gsKnotVector< T > Class Template Reference

Detailed Description

template<typename T>
class gismo::gsKnotVector< T >

Class for representing a knot vector.

Consists of a vector of non-decreasing knots repeated according to their multiplicities and a vector of sums of multiplicities up to the corresponding unique knot. The repeated knots are stored in m_repKnots, whereas the multiplicity sum is stored in m_multSum.

Iterators

There are three means of iterating through the knots:

• iterator, which is just an STL iterator over the repeated knots;
• uiterator, which is similar but iterates over the unique knots (i.e., without multiplicities) and provides additional functionality;
• smart_iterator, which iterates over the repeated knots and provides additional functionality.

Terminology

• repeated knots is the non-decreasing sequence, where each knot is repeated according to its multiplicity.
• unique knots is the increasing sequence of all the distinct values from repeated knots.
• starting knot is the degree-th knot from the beginning, i.e., m_repKnots[m_deg].
• ending knot is the degree-th knot from the end, i.e., m_repKnots[ m_repKnots.size() - m_deg - 1 ] or, equivalently, m_repKnots.end()[-m_deg-1].
• domain is the interval between the starting knot and ending knot.
• knot intervalis the interval between two consecutive unique knots. It is closed from left and open from right, except for the interval [*(domainUEnd()-1), *domainUEnd()], which is considered closed from both sides.

Compatibility notes

Earlier versions contained several functions that have been removed.

• uiterator findElement( const T u ) const : use uFind( const T u ) instead;
• unsigned findspan( T u ) const : use iFind(u) - begin() instead;
• iterator findspanIter( T u ) const : use iFind(u) instead;
• int findElementIndex(T u) const : use uFind(u).uIndex() instead;
• unsigned Uniquefindspan (T u) const : use uFind(u).uIndex() instead;
• gsMatrix<unsigned,1> * findspan (const gsMatrix<T,1> & u) const : call iFind(u(0,i)) - begin() for each i from 0 to u.cols().
• int numKnotSpans() const : use uSize() - 1 instead;
• unsigned spans() const : use uSize() - 1 instead;

Inheritance diagram for gsKnotVector< T >:

Collaboration diagram for gsKnotVector< T >:

Public Types
typedef std::vector< T > ::const_iterator	iterator
typedef internal::gsKnotIterator< T >	smart_iterator
typedef internal::gsUKnotIterator< T >	uiterator

