modularization/src/decition/decition_brain/decision/tinyspline/tinyspline.h

/** @file */

#ifndef TINYSPLINE_H
#define	TINYSPLINE_H

#include <stddef.h>

#ifdef	__cplusplus
extern "C" {
#endif

/******************************************************************************
*                                                                             *
* :: System Dependent Configuration                                           *
*                                                                             *
* The following configuration values must be adapted to your system. Some of  *
* them may be configured with preprocessor definitions. The default values    *
* should be fine for most modern hardware, such as x86, x86_64, and arm.      *
*                                                                             *
******************************************************************************/
#ifdef TINYSPLINE_FLOAT_PRECISION
typedef float tsReal;
#else
typedef double tsReal;
#endif

#define FLT_MAX_ABS_ERROR 1e-5
#define FLT_MAX_REL_ERROR 1e-8



/******************************************************************************
*                                                                             *
* :: Data Types                                                               *
*                                                                             *
* The following section defines all data types available in TinySpline.       *
*                                                                             *
******************************************************************************/
/**
 * Defines the error codes used by various functions to indicate different
 * types of errors. The following code snippet shows how to handle errors:
 *
 *      tsError err = ...                  // any function call here
 *      if (err) {                         // or use err != TS_SUCCESS
 *          printf("we got an error!");
 *          return err;                    // you may want to reuse error codes
 *      }
 */
typedef enum
{
	/* No error. */
	TS_SUCCESS = 0,

	/* Unable to allocate memory (using malloc/realloc). */
	TS_MALLOC = -1,

	/* The dimension of the control points are 0. */
	TS_DIM_ZERO = -2,

	/* Degree of spline >= number of control points. */
	TS_DEG_GE_NCTRLP = -3,

	/* Spline is not defined at knot value u. */
	TS_U_UNDEFINED = -4,

	/* Multiplicity of a knot (s) > order of spline.  */
	TS_MULTIPLICITY = -5,

	/* Decreasing knot vector. */
	TS_KNOTS_DECR = -6,

	/* Unexpected number of knots. */
	TS_NUM_KNOTS = -7,

	/* Spline is not derivable */
	TS_UNDERIVABLE = -8,

	/* len_control_points % dim != 0 */
	TS_LCTRLP_DIM_MISMATCH = -10
} tsError;

/**
 * Describes the structure of the knot vector of a NURBS/B-Spline. For more
 * details, see:
 *
 *     www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-curve.html
 */
typedef enum
{
	/* Not available/Undefined. */
	TS_NONE = 0,

	/* Uniformly spaced knot vector. */
	TS_OPENED = 1,

	/* Uniformly spaced knot vector with clamped end knots. */
	TS_CLAMPED = 2,

	/* Uniformly spaced knot vector with s(u) = order of spline. */
	TS_BEZIERS = 3
} tsBSplineType;

/**
 * Represents a B-Spline which may also be used for NURBS, Bezier curves,
 * lines, and points. NURBS are represented by homogeneous coordinates where
 * the last component of a control point stores the weight of this control
 * point. Bezier curves are B-Splines with 'num_control_points == order' and a
 * clamped knot vector, therefore passing through their first and last control
 * point (a property which does not necessarily apply to B-Splines and NURBS).
 * If a Bezier curve consists of two control points only, we call it a line.
 * Points, ultimately, are just very short lines consisting of a single control
 * point.
 *
 * Two dimensional control points are stored as follows:
 *
 *     [x_0, y_0, x_1, y_1, ..., x_n-1, y_n-1]
 *
 * Tree dimensional control points are stored as follows:
 *
 *     [x_0, y_0, z_0, x_1, y_1, z_1, ..., x_n-1, y_n-1, z_n-1]
 *
 * ... and so on. NURBS are represented by homogeneous coordinates. For
 * example, let's say we have a NURBS in 2D consisting of 11 control points
 * where 'w_i' is the weight of the i'th control point. Then, the corresponding
 * control points are stored as follows:
 *
 *     [x_0*w_0, y_0*w_0, w_0, x_1*w_1, y_1*w_1, w_1, ..., x_10*w_10, y_10*w_10, w_10]
 */
typedef struct
{
	struct tsBSplineImpl *pImpl; /**< The actual implementation. */
} tsBSpline;

