OpenVDB 11.0.0

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Math.h
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1// Copyright Contributors to the OpenVDB Project
2// SPDX-License-Identifier: MPL-2.0
3//
4/// @file Math.h
5/// @brief General-purpose arithmetic and comparison routines, most of which
6/// accept arbitrary value types (or at least arbitrary numeric value types)
7
8#ifndef OPENVDB_MATH_HAS_BEEN_INCLUDED
9#define OPENVDB_MATH_HAS_BEEN_INCLUDED
10
11#include <openvdb/Platform.h>
12#include <openvdb/version.h>
13#include <boost/numeric/conversion/conversion_traits.hpp>
14#include <algorithm> // for std::max()
15#include <cassert>
16#include <cmath> // for std::ceil(), std::fabs(), std::pow(), std::sqrt(), etc.
17#include <cstdlib> // for abs(int)
18#include <cstring> // for memcpy
19#include <random>
20#include <string>
21#include <type_traits> // for std::is_arithmetic
22
23
24// Compile pragmas
25
26// Intel(r) compiler fires remark #1572: floating-point equality and inequality
27// comparisons are unrealiable when == or != is used with floating point operands.
28#if defined(__INTEL_COMPILER)
29 #define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN \
30 _Pragma("warning (push)") \
31 _Pragma("warning (disable:1572)")
32 #define OPENVDB_NO_FP_EQUALITY_WARNING_END \
33 _Pragma("warning (pop)")
34#elif defined(__clang__)
35 #define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN \
36 PRAGMA(clang diagnostic push) \
37 PRAGMA(clang diagnostic ignored "-Wfloat-equal")
38 #define OPENVDB_NO_FP_EQUALITY_WARNING_END \
39 PRAGMA(clang diagnostic pop)
40#else
41 // For GCC, #pragma GCC diagnostic ignored "-Wfloat-equal"
42 // isn't working until gcc 4.2+,
43 // Trying
44 // #pragma GCC system_header
45 // creates other problems, most notably "warning: will never be executed"
46 // in from templates, unsure of how to work around.
47 // If necessary, could use integer based comparisons for equality
48 #define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
49 #define OPENVDB_NO_FP_EQUALITY_WARNING_END
50#endif
51
52
53#ifdef OPENVDB_IS_POD
54#undef OPENVDB_IS_POD
55#endif
56#define OPENVDB_IS_POD(Type) \
57 static_assert(std::is_standard_layout<Type>::value, \
58 #Type" must be a POD type (satisfy StandardLayoutType.)"); \
59 static_assert(std::is_trivial<Type>::value, \
60 #Type" must be a POD type (satisfy TrivialType.)");
61
62 namespace openvdb {
63 OPENVDB_USE_VERSION_NAMESPACE
64 namespace OPENVDB_VERSION_NAME {
65
66/// @brief Return the value of type T that corresponds to zero.
67/// @note A zeroVal<T>() specialization must be defined for each @c ValueType T
68/// that cannot be constructed using the form @c T(0). For example, @c std::string(0)
69/// treats 0 as @c nullptr and throws a @c std::logic_error.
70 template<typename T> inline constexpr T zeroVal() { return T(0); }
71/// Return the @c std::string value that corresponds to zero.
72 template<> inline std::string zeroVal<std::string>() { return ""; }
73/// Return the @c bool value that corresponds to zero.
74 template<> inline constexpr bool zeroVal<bool>() { return false; }
75
76 namespace math {
77
78/// @todo These won't be needed if we eliminate StringGrids.
79//@{
80/// @brief Needed to support the <tt>(zeroVal<ValueType>() + val)</tt> idiom
81/// when @c ValueType is @c std::string
82 inline std::string operator+(const std::string& s, bool) { return s; }
83 inline std::string operator+(const std::string& s, int) { return s; }
84 inline std::string operator+(const std::string& s, float) { return s; }
85 inline std::string operator+(const std::string& s, double) { return s; }
86//@}
87
88/// @brief Componentwise adder for POD types.
89 template<typename Type1, typename Type2>
90 inline auto cwiseAdd(const Type1& v, const Type2 s)
91{
92 OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN
93 return v + s;
94 OPENVDB_NO_TYPE_CONVERSION_WARNING_END
95}
96
97/// @brief Componentwise less than for POD types.