Public Member Functions
void	addConstant (T amount)
	Adds amount to all the knots.
void	addConstant (T start, T amount)
	Adds amount to all the knots, starting at knot start.
void	affineTransformTo (T newBeg, T newEnd)
template<typename iterType >
void	append (iterType begin, iterType end)
	Adds the knots between begin and end to the knot vector.
gsAsConstMatrix< T >	asMatrix () const
	Returns the knots as a matrix of size 1 x size()
T	at (const size_t &i) const
	Returns the value of the i - th knot (counted with repetitions).
iterator	begin () const
	Returns iterator pointing to the beginning of the repeated knots.
iterator	beginAt (mult_t upos) const
virtual gsMatrix< T >	boundingBox ()
virtual knotContainer	breaks () const
	Returns unique knots of the domain (i.e., including the endpoints of the domain).
void	centers_into (gsMatrix< T > &result) const
bool	check () const
	Checks whether the knot vector is in a consistent state.
gsKnotVector *	clone () const
	Returns a pointer to a copy.
std::vector< T >	coarsen (index_t knotRemove=1, index_t knotSkip=1, mult_t mul=-1)
const T *	data () const
	Returns a pointer to the beginning of the vector of knots.
int	deduceDegree () const
int	degree () const
	Returns the degree of the knot vector.
void	degreeDecrease (int const &i=1, bool updateInterior=false)
	Inverse of degreeIncrease.
void	degreeElevate (const short_t &i=1)
void	degreeIncrease (int const &i=1)
void	degreeReduce (const short_t &i)
	Converse to degreeElevate.
std::string	detail () const
	Return a string with detailed information on the knot vector.
void	difference (const gsKnotVector< T > &other, std::vector< T > &result) const
	Computes the difference between this knot-vector and other. More...
virtual short_t	dim () const
	dimension of the domain
iterator	domainBegin () const
	Returns an iterator pointing to the starting knot of the domain.
iterator	domainEnd () const
	Returns an iterator pointing to the end-knot of the domain.
smart_iterator	domainSBegin () const
smart_iterator	domainSEnd () const
uiterator	domainUBegin () const
	Returns a unique iterator pointing to the starting knot of the domain.
uiterator	domainUEnd () const
	Returns a unique iterator pointing to the ending knot of the domain.
virtual gsMatrix< T >	elements ()
	Returns a list of elements.
iterator	end () const
	Returns iterator pointing past the end of the repeated knots.
iterator	endAt (mult_t upos) const
void	erase (const mult_t first, const mult_t last)
	Removes the knots in the range [first,last)
T	first () const
	Get the first knot.
unsigned	firstKnotIndex (const size_t &i) const
const knotContainer &	get () const
	Get a reference to the underlying std::vector of knots.
void	getUniformRefinementKnots (mult_t knotsPerSpan, knotContainer &result, mult_t mult=1) const
gsMatrix< T >	greville () const
T	greville (int i) const
void	greville_into (gsMatrix< T > &result) const
	gsKnotVector ()
	Empty constructor sets the degree to -1 and leaves the knots empty.
	gsKnotVector (knotContainer knots, short_t degree=-1)
	gsKnotVector (short_t degree)
	Sets the degree and leaves the knots uninitialized.
	gsKnotVector (T first, T last, unsigned interior, mult_t mult_ends=1, mult_t mult_interior=1, short_t degree=-1)
template<typename iterType >
	gsKnotVector (short_t deg, const iterType begOfKnots, const iterType endOfKnots)
	gsKnotVector (const knotContainer &uKnots, int degree, int regularity)
bool	has (T knot) const
iterator	iFind (const T u) const
	Returns an iterator to the last occurrence of the knot at the beginning of the knot interval containing u. Note that if u == *domainEnd(), it returns the iterator domainEnd() - 1. Cf. knot interval.
bool	includes (const gsKnotVector< T > &other) const
	Returns true if every knot of other (counted with repetitions) is found within this knot-vector. Also returns true if other is empty. More...
void	increaseMultiplicity (const mult_t i=1, bool boundary=false)
bool	inDomain (const T u) const
	Checks, whether the given value is inside the domain.
void	initClamped (int degree, unsigned numKnots=2, unsigned mult_interior=1)
void	initClamped (T u0, T u1, int degree, unsigned interior=0, unsigned mult_interior=1)
	Resets the knot vector so that it is uniform and has clamped endknots.
void	initGraded (unsigned numKnots, int degree, T grading=0.5, unsigned mult_interior=1)
void	initGraded (T u0, T u1, unsigned interior, int degree, T grading, unsigned mult_interior=1)
	Resets the knot vector so that its knots are graded.
void	initUniform (T first, T last, unsigned interior, unsigned mult_ends, unsigned mult_interior, short_t degree=-1)
void	initUniform (unsigned numKnots, unsigned mult_ends, unsigned mult_interior=1, short_t degree=-1)
void	insert (T knot, mult_t mult=1)
	Inserts knot knot into the knot vector with multiplicity mult.
template<typename iterType >
void	insert (iterType ibeg, iterType iend)
void	insert (const knotContainer &knots, int mult=1)
	Insert knots into the knot vector. More...