/**
 * Represents the output of De Boor's algorithm. De Boor's algorithm is used to
 * evaluate a spline at given knot value 'u' by iteratively computing a net of
 * intermediate values until the result is available:
 *
 *     https://en.wikipedia.org/wiki/De_Boor%27s_algorithm
 *     https://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/de-Boor.html
 *
 * All points of a net are stored in 'points'. The resulting point is the last
 * point in 'points' and, for the sake of convenience, may be accessed with
 * 'result':
 *
 *     tsDeBoorNet net = ...    // evaluate an arbitrary spline and store
 *                              // the resulting net of points in 'net'
 *
 *     ts_deboornet_result(...) // use 'result' to access the resulting point
 *
 * Two dimensional points are stored as follows:
 *
 *     [x_0, y_0, x_1, y_1, ..., x_n-1, y_n-1]
 *
 * Tree dimensional points are stored as follows:
 *
 *     [x_0, y_0, z_0, x_1, y_1, z_1, ..., x_n-1, y_n-1, z_n-1]
 *
 * ... and so on. The output also supports homogeneous coordinates described
 * above.
 *
 * There is a special case in which the evaluation of a knot value 'u' returns
 * two results instead of one. It occurs when the multiplicity of 'u' ( s(u) )
 * is equals to a spline's order, indicating that the spline is discontinuous
 * at 'u'. This is common practice for B-Splines (and NURBS) consisting of
 * connected Bezier curves where the endpoint of curve 'c_i' is equals to the
 * start point of curve 'c_i+1'. The end point of 'c_i' and the start point of
 * 'c_i+1' may still be completely different though, yielding to a spline
 * having a (real and visible) gap at 'u'. Consequently, De Boor's algorithm
 * must return two results if 's(u) == order' in order to give access to the
 * desired points.  In such case, 'points' stores only the two resulting points
 * (there is no net to calculate) and 'result' points to the *first* point in
 * 'points'. Since having (real) gaps in splines is unusual, both points in
 * 'points' are generally equals, making it easy to handle this special case by
 * accessing 'result' as already shown above for regular cases:
 *
 *     tsDeBoorNet net = ...    // evaluate a spline which is discontinuous at
 *                              // the given knot value, yielding to a net with
 *                              // two results
 *
 *     ts_deboornet_result(...) // use 'result' to access the resulting point
 *
 * However, you can access both points if necessary:
 *
 *     tsDeBoorNet net = ...    // evaluate a spline which is discontinuous at
 *                              // the given knot value, yielding to a net with
 *                              // two results
 *
 *     ts_deboornet_result(...)[0] ...       // stores the first component of
 *                                           // the first point
 *
 *     ts_deboornet_result(...)[dim(spline)] // stores the first component of
 *                                           // the second point
 *
 * As if this wasn't complicated enough, there is an exception for this special
 * case, yielding to exactly one result (just like the regular case) even if
 * 's(u) == order'. It occurs when 'u' is the lower or upper bound of a
 * spline's domain. For instance, if 'b' is a spline with domain [0, 1] and is
 * evaluated at 'u = 0' or 'u = 1' then 'result' is *always* a single point
 * regardless of the multiplicity of 'u'. Hence, handling this exception is
 * straightforward:
 *
 *     tsDeBoorNet net = ...    // evaluate a spline at the lower or upper
 *                              // bound of its domain, for instance, 0 or 1
 *
 *     ts_deboornet_result(...) // use 'result' to access the resulting point
 *
 * In summary, we have three different types of evaluation. 1) The regular case
 * returning all points of the net we used to calculate the resulting point. 2)
 * The special case returning exactly two points which is required for splines
 * with (real) gaps. 3) The exception of 2) returning exactly one point even if
 * 's(u) == order'. All in all this looks quite complex (and actually it is)
 * but for most applications you do not have to deal with it. Just use 'result'
 * to access the outcome of De Boor's algorithm.
 */
typedef struct
{
	struct tsDeBoorNetImpl *pImpl; /**< The actual implementation. */
} tsDeBoorNet;



/******************************************************************************
*                                                                             *
* :: Field Access Functions                                                   *
*                                                                             *
* The following section contains getter and setter functions for the internal *
* state of the structs listed above.                                          *
*                                                                             *
******************************************************************************/
/**
 * Returns the degree of \p spline.
 *
 * @param spline
 * 	The spline whose degree is read.
 * @return
 * 	The degree of \p spline.
 */
size_t ts_bspline_degree(const tsBSpline *spline);

/**
 * Sets the degree of \p spline.
 *
 * @param spline
 * 	The spline whose degree is set.
 * @param deg
 * 	The degree to be set.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_DEG_GE_NCTRLP
 * 	If \p degree >= ts_bspline_get_control_points(spline).
 */
tsError ts_bspline_set_degree(tsBSpline *spline, size_t deg);

/**
 * Returns the order (degree + 1) of \p spline.
 *
 * @param spline
 * 	The spline whose order is read.
 * @return
 * 	The order of \p spline.
 */
size_t ts_bspline_order(const tsBSpline *spline);