98 template<typename Type1, typename Type2>
99 inline bool cwiseLessThan(const Type1& a, const Type2& b)
100{
101 OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN
102 return a < b;
103 OPENVDB_NO_TYPE_CONVERSION_WARNING_END
104}
105
106/// @brief Componentwise greater than for POD types.
107 template<typename Type1, typename Type2>
108 inline bool cwiseGreaterThan(const Type1& a, const Type2& b)
109{
110 OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN
111 return a > b;
112 OPENVDB_NO_TYPE_CONVERSION_WARNING_END
113}
114
115
116
117/// @brief Pi constant taken from Boost to match old behaviour
118/// @note Available in C++20
119 template <typename T> inline constexpr T pi() { return 3.141592653589793238462643383279502884e+00; }
120 template <> inline constexpr float pi() { return 3.141592653589793238462643383279502884e+00F; }
121 template <> inline constexpr double pi() { return 3.141592653589793238462643383279502884e+00; }
122 template <> inline constexpr long double pi() { return 3.141592653589793238462643383279502884e+00L; }
123
124
125/// @brief Return the unary negation of the given value.
126/// @note A negative<T>() specialization must be defined for each ValueType T
127/// for which unary negation is not defined.
128 template<typename T> inline T negative(const T& val)
129{
130// disable unary minus on unsigned warning
131#if defined(_MSC_VER)
132#pragma warning(push)
133#pragma warning(disable:4146)
134#endif
135 return T(-val);
136#if defined(_MSC_VER)
137#pragma warning(pop)
138#endif
139}
140/// Return the negation of the given boolean.
141 template<> inline bool negative(const bool& val) { return !val; }
142/// Return the "negation" of the given string.
143 template<> inline std::string negative(const std::string& val) { return val; }
144
145
146//@{
147/// Tolerance for floating-point comparison
148 template<typename T> struct Tolerance { static T value() { return zeroVal<T>(); } };
149 template<> struct Tolerance<float> { static float value() { return 1e-8f; } };
150 template<> struct Tolerance<double> { static double value() { return 1e-15; } };
151//@}
152
153//@{
154/// Delta for small floating-point offsets
155 template<typename T> struct Delta { static T value() { return zeroVal<T>(); } };
156 template<> struct Delta<float> { static float value() { return 1e-5f; } };
157 template<> struct Delta<double> { static double value() { return 1e-9; } };
158//@}
159
160
161// ==========> Random Values <==================
162
163/// @brief Simple generator of random numbers over the range [0, 1)
164/// @details Thread-safe as long as each thread has its own Rand01 instance
165 template<typename FloatType = double, typename EngineType = std::mt19937>
166 class Rand01
167{
168 private:
169 EngineType mEngine;
170 std::uniform_real_distribution<FloatType> mRand;
171
172 public:
173 using ValueType = FloatType;
174
175 /// @brief Initialize the generator.
176 /// @param engine random number generator
177 Rand01(const EngineType& engine): mEngine(engine) {}
178
179 /// @brief Initialize the generator.
180 /// @param seed seed value for the random number generator
181 Rand01(unsigned int seed): mEngine(static_cast<typename EngineType::result_type>(seed)) {}
182
183 /// Set the seed value for the random number generator
184 void setSeed(unsigned int seed)
185 {
186 mEngine.seed(static_cast<typename EngineType::result_type>(seed));
187 }
188
189 /// Return a const reference to the random number generator.
190 const EngineType& engine() const { return mEngine; }
191
192 /// Return a uniformly distributed random number in the range [0, 1).
193 FloatType operator()() { return mRand(mEngine); }
194};
195
196 using Random01 = Rand01<double, std::mt19937>;
197
198
199/// @brief Simple random integer generator
200/// @details Thread-safe as long as each thread has its own RandInt instance
201 template<typename IntType = int, typename EngineType = std::mt19937>
202 class RandInt
203{
204 private:
205 using Distr = std::uniform_int_distribution<IntType>;
206 EngineType mEngine;
207 Distr mRand;
208
209 public:
210 /// @brief Initialize the generator.
211 /// @param engine random number generator
212 /// @param imin,imax generate integers that are uniformly distributed over [imin, imax]
213 RandInt(const EngineType& engine, IntType imin, IntType imax):
214 mEngine(engine),
215 mRand(std::min(imin, imax), std::max(imin, imax))
216 {}
217
218 /// @brief Initialize the generator.
