are equivalent for small angles. Therefore, in order to check whether two unit vectors are perpendicular : gp_Dir D1, D2 ; D1 = ... D2 = ... you can use : If ( std::abs( D1.D2 ) < Precision::Angular() ) ... (although the function IsNormal does exist).
static constexpr double	Confusion ()
	Returns the recommended precision value when checking coincidence of two points in real space. The tolerance of confusion is used for testing a 3D distance :
static constexpr double	SquareConfusion ()
	Returns square of Confusion. Created for speed and convenience.
static constexpr double	Computational ()
	Returns a precision value at machine epsilon level, used for low-level numerical computations and floating-point comparisons. Unlike the geometric tolerances (Confusion, Intersection, Approximation) which are application-level values for modeling operations, this value represents the fundamental limit of floating-point arithmetic precision.
static constexpr double	SquareComputational ()
	Returns square of Computational. Created for speed and convenience when comparing squared values.
static constexpr double	Intersection ()
	Returns the precision value in real space, frequently used by intersection algorithms to decide that a solution is reached. This function provides an acceptable level of precision for an intersection process to define the adjustment limits. The tolerance of intersection is designed to ensure that a point computed by an iterative algorithm as the intersection between two curves is indeed on the intersection. It is obvious that two tangent curves are close to each other, on a large distance. An iterative algorithm of intersection may find points on these curves within the scope of the confusion tolerance, but still far from the true intersection point. In order to force the intersection algorithm to continue the iteration process until a correct point is found on the tangent objects, the tolerance of intersection must be smaller than the tolerance of confusion. On the other hand, the tolerance of intersection must be large enough to minimize the time required by the process to converge to a solution. The tolerance of intersection is equal to : Precision::Confusion() / 100. (that is, 1.e-9).
static constexpr double	Approximation ()
	Returns the precision value in real space, frequently used by approximation algorithms. This function provides an acceptable level of precision for an approximation process to define adjustment limits. The tolerance of approximation is designed to ensure an acceptable computation time when performing an approximation process. That is why the tolerance of approximation is greater than the tolerance of confusion. The tolerance of approximation is equal to : Precision::Confusion() * 10. (that is, 1.e-6). You may use a smaller tolerance in an approximation algorithm, but this option might be costly.
static constexpr double	Parametric (const double P, const double T)
	Convert a real space precision to a parametric space precision. <T> is the mean value of the length of the tangent of the curve or the surface.
static constexpr double	PConfusion (const double T)
	Returns a precision value in parametric space, which may be used :
static constexpr double	SquarePConfusion ()
	Returns square of PConfusion. Created for speed and convenience.
static constexpr double	PIntersection (const double T)
	Returns a precision value in parametric space, which may be used by intersection algorithms, to decide that a solution is reached. The purpose of this function is to provide an acceptable level of precision in parametric space, for an intersection process to define the adjustment limits. The parametric tolerance of intersection is designed to give a mean value in relation with the dimension of the curve or the surface. It considers that a variation of parameter equal to 1. along a curve (or an isoparametric curve of a surface) generates a segment whose length is equal to 100. (default value), or T. The parametric tolerance of intersection is equal to :
static constexpr double	PApproximation (const double T)
	Returns a precision value in parametric space, which may be used by approximation algorithms. The purpose of this function is to provide an acceptable level of precision in parametric space, for an approximation process to define the adjustment limits. The parametric tolerance of approximation is designed to give a mean value in relation with the dimension of the curve or the surface. It considers that a variation of parameter equal to 1. along a curve (or an isoparametric curve of a surface) generates a segment whose length is equal to 100. (default value), or T. The parametric tolerance of intersection is equal to :
static constexpr double	Parametric (const double P)
	Convert a real space precision to a parametric space precision on a default curve.
static constexpr double	PConfusion ()
	Used to test distances in parametric space on a default curve.
static constexpr double	PIntersection ()
	Used for Intersections in parametric space on a default curve.
static constexpr double	PApproximation ()
	Used for Approximations in parametric space on a default curve.
static bool	IsInfinite (const double R)
	Returns True if R may be considered as an infinite number. Currently std::abs(R) > 1e100.
static constexpr bool	IsPositiveInfinite (const double R)
	Returns True if R may be considered as a positive infinite number. Currently R > 1e100.
static constexpr bool	IsNegativeInfinite (const double R)
	Returns True if R may be considered as a negative infinite number. Currently R < -1e100.
static constexpr double	Infinite ()
	Returns a big number that can be considered as infinite. Use -Infinite() for a negative big number.