/**
 * Sets the order (degree + 1) of \p spline.
 *
 * @param spline
 * 	The spline whose order is set.
 * @param order
 * 	The order to be set.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_DEG_GE_NCTRLP
 * 	If \p order > ts_bspline_get_control_points(spline) or if \p order == 0
 * 	( due to the underflow resulting from: order - 1 => 0 - 1 => INT_MAX
 * 	which always is >= ts_bspline_get_control_points(spline) ).
 */
tsError ts_bspline_set_order(tsBSpline *spline, size_t order);

/**
 * Returns the dimension of \p spline. The dimension of a spline describes the
 * number of components for each point in ts_bspline_get_control_points(spline).
 * One-dimensional splines are possible, albeit their benefit might be
 * questionable.
 *
 * @param spline
 * 	The spline whose dimension is read.
 * @return
 * 	The dimension of \p spline.
 */
size_t ts_bspline_dimension(const tsBSpline *spline);

/**
 * Sets the dimension of \p spline. The following conditions must be satisfied:
 *
 * 	(1) dim >= 1
 * 	(2) len_control_points % dim == 0
 *
 * with _len_control_points_ being the length of the control point array of \p
 * spline. The dimension of a spline describes the number of components for
 * each point in ts_bspline_get_control_points(spline). One-dimensional splines
 * are possible, albeit their benefit might be questionable.
 *
 * @param spline
 * 	The spline whose dimension is set.
 * @param dim
 * 	The dimension to be set.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_DIM_ZERO
 * 	If \p dimension == 0.
 * @return TS_LCTRLP_DIM_MISMATCH
 * 	If len_control_points % \p dim != 0
 */
tsError ts_bspline_set_dimension(tsBSpline *spline, size_t dim);

/**
 * Returns the length of the control point array of \p spline.
 *
 * @param spline
 * 	The spline with its control point array whose length is read.
 * @return
 * 	The length of the control point array of \p spline.
 */
size_t ts_bspline_len_control_points(const tsBSpline *spline);

/**
 * Returns the number of control points of \p spline.
 *
 * @param spline
 * 	The spline whose number of control points is read.
 * @return
 * 	The number of control points of \p spline.
 */
size_t ts_bspline_num_control_points(const tsBSpline *spline);

/**
 * Returns the size of the control point array of \p spline. This function may
 * be useful when copying control points using memcpy or memmove.
 *
 * @param spline
 * 	The spline with its control point array whose size is read.
 * @return
 * 	The size of the control point array of \p spline.
 */
size_t ts_bspline_sof_control_points(const tsBSpline *spline);

/**
 * Returns a deep copy of the control points of \p spline.
 *
 * @param spline
 * 	The spline whose control points are read.
 * @param ctrlp
 * 	The output array.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_bspline_control_points(const tsBSpline *spline, tsReal **ctrlp);

/**
 * Sets the control points of \p spline. Creates a deep copy of \p ctrlp.
 *
 * @param spline
 * 	The spline whose control points are set.
 * @param ctrlp
 * 	The values to deep copy.
 * @return TS_SUCCESS
 * 	On success.
 */
tsError ts_bspline_set_control_points(tsBSpline *spline, const tsReal *ctrlp);

/**
 * Returns the number of knots of \p spline.
 *
 * @param spline
 * 	The spline whose number of knots is read.
 * @return
 * 	The number of knots of \p spline.
 */
size_t ts_bspline_num_knots(const tsBSpline *spline);

/**
 * Returns the size of the knot array of \p spline. This function may be useful
 * when copying knots using memcpy or memmove.
 *
 * @param spline
 * 	The spline with its knot array whose size is read.
 * @return TS_SUCCESS
 * 	The size of the knot array of \p spline.
 */
size_t ts_bspline_sof_knots(const tsBSpline *spline);

/**
 * Returns a deep copy of the knots of \p spline.
 *
 * @param spline
 * 	The spline whose knots are read.
 * @param knots
 * 	The output array.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_bspline_knots(const tsBSpline *spline, tsReal **knots);

/**
 * Sets the knots of \p spline. Creates a deep copy of \p knots.
 *
 * @param spline
 * 	The spline whose knots are set.
 * @param knots
 * 	The values to deep copy.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_KNOTS_DECR
 * 	If the knot vector is decreasing.
 * @return TS_MULTIPLICITY
 * 	If there is a knot with multiplicity > order
 */
tsError ts_bspline_set_knots(tsBSpline *spline, const tsReal *knots);

/* ------------------------------------------------------------------------- */

/**
 * Returns the knot (sometimes also called 'u' or 't') of \p net.
 *
 * @param net
 * 	The net whose knot is read.
 * @return
 * 	The knot of \p net.
 */
tsReal ts_deboornet_knot(const tsDeBoorNet *net);

/**
 * Returns the index [u_k, u_k+1) with u being the knot of \p net.
 *
 * @param net
 * 	The net whose index is read.
 * @return
 * 	The index [u_k, u_k+1) with u being the knot of \p net.
 */
size_t ts_deboornet_index(const tsDeBoorNet *net);