bool	isOpen () const
bool	isUniform (T tol=1e-9) const
	Checks whether the knot vector is uniform.
gsKnotVector	knotIntersection (const gsKnotVector &b) const
	Computes the intersection of knot-vectors this and b.
unsigned	knotsUntilSpan (const size_t &i) const
gsKnotVector	knotUnion (const gsKnotVector &b) const
	Computes the union of knot-vectors this and b. More...
T	last () const
	Get the last knot.
unsigned	lastKnotIndex (const size_t &i) const
mult_t	maxInteriorMultiplicity () const
	Returns the maximum multiplicity in the interior.
T	maxIntervalLength () const
	Returns the maximum interval length of the knot sequence.
virtual gsMatrix< T >	mesh ()
	Returns the mesh..
mult_t	minInteriorMultiplicity () const
	Returns the minimum multiplicity in the interior.
T	minIntervalLength () const
	Returns the minimum interval length of the knot sequence.
mult_t	multFirst () const
	Returns the multiplicity of the first knot.
multContainer	multiplicities () const
	Returns vector of multiplicities of the knots.
mult_t	multiplicity (T u) const
mult_t	multiplicityIndex (mult_t i) const
mult_t	multLast () const
	Returns the multiplicity of the last knot.
const mult_t *	multSumData () const
size_t	numElements () const
	Number of knot intervals inside domain.
index_t	numLeftGhosts () const
index_t	numRightGhosts () const
	operator const knotContainer & () const
	Cast to the full vector of knot values (with repetitions).
bool	operator!= (const gsKnotVector< T > &other) const
	Compares with another knot vector.
const T &	operator() (const mult_t i) const
	Provides the knot with unique index i.
bool	operator== (const gsKnotVector< T > &other) const
	Compare with another knot vector.
const T &	operator[] (const mult_t i) const
	Provides the i-th knot (numbered including repetitions).
std::ostream &	print (std::ostream &os=gsInfo) const
reverse_iterator	rbegin () const
	Returns reverse iterator pointing past the end of the repeated knots.
void	reduceMultiplicity (const mult_t i=1, bool boundary=false)
template<typename iterType >
void	refineSpans (iterType begin, iterType end, mult_t knotsPerSpan)
	Inserts knotsPerSpan knots between each two knots between begin and end.
void	refineSpans (const std::vector< unsigned > &spanIndices, mult_t knotsPerSpan=1)
	Because of type compatibility, cf. the other version.
void	refineSpans (const multContainer &spanIndices, mult_t knotsPerSpan=1)
void	remove (uiterator uit, mult_t mult=1)
void	remove (const T knot, mult_t mult=1)
	Decreases multiplicity of the knot knot by mult.
reverse_iterator	rend () const
	Returns reverse iterator pointing to the beginning of the repeated knots.
void	reverse ()
	Better directly use affineTransformTo.
reverse_smart_iterator	rsbegin () const
	Returns the reverse smart iterator pointing past the end of the repeated knots.
reverse_smart_iterator	rsend () const
	Returns the reverse smart iterator pointing to the beginning of the repeated knots.
smart_iterator	sbegin () const
	Returns the smart iterator pointing to the beginning of the repeated knots.
smart_iterator	send () const
	Returns the smart iterator pointing past the end of the repeated knots.
void	set_degree (short_t p)
	Sets the degree to p.
size_t	size () const
	Number of knots (including repetitions).
void	supportIndex_into (const mult_t &i, gsMatrix< index_t > &result) const
void	swap (gsKnotVector &other)
	Swaps with other knot vector.
void	symDifference (const gsKnotVector< T > &other, std::vector< T > &result) const
	Computes the symmetric difference between this knot-vector and other. More...
void	transform (T c, T d)
	See affineTransform().
void	trimLeft (const mult_t numKnots)
	Removes the left-most numKnots from the knot-vector.
void	trimRight (const mult_t numKnots)
	Removes the right-most numKnots from the knot-vector.
unsigned	u_multiplicityIndex (size_t const &i) const
uiterator	ubegin () const
	Returns unique iterator pointing to the beginning of the unique knots.
uiterator	uend () const
	Returns unique iterator pointing past the end of the unique knots.
uiterator	uFind (const T u) const
	Returns the uiterator pointing to the knot at the beginning of the knot interval containing u. Note that if u == *domainUEnd(), it returns the uiterator domainUEnd() - 1. Cf. knot interval.
uiterator	uLowerBound (const T u) const
	Returns a uiterator pointing to the first knot which does not compare less than u. More...
void	uniformRefine (mult_t numKnots=1, mult_t mult=1)
knotContainer	unique () const
	Returns unique knots.
reverse_uiterator	urbegin () const
	Returns reverse unique iterator pointing past the end of the unique knots.
reverse_uiterator	urend () const
	Returns reverse unique iterator pointing to the beginning of the unique knots.
size_t	uSize () const
	Number of unique knots (i.e., without repetitions).
uiterator	uUpperBound (const T u) const
	Returns an iterator pointing to the first knot which compares greater than u. More...
T	uValue (const size_t &i) const
	Provides the knot with unique index i.