Detailed Description

The Precision package offers a set of functions defining precision criteria for use in conventional situations when comparing two numbers. Generalities It is not advisable to use floating number equality. Instead, the difference between numbers must be compared with a given precision, i.e. : double x1, x2 ; x1 = ... x2 = ... If ( x1 == x2 ) ... should not be used and must be written as indicated below: double x1, x2 ; double Precision = ... x1 = ... x2 = ... If ( Abs ( x1 - x2 ) < Precision ) ... Likewise, when ordering floating numbers, you must take the following into account : double x1, x2 ; double Precision = ... x1 = ... ! a large number x2 = ... ! another large number If ( x1 < x2 - Precision ) ... is incorrect when x1 and x2 are large numbers ; it is better to write : double x1, x2 ; double Precision = ... x1 = ... ! a large number x2 = ... ! another large number If ( x2 - x1 > Precision ) ... Precision in Cas.Cade Generally speaking, the precision criterion is not implicit in Cas.Cade. Low-level geometric algorithms accept precision criteria as arguments. As a rule, they should not refer directly to the precision criteria provided by the Precision package. On the other hand, high-level modeling algorithms have to provide the low-level geometric algorithms that they call, with a precision criteria. One way of doing this is to use the above precision criteria. Alternatively, the high-level algorithms can have their own system for precision management. For example, the Topology Data Structure stores precision criteria for each elementary shape (as a vertex, an edge or a face). When a new topological object is constructed, the precision criteria are taken from those provided by the Precision package, and stored in the related data structure. Later, a topological algorithm which analyses these objects will work with the values stored in the data structure. Also, if this algorithm is to build a new topological object, from these precision criteria, it will compute a new precision criterion for the new topological object, and write it into the data structure of the new topological object. The different precision criteria offered by the Precision package, cover the most common requirements of geometric algorithms, such as intersections, approximations, and so on. The choice of precision depends on the algorithm and on the geometric space. The geometric space may be :

• a "real" 2D or 3D space, where the lengths are measured in meters, millimeters, microns, inches, etc ..., or
• a "parametric" space, 1D on a curve or 2D on a surface, where lengths have no dimension. The choice of precision criteria for real space depends on the choice of the product, as it is based on the accuracy of the machine and the unit of measurement. The choice of precision criteria for parametric space depends on both the accuracy of the machine and the dimensions of the curve or the surface, since the parametric precision criterion and the real precision criterion are linked : if the curve is defined by the equation P(t), the inequation : Abs ( t2 - t1 ) < ParametricPrecision means that the parameters t1 and t2 are considered to be equal, and the inequation : Distance ( P(t2) , P(t1) ) < RealPrecision means that the points P(t1) and P(t2) are considered to be coincident. It seems to be the same idea, and it would be wonderful if these two inequations were equivalent. Note that this is rarely the case ! What is provided in this package? The Precision package provides :
• a set of real space precision criteria for the algorithms, in view of checking distances and angles,
• a set of parametric space precision criteria for the algorithms, in view of checking both :
• the equality of parameters in a parametric space,
• or the coincidence of points in the real space, by using parameter values,
• the notion of infinite value, composed of a value assumed to be infinite, and checking tests designed to verify if any value could be considered as infinite. All the provided functions are very simple. The returned values result from the adaptation of the applications developed by the Open CASCADE company to Open CASCADE algorithms. The main interest of these functions lies in that it incites engineers developing applications to ask questions on precision factors. Which one is to be used in such or such case ? Tolerance criteria are context dependent. They must first choose :
• either to work in real space,
• or to work in parametric space,
• or to work in a combined real and parametric space. They must next decide which precision factor will give the best answer to the current problem. Within an application environment, it is crucial to master precision even though this process may take a great deal of time.

Member Function Documentation

Angular()

static constexpr double Precision::Angular ( )

inlinestaticconstexpr

Returns the recommended precision value when checking the equality of two angles (given in radians). double Angle1 = ... , Angle2 = ... ; If ( std::abs( Angle2 - Angle1 ) < Precision::Angular() ) ... The tolerance of angular equality may be used to check the parallelism of two vectors : gp_Vec V1, V2 ; V1 = ... V2 = ... If ( V1.IsParallel (V2, Precision::Angular() ) ) ... The tolerance of angular equality is equal to 1.e-12. Note : The tolerance of angular equality can be used when working with scalar products or cross products since sines and angles are equivalent for small angles. Therefore, in order to check whether two unit vectors are perpendicular : gp_Dir D1, D2 ; D1 = ... D2 = ... you can use : If ( std::abs( D1.D2 ) < Precision::Angular() ) ... (although the function IsNormal does exist).