/**
 * Returns the multiplicity of the knot of \p net.
 *
 * @param net
 * 	The net whose multiplicity is read.
 * @return
 * 	The multiplicity of the knot of \p net.
 */
size_t ts_deboornet_multiplicity(const tsDeBoorNet *net);

/**
 * Returns the number of insertion that were necessary to evaluate the knot of
 * \p net.
 *
 * @param net
 * 	The net with its knot whose number of insertions is read.
 * @return
 * 	The number of insertions that were necessary to evaluate the knot of \p
 * 	net.
 */
size_t ts_deboornet_num_insertions(const tsDeBoorNet *net);

/**
 * Returns the dimension of \p net. The dimension of a net describes the number
 * of components for each point in ts_bspline_get_points(spline).
 * One-dimensional nets are possible, albeit their benefit might be
 * questionable.
 *
 * @param net
 * 	The net whose dimension is read.
 * @return
 * 	The dimension of \p net.
 */
size_t ts_deboornet_dimension(const tsDeBoorNet *net);

/**
 * Returns the length of the point array of \p net.
 *
 * @param net
 * 	The net with its point array whose length is read.
 * @return
 * 	The length of the point array of \p net.
 */
size_t ts_deboornet_len_points(const tsDeBoorNet *net);

/**
 * Returns the number of points of \p net.
 *
 * @param net
 * 	The net whose number of points is read.
 * @return
 * 	The number of points of \p net.
 */
size_t ts_deboornet_num_points(const tsDeBoorNet *net);

/**
 * Returns the size of the point array of \p net. This function may be useful
 * when copying points using memcpy or memmove.
 *
 * @param net
 * 	The net with its point array whose size is read.
 * @return
 * 	The size of the point array of \p net.
 */
size_t ts_deboornet_sof_points(const tsDeBoorNet *net);

/**
 * Returns a deep copy of the points of \p net.
 *
 * @param net
 * 	The net whose points is read.
 * @param points
 * 	The output array.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_deboornet_points(const tsDeBoorNet *net, tsReal **points);

/**
 * Returns the length of the result array of \p net.
 *
 * @param net
 * 	The net with its result array whose length is read.
 * @return
 * 	The length of the result array of \p net.
 */
size_t ts_deboornet_len_result(const tsDeBoorNet *net);

/**
 * Returns the number of points in the result array of \p net
 * (1 <= num_result <= 2).
 *
 * @param net
 * 	The net with its result array whose number of points is read.
 * @return
 * 	The number of points in the result array of \p net.
 */
size_t ts_deboornet_num_result(const tsDeBoorNet *net);

/**
 * Returns the size of the result array of \p net. This function may be useful
 * when copying results using memcpy or memmove.
 *
 * @param net
 * 	The net with its result array whose size is read.
 * @return TS_SUCCESS
 * 	The size of the result array of \p net.
 */
size_t ts_deboornet_sof_result(const tsDeBoorNet *net);

/**
 * Returns a deep copy of the result of \p net.
 *
 * @param net
 * 	The net whose result is read.
 * @param result
 * 	The output array.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_deboornet_result(const tsDeBoorNet *net, tsReal **result);



/******************************************************************************
*                                                                             *
* :: Constructors, Destructors, Copy, and Move Functions                      *
*                                                                             *
* The following section contains functions to create and delete instances of  *
* the data types listed above. Additionally, each data type has a copy and    *
* move function.                                                              *
*                                                                             *
******************************************************************************/
/**
 * The default constructor of tsBSpline.
 *
 * All values of \p \_spline\_ are set to 0/NULL.
 * 
 * @param \_spline\_
 * 	The spline whose values are set 0/NULL.
 */
void ts_bspline_default(tsBSpline *_spline_);

/**
 * A convenient constructor for tsBSpline.
 *
 * On error, all values of \p \_spline\_ are set to 0/NULL.
 *
 * @param n_ctrlp
 * 	The number of control points of \p \_spline\_.
 * @param dim
 * 	The dimension of each control point in \p \_spline\_.
 * @param deg
 * 	The degree of \p \_spline\_.
 * @param type
 * 	How to setup the knot vector of \p \_spline\_.
 * @param \_spline\_
 * 	The output parameter storing the result of this function.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_DIM_ZERO
 * 	If \p deg == 0.
 * @return TS_DEG_GE_NCTRLP
 * 	If \p deg >= \p n_ctrlp.
 * @return TS_NUM_KNOTS
 * 	If \p type == ::TS_BEZIERS and (\p n_ctrlp % \p deg + 1) != 0.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_bspline_new(size_t n_ctrlp, size_t dim, size_t deg,
	tsBSplineType type, tsBSpline *_spline_);