Static Public Member Functions
static bool	isConsistent (const knotContainer &repKnots, const multContainer &multSums)
	Sanity check.

Member Typedef Documentation

typedef std::vector<T>::const_iterator iterator

This iterator is an iterator over the repeated knots. Can be obtained by calling [r]begin() and [r]end().

typedef internal::gsKnotIterator<T> smart_iterator

This iterator is an iterator over repeated knots with additional functions. Can be obtained by calling [r]sbegin() and [r]send().

typedef internal::gsUKnotIterator<T> uiterator

This iterator is an iterator over the unique knots. Has additional functions. Can be obtained by calling [r]ubegin() and [r]uend().

Constructor & Destructor Documentation

gsKnotVector ( knotContainer  knots, short_t  degree = -1  )

explicit

Constructs knot vector from the given knots (repeated according to multiplicities) and deduces the degree from the multiplicity of the endknots.

gsKnotVector	(	T	first,
		T	last,
		unsigned	interior,
		mult_t	mult_ends = 1,
		mult_t	mult_interior = 1,
		short_t	degree = -1
	)

Parameters

first	starting parameter
last	end parameter parameter
interior	number of interior knots
mult_ends	multiplicity at the two end knots
mult_interior	multiplicity at the interior knots
degree	Degree of the knot. (default: -1)

gsKnotVector ( short_t  deg, const iterType  begOfKnots, const iterType  endOfKnots  )

inline

Constructs knot vector from the given degree and iterators marking its endpoints.

Parameters

deg	Degree of the knot.
begOfKnots	iterator pointing to the beginning of the knots,
endOfKnots	iterator pointing past the end of the knots. The knots between begOfKnots and endOfKnots are assumed to be repeated according to their multiplicities and sorted.

gsKnotVector	(	const knotContainer &	uKnots,
		int	degree,
		int	regularity
	)

Parameters

uKnots	unique knots (assumed to be sorted),
degree	-> endknots have multiplicity degree + 1,
regularity	-> internal knots have multiplicity degree - regularity

Member Function Documentation

void affineTransformTo	(	T	newBeg,
		T	newEnd
	)

Scales and translates the knot vector so that its new endpoints are newBeg and newEnd.

gsKnotVector< T >::iterator beginAt

(

mult_t

upos

)

const

Returns an iterator pointing to the first appearance of the knot with unique index (ie. counted without repetitions, left ghosts mapped to negatives) equal to upos.

virtual gsMatrix<T> boundingBox ( )

inlinevirtualinherited

Returns a bounding box for the domain eg. This coincides to the domain in case of tensor-product domains

void centers_into

(

gsMatrix< T > &

result

)

const

Writes the center points of each element (knot-span inside the domain) to result.

std::vector< T > coarsen	(	index_t	knotRemove = 1,
		index_t	knotSkip = 1,
		mult_t	mul = -1
	)

Coarsen the knot vector: for each group of knotRemove consecutive interior unique knots, reduce their multiplicity by mul (or remove completely, if mul = -1), then skip knotSkip consecutive unique knots and go on