Approximation()

static constexpr double Precision::Approximation ( )

inlinestaticconstexpr

Returns the precision value in real space, frequently used by approximation algorithms. This function provides an acceptable level of precision for an approximation process to define adjustment limits. The tolerance of approximation is designed to ensure an acceptable computation time when performing an approximation process. That is why the tolerance of approximation is greater than the tolerance of confusion. The tolerance of approximation is equal to : Precision::Confusion() * 10. (that is, 1.e-6). You may use a smaller tolerance in an approximation algorithm, but this option might be costly.

Computational()

static constexpr double Precision::Computational ( )

inlinestaticconstexpr

Returns a precision value at machine epsilon level, used for low-level numerical computations and floating-point comparisons. Unlike the geometric tolerances (Confusion, Intersection, Approximation) which are application-level values for modeling operations, this value represents the fundamental limit of floating-point arithmetic precision.

Typical use cases include:

• Checking if squared magnitudes are effectively zero (e.g., vector.SquareMagnitude() < SquareComputational())
• Division-by-zero protection in numerical algorithms
• Convergence criteria in iterative solvers at machine precision level
• Detecting numerical degeneracies in low-level computations

The computational tolerance is equal to DBL_EPSILON (approximately 2.22e-16), which is the smallest positive value such that 1.0 + eps != 1.0 in double precision floating-point arithmetic. This is the fundamental machine epsilon for double (double) type.

Note: This value should NOT be used for geometric comparisons. Use Precision::Confusion() for comparing geometric distances, Precision::Angular() for angles, etc.

Confusion()

static constexpr double Precision::Confusion ( )

inlinestaticconstexpr

Returns the recommended precision value when checking coincidence of two points in real space. The tolerance of confusion is used for testing a 3D distance :

• Two points are considered to be coincident if their distance is smaller than the tolerance of confusion. gp_Pnt P1, P2 ; P1 = ... P2 = ... if ( P1.IsEqual ( P2 , Precision::Confusion() ) ) then ...
• A vector is considered to be null if it has a null length : gp_Vec V ; V = ... if ( V.Magnitude() < Precision::Confusion() ) then ... The tolerance of confusion is equal to 1.e-7. The value of the tolerance of confusion is also used to define :
• the tolerance of intersection, and
• the tolerance of approximation. Note : As a rule, coordinate values in Cas.Cade are not dimensioned, so 1. represents one user unit, whatever value the unit may have : the millimeter, the meter, the inch, or any other unit. Let's say that Cas.Cade algorithms are written to be tuned essentially with mechanical design applications, on the basis of the millimeter. However, these algorithms may be used with any other unit but the tolerance criterion does no longer have the same signification. So pay particular attention to the type of your application, in relation with the impact of your unit on the precision criterion.
• For example in mechanical design, if the unit is the millimeter, the tolerance of confusion corresponds to a distance of 1 / 10000 micron, which is rather difficult to measure.
• However in other types of applications, such as cartography, where the kilometer is frequently used, the tolerance of confusion corresponds to a greater distance (1 / 10 millimeter). This distance becomes easily measurable, but only within a restricted space which contains some small objects of the complete scene.

Infinite()

static constexpr double Precision::Infinite ( )

inlinestaticconstexpr

Returns a big number that can be considered as infinite. Use -Infinite() for a negative big number.

Intersection()

static constexpr double Precision::Intersection ( )

inlinestaticconstexpr

Returns the precision value in real space, frequently used by intersection algorithms to decide that a solution is reached. This function provides an acceptable level of precision for an intersection process to define the adjustment limits. The tolerance of intersection is designed to ensure that a point computed by an iterative algorithm as the intersection between two curves is indeed on the intersection. It is obvious that two tangent curves are close to each other, on a large distance. An iterative algorithm of intersection may find points on these curves within the scope of the confusion tolerance, but still far from the true intersection point. In order to force the intersection algorithm to continue the iteration process until a correct point is found on the tangent objects, the tolerance of intersection must be smaller than the tolerance of confusion. On the other hand, the tolerance of intersection must be large enough to minimize the time required by the process to converge to a solution. The tolerance of intersection is equal to : Precision::Confusion() / 100. (that is, 1.e-9).

IsInfinite()

static bool Precision::IsInfinite ( const double R )

inlinestatic

Returns True if R may be considered as an infinite number. Currently std::abs(R) > 1e100.

IsNegativeInfinite()

static constexpr bool Precision::IsNegativeInfinite ( const double R )

inlinestaticconstexpr

Returns True if R may be considered as a negative infinite number. Currently R < -1e100.

IsPositiveInfinite()

static constexpr bool Precision::IsPositiveInfinite ( const double R )

inlinestaticconstexpr

Returns True if R may be considered as a positive infinite number. Currently R > 1e100.

PApproximation() [1/2]

static constexpr double Precision::PApproximation ( )

inlinestaticconstexpr

Used for Approximations in parametric space on a default curve.