219 /// @param seed seed value for the random number generator
220 /// @param imin,imax generate integers that are uniformly distributed over [imin, imax]
221 RandInt(unsigned int seed, IntType imin, IntType imax):
222 mEngine(static_cast<typename EngineType::result_type>(seed)),
223 mRand(std::min(imin, imax), std::max(imin, imax))
224 {}
225
226 /// Change the range over which integers are distributed to [imin, imax].
227 void setRange(IntType imin, IntType imax)
228 {
229 mRand = Distr(std::min(imin, imax), std::max(imin, imax));
230 }
231
232 /// Set the seed value for the random number generator
233 void setSeed(unsigned int seed)
234 {
235 mEngine.seed(static_cast<typename EngineType::result_type>(seed));
236 }
237
238 /// Return a const reference to the random number generator.
239 const EngineType& engine() const { return mEngine; }
240
241 /// Return a randomly-generated integer in the current range.
242 IntType operator()() { return mRand(mEngine); }
243
244 /// @brief Return a randomly-generated integer in the new range [imin, imax],
245 /// without changing the current range.
246 IntType operator()(IntType imin, IntType imax)
247 {
248 const IntType lo = std::min(imin, imax), hi = std::max(imin, imax);
249 return mRand(mEngine, typename Distr::param_type(lo, hi));
250 }
251};
252
253 using RandomInt = RandInt<int, std::mt19937>;
254
255
256// ==========> Clamp <==================
257
258/// Return @a x clamped to [@a min, @a max]
259 template<typename Type>
260 inline Type
261 Clamp(Type x, Type min, Type max)
262{
263 assert( !(min>max) );
264 return x > min ? x < max ? x : max : min;
265}
266
267
268/// Return @a x clamped to [0, 1]
269 template<typename Type>
270 inline Type
271 Clamp01(Type x) { return x > Type(0) ? x < Type(1) ? x : Type(1) : Type(0); }
272
273
274/// Return @c true if @a x is outside [0,1]
275 template<typename Type>
276 inline bool
277 ClampTest01(Type &x)
278{
279 if (x >= Type(0) && x <= Type(1)) return false;
280 x = x < Type(0) ? Type(0) : Type(1);
281 return true;
282}
283
284/// @brief Return 0 if @a x < @a 0, 1 if @a x > 1 or else (3 &minus; 2 @a x) @a x&sup2;.
285 template<typename Type>
286 inline Type
287 SmoothUnitStep(Type x)
288{
289 return x > 0 ? x < 1 ? (3-2*x)*x*x : Type(1) : Type(0);
290}
291
292/// @brief Return 0 if @a x < @a min, 1 if @a x > @a max or else (3 &minus; 2 @a t) @a t&sup2;,
293/// where @a t = (@a x &minus; @a min)/(@a max &minus; @a min).
294 template<typename Type>
295 inline Type
296 SmoothUnitStep(Type x, Type min, Type max)
297{
298 assert(min < max);
299 return SmoothUnitStep((x-min)/(max-min));
300}
301
302
303// ==========> Absolute Value <==================
304
305
306//@{
307/// Return the absolute value of the given quantity.
308 inline int32_t Abs(int32_t i) { return std::abs(i); }
309 inline int64_t Abs(int64_t i)
310{
311 static_assert(sizeof(decltype(std::abs(i))) == sizeof(int64_t),
312 "std::abs(int64) broken");
313 return std::abs(i);
314}
315 inline float Abs(float x) { return std::fabs(x); }
316 inline double Abs(double x) { return std::fabs(x); }
317 inline long double Abs(long double x) { return std::fabs(x); }
318 inline uint32_t Abs(uint32_t i) { return i; }
319 inline uint64_t Abs(uint64_t i) { return i; }
320 inline bool Abs(bool b) { return b; }
321// On systems like macOS and FreeBSD, size_t and uint64_t are different types
322 template <typename T>
323 inline typename std::enable_if<std::is_same<T, size_t>::value, T>::type
324 Abs(T i) { return i; }
325//@}
326
327
328////////////////////////////////////////
329
330
331// ==========> Value Comparison <==================
332
333
334/// Return @c true if @a x is exactly equal to zero.