Parameters

knotRemove	Number of consecutive knots to remove
knotSkip	Number of consecutive knots which stay untouched in between
mul	Multiplicity drop for each processed knot (or -1, full remove)

int deduceDegree

(

)

const

Attempts to deduce the degree from the multiplicities of the endpoints (assuming clamped knots).

void degreeElevate

(

const short_t &

i = 1

)

Elevate the degree. I.e., increase the multiplicity of all the knots by one and increment the degree.

void degreeIncrease ( int const &  i = 1 )

inline

Increase the degree keeping interior knots intact (add clamped knots only). Cf. degreeElevate.

void difference	(	const gsKnotVector< T > &	other,
		std::vector< T > &	result
	)		const

Computes the difference between this knot-vector and other.

All knots in this which are not found in other (counted with repetitions), are stored in result

smart_iterator domainSBegin ( ) const

inline

Returns a smart iterator pointing to the starting knot of the domain

smart_iterator domainSEnd ( ) const

inline

Returns a smart iterator pointing to the ending knot of the domain

gsKnotVector< T >::iterator endAt

(

mult_t

upos

)

const

Returns an iterator pointing one past the last appearance of the knot with cardinal index (ie. counted without repetitions, left ghosts mapped to negatives) equal to upos.

unsigned firstKnotIndex ( const size_t &  i ) const

inline

Returns the first knot-index of cardinal index i (i.e. counted without repetitions)

void getUniformRefinementKnots	(	mult_t	knotsPerSpan,
		knotContainer &	result,
		mult_t	mult = 1
	)		const

Compute the new knots needed for uniform refinement with the given number of knots per span and return them in result. TODO: Think who is in charge of uniform refinement: basis or knot vector?

gsMatrix<T> greville ( ) const

inline

Returns the greville points of the B-splines defined on this knot vector.

T greville

(

int

i

)

const

Returns Greville abscissa of the i - the B-spline defined on the knot vector.

void greville_into

(

gsMatrix< T > &

result

)

const

Writes Greville abscissae of the B-splines defined on this knot vector to result.

bool has ( T  knot ) const

inline

True iff the knot exists in the vector

Parameters

knot	parameter value.

bool includes

(

const gsKnotVector< T > &

other

)

const

Returns true if every knot of other (counted with repetitions) is found within this knot-vector. Also returns true if other is empty.

Compresses the knot-vector by making the knot-sequence strictly increasing

void increaseMultiplicity	(	const mult_t	i = 1,
		bool	boundary = false
	)

Increase the multiplicity of all the knots by i. If boundary is set to false, the boundary knots are not adjusted

void initClamped	(	int	degree,
		unsigned	numKnots = 2,
		unsigned	mult_interior = 1
	)

Resets the knot vector so that it is uniform from 0 to 1 and the multiplicities of the endpoints are chosen according to the degree.

void initGraded ( unsigned  numKnots, int  degree, T  grading = 0.5, unsigned  mult_interior = 1  )

inline

Resets the knot vector so that its knots are graded towards 0 according to the grading parameter and the endpoints are 0 and 1.

void initUniform	(	T	first,
		T	last,
		unsigned	interior,
		unsigned	mult_ends,
		unsigned	mult_interior,
		short_t	degree = -1
	)

Resets the knot vector so that it has interior knots between first and last, each of them repeated mult_interior times (and the endpoints repeated mult_ends times) and the degree is equal to degree

void initUniform ( unsigned  numKnots, unsigned  mult_ends, unsigned  mult_interior = 1, short_t  degree = - 1  )

inline

Resets the knot vector so that it has numKnots knots between 0 and 1, each repeated mult_interior times (whereas 0 and 1 are repeated mult_ends times each) and the degree is equal to degree.

void insert ( iterType  ibeg, iterType  iend  )

inline

Inserts each knot between ibeg and iend with multiplicity 1. Knots can be repeated in the sequence but they must be sorted. The complexity is size() + (iend-ibeg) comparisons

void insert ( const knotContainer &  knots, int  mult = 1  )

inline

Insert knots into the knot vector.