This is Precision::Parametric(Precision::Approximation())

PApproximation() [2/2]

static constexpr double Precision::PApproximation ( const double T )

inlinestaticconstexpr

Returns a precision value in parametric space, which may be used by approximation algorithms. The purpose of this function is to provide an acceptable level of precision in parametric space, for an approximation process to define the adjustment limits. The parametric tolerance of approximation is designed to give a mean value in relation with the dimension of the curve or the surface. It considers that a variation of parameter equal to 1. along a curve (or an isoparametric curve of a surface) generates a segment whose length is equal to 100. (default value), or T. The parametric tolerance of intersection is equal to :

• Precision::Approximation() / 100., or Precision::Approximation() / T.

Parametric() [1/2]

static constexpr double Precision::Parametric ( const double P )

inlinestaticconstexpr

Convert a real space precision to a parametric space precision on a default curve.

Value is Parametric(P,1.e+2)

Parametric() [2/2]

static constexpr double Precision::Parametric ( const double P, const double T )

inlinestaticconstexpr

Convert a real space precision to a parametric space precision. <T> is the mean value of the length of the tangent of the curve or the surface.

Value is P / T

PConfusion() [1/2]

static constexpr double Precision::PConfusion ( )

inlinestaticconstexpr

Used to test distances in parametric space on a default curve.

This is Precision::Parametric(Precision::Confusion())

PConfusion() [2/2]

static constexpr double Precision::PConfusion ( const double T )

inlinestaticconstexpr

Returns a precision value in parametric space, which may be used :

• to test the coincidence of two points in the real space, by using parameter values, or
• to test the equality of two parameter values in a parametric space. The parametric tolerance of confusion is designed to give a mean value in relation with the dimension of the curve or the surface. It considers that a variation of parameter equal to 1. along a curve (or an isoparametric curve of a surface) generates a segment whose length is equal to 100. (default value), or T. The parametric tolerance of confusion is equal to :
• Precision::Confusion() / 100., or Precision::Confusion() / T. The value of the parametric tolerance of confusion is also used to define :
• the parametric tolerance of intersection, and
• the parametric tolerance of approximation. Warning It is rather difficult to define a unique precision value in parametric space.
• First consider a curve (c) ; if M is the point of parameter u and M' the point of parameter u+du on the curve, call 'parametric tangent' at point M, for the variation du of the parameter, the quantity : T(u,du)=MM'/du (where MM' represents the distance between the two points M and M', in the real space).
• Consider the other curve resulting from a scaling transformation of (c) with a scale factor equal to

1. The 'parametric tangent' at the point of parameter u of this curve is ten times greater than the previous one. This shows that for two different curves, the distance between two points on the curve, resulting from the same variation of parameter du, may vary considerably.

• Moreover, the variation of the parameter along the curve is generally not proportional to the curvilinear abscissa along the curve. So the distance between two points resulting from the same variation of parameter du, at two different points of a curve, may completely differ.
• Moreover, the parameterization of a surface may generate two quite different 'parametric tangent' values in the u or in the v parametric direction.
• Last, close to the poles of a sphere (the points which correspond to the values -Pi/2. and Pi/2. of the v parameter) the u parameter may change from 0 to 2.Pi without impacting on the resulting point. Therefore, take great care when adjusting a parametric tolerance to your own algorithm.

PIntersection() [1/2]

static constexpr double Precision::PIntersection ( )

inlinestaticconstexpr

Used for Intersections in parametric space on a default curve.

This is Precision::Parametric(Precision::Intersection())

PIntersection() [2/2]

static constexpr double Precision::PIntersection ( const double T )

inlinestaticconstexpr

Returns a precision value in parametric space, which may be used by intersection algorithms, to decide that a solution is reached. The purpose of this function is to provide an acceptable level of precision in parametric space, for an intersection process to define the adjustment limits. The parametric tolerance of intersection is designed to give a mean value in relation with the dimension of the curve or the surface. It considers that a variation of parameter equal to 1. along a curve (or an isoparametric curve of a surface) generates a segment whose length is equal to 100. (default value), or T. The parametric tolerance of intersection is equal to :

• Precision::Intersection() / 100., or Precision::Intersection() / T.

SquareComputational()

static constexpr double Precision::SquareComputational ( )

inlinestaticconstexpr

Returns square of Computational. Created for speed and convenience when comparing squared values.

SquareConfusion()

static constexpr double Precision::SquareConfusion ( )

inlinestaticconstexpr

Returns square of Confusion. Created for speed and convenience.

SquarePConfusion()

static constexpr double Precision::SquarePConfusion ( )

inlinestaticconstexpr

Returns square of PConfusion. Created for speed and convenience.

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The documentation for this class was generated from the following file:

• Precision.hxx