/**
 * The copy constructor of tsBSpline.
 *
 * Creates a deep copy of \p original and stores the result in \p \_copy\_.
 *
 * On error, all values of \p \_copy\_ are set to 0/NULL. Does nothing, if
 * \p original == \p \_copy\_.
 *
 * @param original
 * 	The spline to deep copy.
 * @param \_copy\_
 * 	The output parameter storing the copied values of \p original.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_bspline_copy(const tsBSpline *original, tsBSpline *_copy_);

/**
 * The move constructor of tsBSpline.
 *
 * Moves all values from \p from to \p \_to\_ and calls ::ts_bspline_default
 * on \p from afterwards. Does nothing, if \p from == \p \_to\_.
 * 
 * @param from
 * 	The spline whose values are moved to \p \_to\_.
 * @param \_to\_
 * 	The output parameter storing the moved values of \p from.
 */
void ts_bspline_move(tsBSpline *from, tsBSpline *_to_);

/**
 * The destructor of tsBSpline.
 *
 * Frees all dynamically allocated memory in \p \_spline\_ and calls
 * ::ts_bspline_default afterwards.
 * 
 * @param \_spline\_
 * 	The spline to free.
 */
void ts_bspline_free(tsBSpline *_spline_);

/* ------------------------------------------------------------------------- */

/**
 * The default constructor of tsDeBoorNet.
 *
 * All values of \p \_deBoorNet\_ are set to 0/NULL.
 * 
 * @param \_deBoorNet\_
 * 	The net whose values are set 0/NULL.
 */
void ts_deboornet_default(tsDeBoorNet *_deBoorNet_);

/**
 * The copy constructor of tsDeBoorNet.
 *
 * Creates a deep copy of \p original and stores the result in \p \_copy\_.
 *
 * On error, all values of \p _copy_ are set to 0/NULL. Does nothing, if
 * \p original == \p \_copy\_.
 *
 * @param original
 * 	The net to deep copy.
 * @param \_copy\_
 * 	The output parameter storing the copied values of \p original.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_deboornet_copy(const tsDeBoorNet *original, tsDeBoorNet *_copy_);

/**
 * The move constructor of tsDeBoorNet.
 *
 * Moves all values from \p from to \p \_to\_ and calls ::ts_deboornet_default
 * on \p from afterwards. Does nothing, if \p from == \p \_to\_.
 * 
 * @param from
 * 	The net whose values are moved to \p \_to\_.
 * @param \_to\_
 * 	The output parameter storing the moved values of \p from.
 */
void ts_deboornet_move(tsDeBoorNet *from, tsDeBoorNet *_to_);

/**
 * The destructor of tsDeBoorNet.
 *
 * Frees all dynamically allocated memory in \p \_deBoorNet\_ and calls
 * ::ts_deboornet_default afterwards.
 * 
 * @param \_deBoorNet\_
 * 	The net to free.
 */
void ts_deboornet_free(tsDeBoorNet *_deBoorNet_);



/******************************************************************************
*                                                                             *
* :: Interpolation and Approximation Functions                                *
*                                                                             *
* The following section contains functions to interpolate and approximate     *
* arbitrary splines.                                                          *
*                                                                             *
******************************************************************************/
/**
 * Performs a cubic spline interpolation using the thomas algorithm, see:
 * 
 *     https://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm
 *     http://www.math.ucla.edu/~baker/149.1.02w/handouts/dd_splines.pdf
 *     http://www.bakoma-tex.com/doc/generic/pst-bspline/pst-bspline-doc.pdf
 *
 * The resulting spline is a sequence of bezier curves connecting each point
 * in \p points. Each bezier curve _b_ is of degree 3 with \p dim being the
 * dimension of the each control point in _b_. The total number of control
 * points is (\p n - 1) * 4.
 *
 * On error, all values of \p \_spline\_ are set to 0/NULL.
 *
 * Note: \p n is the number of points in \p points and not the length of
 * \p points. For instance, the follwing point vector yields to \p n = 4 and
 * \p dim = 2:
 * 
 *     [x0, y0, x1, y1, x2, y2, x3, y3]
 *
 * @param points
 * 	The points to interpolate.
 * @param n
 * 	The number of points in \p points.
 * @param dim
 * 	The dimension of each control point in \p \_spline\_.
 * @param \_spline\_
 * 	The output parameter storing the result of this function.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_DIM_ZERO
 * 	If \p dim == 0.
 * @return TS_DEG_GE_NCTRLP
 * 	If \p n < 2.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_bspline_interpolate_cubic(const tsReal *points, size_t n,
	size_t dim, tsBSpline *_spline_);