Parameters

knots	parameter values of the new knots, stored in a std::vector
mult	multiplicity of the new knots

bool isOpen ( ) const

inline

Returns true iff the knot is open (i.e., both endpoints have multiplicities equal to degree+1).

unsigned knotsUntilSpan ( const size_t &  i ) const

inline

Returns the multiplicity sum at unique index i (i.e. counted without repetitions)

gsKnotVector< T > knotUnion

(

const gsKnotVector< T > &

b

)

const

Computes the union of knot-vectors this and b.

Example: gsKnotVector<> testKv1(0, 1, 2, 2, 1); // 0 0 1/3 2/3 1 1 gsKnotVector<> testKv2(0, 1, 1, 2, 2); // 0 0 1/2 1/2 1 1 gsKnotVector<T> kv = kv1.knotUnion(kv2);

Results in: 0 0 1/3 1/2 1/2 2/3 1 1

unsigned lastKnotIndex ( const size_t &  i ) const

inline

Returns the last knot-index of cardinal index i (i.e. counted without repetitions)

gsKnotVector< T >::mult_t multiplicity

(

T

u

)

const

Returns the multiplicity of the knot-value u (or zero if the value is not a knot)

gsKnotVector< T >::mult_t multiplicityIndex

(

mult_t

i

)

const

Returns the multiplicity of the knot number i (counter with repetitions).

const mult_t* multSumData ( ) const

inline

Returns a pointer to the beginning of the vector of running multiplicities.

index_t numLeftGhosts ( ) const

inline

Computes the number of left ghosts, i.e., of the knots to the left of the domain beginning.

index_t numRightGhosts ( ) const

inline

Computes the number of right ghosts, i.e., of the knots to the right of the domain end.

std::ostream & print ( std::ostream &  os = gsInfo ) const

virtual

Print the knot vector to the given stream. TODO: Improve.

Implements gsDomain< T >.

void reduceMultiplicity	(	const mult_t	i = 1,
		bool	boundary = false
	)

Reduce the multiplicity of all knots by i. If boundary is set to false, the boundary knots are not adjusted

void refineSpans	(	const multContainer &	spanIndices,
		mult_t	knotsPerSpan = 1
	)

Inserts knotsPerSpan knots into the spans corresponding to indices listed in span Indices.

void remove	(	uiterator	uit,
		mult_t	mult = 1
	)

Decreases multiplicity of the knot pointed by uit by mult; removes it altogether, should its multiplicity be decreased to zero.

void supportIndex_into	(	const mult_t &	i,
		gsMatrix< index_t > &	result
	)		const

Writes unique indices of the knots of the endpoints of i - th B-spline defined on this knot vector to result.

void symDifference	(	const gsKnotVector< T > &	other,
		std::vector< T > &	result
	)		const

Computes the symmetric difference between this knot-vector and other.

All knots which do not exist in both knot-vectors, this and other (counted with repetitions), are stored in result

unsigned u_multiplicityIndex ( size_t const &  i ) const

inline

Get the multiplicity of the unique knot indexed i

Parameters

i	index of the knot (without repetitions)

gsKnotVector< T >::uiterator uLowerBound

(

const T

u

)

const

Returns a uiterator pointing to the first knot which does not compare less than u.

Unlike uUpperBound, the value pointed by the iterator returned by this function may also be equivalent to u, and not only greater.

void uniformRefine	(	mult_t	numKnots = 1,
		mult_t	mult = 1
	)

Inserts numKnots (and each of them mult - times) between each two knots.

gsKnotVector< T >::uiterator uUpperBound

(

const T

u

)

const

Returns an iterator pointing to the first knot which compares greater than u.

Unlike uLowerBound, the value pointed by the iterator returned by this function cannot be equal to u, only greater.