/******************************************************************************
*                                                                             *
* :: Query Functions                                                          *
*                                                                             *
* The following section contains functions to query splines.                  *
*                                                                             *
******************************************************************************/
/**
 * Evaluates \p spline at knot value \p u and stores the result in
 * \p \_deBoorNet\_.
 *
 * On error, all values of \p \_deBoorNet\_ are set to 0/NULL.
 *
 * @param spline
 * 	The spline to evaluate.
 * @param u
 * 	The knot value to evaluate.
 * @param \_deBoorNet\_
 * 	The output parameter storing the evaluation result.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MULTIPLICITY
 * 	If multiplicity s(\p u) > order of \p spline.
 * @return TS_U_UNDEFINED
 * 	If \p spline is not defined at knot value \p u.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_bspline_eval(const tsBSpline *spline, tsReal u,
	tsDeBoorNet *_deBoorNet_);



/******************************************************************************
*                                                                             *
* :: Transformation functions                                                 *
*                                                                             *
* TinySpline is a library focusing on transformations. That is, most          *
* functions are used to transform splines by modifying their state, e.g.,     *
* their number of control points, their degree, and so on. Accordingly, each  *
* transformation functions specifies an input and output parameter (along     *
* with the other parameters required to calculate the actual transformation). *
* By passing a different pointer to the output parameter, the transformation  *
* result is calculated and stored without changing the state of the input.    *
* This is in particular useful when dealing with errors as the original state *
* will never be modified. For instance, let's have a look at the following    *
* code snippet:                                                               *
*                                                                             *
*     tsBSpline in = ...    // an arbitrary spline                            *
*     tsBSpline out;        // result of transformation                       *
*                                                                             *
*     // Subdivide 'in' into sequence of bezier curves and store the result   *
*     // in 'out'. Does not change 'in' in any way.                           *
*     tsError err = ts_bspline_to_beziers(&in, &out);                         *
*     if (err != TS_SUCCESS) {                                                *
*         // fortunately, 'in' has not been changed                           *
*     }                                                                       *
*                                                                             *
* Even if 'ts_bspline_to_beziers' fails, the state of 'in' has not been       *
* changed allowing you to handle the error properly.                          *
*                                                                             *
* Unless stated otherwise, the order of the parameters for transformation     *
* functions is:                                                               *
*                                                                             *
*     function(input, [additional_input], output, [additional_output])        *
*                                                                             *
* 'additional_input' are parameters required to calculate the actual          *
* transformation. 'additional_output' are parameters storing further result.  *
*                                                                             *
* Note: None of TinySpline's transformation functions frees the memory of the *
*       output parameter. Thus, when using the same output parameter multiple *
*       times, make sure to free memory before each call. Otherwise, you will *
*       have a bad time with memory leaks:                                    *
*                                                                             *
*     tsBSpline in = ...                  // an arbitrary spline              *
*     tsBSpline out;                      // result of transformations        *
*                                                                             *
*     ts_bspline_to_beziers(&in, &out);   // first transformation             *
*     ...                                 // some code                        *
*     ts_bspline_free(&out);              // avoid memory leak.               *
*     ts_bspline_buckle(&in, &out);       // next transformation              *
*                                                                             *
* If you want to modify your input directly without having a separate output, *
* pass it as input and output at once:                                        *
*                                                                             *
*     tsBSpline s = ...                       // an arbitrary spline          *
*     tsReal *knots = ...                     // a knot vector                *
*                                                                             *
*     ts_bspline_set_knots(&s, knots, &s);    // copy 'knots' into 's'        *
*                                                                             *
* Note: If a transformation function fails *and* input != output, all fields  *
*       of the output parameter are set to 0/NULL. If input == output, your   *
*       input may have an invalid state in case of errors.                    *
*                                                                             *
******************************************************************************/
/**
 * Computes the derivative of \p spline, see:
 * 
 *     http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-derv.html
 *
 * The derivative of a spline _s_ of degree _d_ with _m_ control points and
 * _n_ knots is another spline _s'_ of degree _d-1_ with _m-1_ control points
 * and _n-2_ knots, defined over _s_ as:
 * 
 * \f{eqnarray*}{
 *   s'(u) &=& \sum_{i=0}^{n-1} N_{i+1,p-1}(u) *
 * 		(P_{i+1} - P_{i}) * p / (u_{i+p+1}-u_{i+1}) \\
 *         &=& \sum_{i=1}^{n} N_{i,p-1}(u) *
 * 		(P_{i} - P_{i-1}) * p / (u_{i+p}-u_{i})
 * \f}
 * 
 * If _s_ has a clamped knot vector, it can be shown that:
 * 
 * \f{eqnarray*}{
 *   s'(u) &=& \sum_{i=0}^{n-1} N_{i,p-1}(u) *
 * 		(P_{i+1} - P_{i}) * p / (u_{i+p+1}-u_{i+1})
 * \f}
 * 
 * where the multiplicity of the first and the last knot value _u_ is _p_
 * rather than _p+1_.
 *
 * On error, (and if \p spline != \p \_derivative\_) all values of
 * \p \_derivative\_ are set to 0/NULL.
 *
 * @param spline
 * 	The spline to derive.
 * @param \_derivative\_
 *	The output parameter storing the derivative of \p spline.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_UNDERIVABLE
 * 	If \p spline->deg < 1, \p spline->n_ctrlp < 2, or the multiplicity of
 * 	an internal knot of \p spline is greater than the degree of \p spline.
 * 	NOTE: This will be fixed in the future.
 * @return TS_MALLOC
 * 	If allocating memory failed.
 */
tsError ts_bspline_derive(const tsBSpline *spline, tsBSpline *_derivative_);

/**
 * Fills the knot vector of \p spline according to \p type with minimum knot
 * value \p min to maximum knot value \p max and stores the result in
 * \p \_result\_. Creates a deep copy of \p spline, if
 * \p spline != \p \_result\_.
 *
 * On error, (and if \p spline != \p \_result\_) all values of \p \_result\_
 * are set to 0/NULL.
 *
 * @param spline
 * 	The spline to deep copy (if \p spline != \p \_result\_) and whose knot
 * 	vector is filled according to \p type with minimum knot value \p min
 * 	and maximum knot value \p max.
 * @param type
 * 	How to fill the knot vector of \p \_result\_.
 * @param min
 * 	The minimum knot value of the knot vector of \p \_result\_.
 * @param max
 * 	The maximum knot value of the knot vector of \p \_result\_.
 * @param \_result\_
 * 	The output parameter storing the result of this function.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_DEG_GE_NCTRLP
 * 	If \p spline->n_knots < 2*(\p original->deg+1). We can reuse this
 * 	error code because \p spline->n_knots < 2*(\p spline->deg+1) implies
 * 	\p spline->deg >= \p spline->n_ctrlp. Furthermore, using
 * 	TS_DEG_GE_NCTRLP instead of TS_NUM_KNOTS ensures that TS_NUM_KNOTS is
 * 	not used twice for this function. To be more fail-safe,
 * 	\p spline->deg+1 instead of \p spline->order is used, to make sure
 * 	that \p spline->deg+1 >= 1.
 * @return TS_NUM_KNOTS
 * 	If \p type == TS_BEZIERS and
 * 	\p spline->n_knots % \p spline->order != 0.
 * @return TS_KNOTS_DECR
 * 	If \p min >= \p max. (::ts_fequals is used to determine whether
 * 	\p min == \p max).
 * @return TS_MALLOC
 * 	If \p spline != \p \_result\_ and allocating memory failed.
 */
tsError ts_bspline_fill_knots(const tsBSpline *spline, tsBSplineType type,
	tsReal min, tsReal max, tsBSpline *_result_);

/**
 * Inserts the knot value \p u \p n times into \p spline and stores the result
 * in \p \_result\_. Creates a deep copy of \p spline, if
 * \p spline != \p \_result\_.
 * 
 * On error, (and if \p spline != \p \_result\_) all values of \p \_result\_
 * are set to 0/NULL.
 * 
 * @param spline
 * 	The spline to deep copy (if \p spline != \p \_result\_) and whose knot
 * 	vector is extended with \p u \p n times.
 * @param u
 * 	The knot value to insert.
 * @param n
 * 	How many times \p u should be inserted.
 * @param \_result\_
 * 	The output parameter storing the updated knot vector.
 * @param \_k\_
 * 	The output parameter storing the last index of \p u in \p \_result\_.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If \p spline != \p \_result\_ and allocating memory failed.
 */
tsError ts_bspline_insert_knot(const tsBSpline *spline, tsReal u, size_t n,
	tsBSpline *_result_, size_t *_k_);

/**
 * Resizes \p spline by \p n (number of control points) and stores the result
 * in \p \_resized\_. Creates a deep copy of \p spline, if
 * \p spline != \p \_result\_. If \p back != 0 \p spline is resized at the
 * end. If \p back == 0 \p spline is resized at front.
 *
 * On error, (and if \p spline != \p \_result\_) all values of \p \_result\_
 * are set to 0/NULL.
 *
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_DEG_GE_NCTRLP
 * 	If the degree of \p \_resized\_ would be >= the number of the control
 * 	points of \p \_resized\_.
 * @return TS_DIM_ZERO
 * 	If \p spline != \p \_result\_ and \p spline->dim == 0.
 * @return TS_DEG_GE_NCTRLP
 * 	If \p spline != \p \_result\_ and
 * 	\p spline->deg >= \p spline->n_ctrlp.
 * @return TS_MALLOC
 * 	If \p spline != \p \_result\_ and allocating memory failed.
 */
tsError ts_bspline_resize(const tsBSpline *spline, int n, int back,
	tsBSpline *_resized_);

/**
 * Splits \p spline at \p u and stores the result in \p \_split\_. That is,
 * \p u is inserted _n_ times such that s(\p u) == \p \_split\_->order.
 * Creates a deep copy of \p spline, if \p spline != \p \_split\_.
 * 
 * On error, (and if \p spline != \p \_split\_) all values of \p \_split\_
 * are set to 0/NULL.
 * 
 * @param spline
 * 	The spline to deep copy (if \p spline != \p \_result\_) and split.
 * @param u
 * 	The split point.
 * @param \_split\_
 * 	The output parameter storing the split spline.
 * @param \_k\_
 * 	The output parameter storing the last index of \p u in \p \_split\_.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If \p spline != \p \_split\_ and allocating memory failed.
 */
tsError ts_bspline_split(const tsBSpline *spline, tsReal u, tsBSpline *_split_,
	size_t *_k_);

/**
 * Buckles \p spline by \p b and stores the result in \p \_buckled\_. Creates
 * a deep copy of \p spline, if \p spline != \p \_buckled\_.
 *
 * This function is based on:
 * 
 *      Holten, Danny. "Hierarchical edge bundles: Visualization of adjacency
 *      relations in hierarchical data." Visualization and Computer Graphics,
 *      IEEE Transactions on 12.5 (2006): 741-748.
 * 
 * Holten calls it "straightening" (page 744, equation 1).
 *
 * Usually, the range of \p b is: 0.0 <= \p b <= 1.0 with 0 yielding to a line
 * connecting the first and the last control point (no buckle) and 1 keeping
 * the original shape (maximum buckle). If \b < 0 or \b > 1 the behaviour is
 * undefined, though, it will not result in an error.
 *
 * On error, (and if \p spline != \p \_buckled\_) all values of \p \_buckled\_
 * are set to 0/NULL.
 *
 * @param spline
 * 	The spline to buckle by \p b.
 * @param b
 * 	The buckle factor (usually 0.0 <= \p b <= 1.0).
 * @param \_buckled\_
 * 	The output parameter storing the buckled spline.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If \p spline != \p \_buckled\_ and allocating memory failed.
 */
tsError ts_bspline_buckle(const tsBSpline *spline, tsReal b,
	tsBSpline *_buckled_);

/**
 * Subdivides \p spline into a sequence of Bezier curvs by splitting it at
 * each internal knot value. Creates a deep copy of \p spline, if
 * \p spline != \p \_beziers\_.
 * 
 * On error, (and if \p spline != \p \_beziers\_) all values of \p \_beziers\_
 * are set to 0/NULL.
 * 
 * @param spline
 * 	The spline to subdivide into a sequence of Bezier curves.
 * @param \_beziers\_
 * 	The output parameter storing the sequence of Bezier curves.
 * @return TS_SUCCESS
 * 	On success.
 * @return TS_MALLOC
 * 	If \p spline != \p \_beizers\_ and allocating memory failed.
 */
tsError ts_bspline_to_beziers(const tsBSpline *spline, tsBSpline *_beziers_);



/******************************************************************************
*                                                                             *
* :: Utility Functions                                                        *
*                                                                             *
* The following section contains utility functions used by TinySpline which   *
* also may be helpful when using this library.                                *
*                                                                             *
******************************************************************************/
/**
 * Compares the tsReal values \p x and \p y using an absolute and relative
 * epsilon environment.
 *
 * @param x
 * 	The x value to compare.
 * @param y
 * 	The y value to compare.
 * @return 1
 * 	If \p x is equals to \p y.
 * @return 0
 * 	Otherwise.
 */
int ts_fequals(tsReal x, tsReal y);

/**
 * Returns the error message associated to \p err. Returns "unknown error" if
 * \p err is no associated (indicating a bug) or is TS_SUCCESS (which is not
 * an actual error).
 */
const char* ts_enum_str(tsError err);

/**
 * Returns the error code associated to \p str or TS_SUCCESS if \p str is not
 * associated. Keep in mind that by concept "unknown error" is not associated,
 * though, TS_SUCCESS is returned.
 */
tsError ts_str_enum(const char *str);

/**
 * Fills the given array \p arr with \p val from \p arr+0 to \p arr+ \p num
 * (exclusive).
 */
void ts_arr_fill(tsReal *arr, size_t num, tsReal val);

/**
 * Returns the euclidean distance of \p x and \p y consisting of \p dim
 * components, respectively.
 *
 * @param x
 * 	The x value.
 * @param y
 * 	The y value.
 * @param dim
 * 	The dimension of \p x and \p y.
 * @return
 * 	The euclidean distanc of \p x and \p y.
 */
tsReal ts_ctrlp_dist2(const tsReal *x, const tsReal *y, size_t dim);



#ifdef	__cplusplus
}
#endif

#endif	/* TINYSPLINE_H */