335 template<typename Type>
336 inline bool
337 isZero(const Type& x)
338{
339 OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
340 return x == zeroVal<Type>();
341 OPENVDB_NO_FP_EQUALITY_WARNING_END
342}
343
344
345/// @brief Return @c true if @a x is equal to zero to within
346/// the default floating-point comparison tolerance.
347 template<typename Type>
348 inline bool
349 isApproxZero(const Type& x)
350{
351 const Type tolerance = Type(zeroVal<Type>() + Tolerance<Type>::value());
352 return !(x > tolerance) && !(x < -tolerance);
353}
354
355/// Return @c true if @a x is equal to zero to within the given tolerance.
356 template<typename Type>
357 inline bool
358 isApproxZero(const Type& x, const Type& tolerance)
359{
360 return !(x > tolerance) && !(x < -tolerance);
361}
362
363
364/// Return @c true if @a x is less than zero.
365 template<typename Type>
366 inline bool
367 isNegative(const Type& x) { return x < zeroVal<Type>(); }
368
369// Return false, since bool values are never less than zero.
370 template<> inline bool isNegative<bool>(const bool&) { return false; }
371
372
373/// Return @c true if @a x is finite.
374 inline bool
375 isFinite(const float x) { return std::isfinite(x); }
376
377/// Return @c true if @a x is finite.
378template<typename Type, typename std::enable_if<std::is_arithmetic<Type>::value, int>::type = 0>
379 inline bool
380 isFinite(const Type& x) { return std::isfinite(static_cast<double>(x)); }
381
382
383/// Return @c true if @a x is an infinity value (either positive infinity or negative infinity).
384 inline bool
385 isInfinite(const float x) { return std::isinf(x); }
386
387/// Return @c true if @a x is an infinity value (either positive infinity or negative infinity).
388template<typename Type, typename std::enable_if<std::is_arithmetic<Type>::value, int>::type = 0>
389 inline bool
390 isInfinite(const Type& x) { return std::isinf(static_cast<double>(x)); }
391
392
393/// Return @c true if @a x is a NaN (Not-A-Number) value.
394 inline bool
395 isNan(const float x) { return std::isnan(x); }
396
397/// Return @c true if @a x is a NaN (Not-A-Number) value.
398template<typename Type, typename std::enable_if<std::is_arithmetic<Type>::value, int>::type = 0>
399 inline bool
400 isNan(const Type& x) { return std::isnan(static_cast<double>(x)); }
401
402
403/// Return @c true if @a a is equal to @a b to within the given tolerance.
404 template<typename Type>
405 inline bool
406 isApproxEqual(const Type& a, const Type& b, const Type& tolerance)
407{
408 return !cwiseGreaterThan(Abs(a - b), tolerance);
409}
410
411/// @brief Return @c true if @a a is equal to @a b to within
412/// the default floating-point comparison tolerance.
413 template<typename Type>
414 inline bool
415 isApproxEqual(const Type& a, const Type& b)
416{
417 const Type tolerance = Type(zeroVal<Type>() + Tolerance<Type>::value());
418 return isApproxEqual(a, b, tolerance);
419}
420
421#define OPENVDB_EXACT_IS_APPROX_EQUAL(T) \
422 template<> inline bool isApproxEqual<T>(const T& a, const T& b) { return a == b; } \
423 template<> inline bool isApproxEqual<T>(const T& a, const T& b, const T&) { return a == b; } \
424 /**/
425
426 OPENVDB_EXACT_IS_APPROX_EQUAL(bool)
427 OPENVDB_EXACT_IS_APPROX_EQUAL(std::string)
428
429
430/// @brief Return @c true if @a a is larger than @a b to within
431/// the given tolerance, i.e., if @a b - @a a < @a tolerance.
432template<typename Type>
433inline bool
434 isApproxLarger(const Type& a, const Type& b, const Type& tolerance)
435{
436 return (b - a < tolerance);
437}
438
439
440/// @brief Return @c true if @a a is exactly equal to @a b.
441 template<typename T0, typename T1>
442 inline bool
443 isExactlyEqual(const T0& a, const T1& b)
444{
445 OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
446 return a == b;
447 OPENVDB_NO_FP_EQUALITY_WARNING_END
448}
449
450
451 template<typename Type>
452 inline bool
453 isRelOrApproxEqual(const Type& a, const Type& b, const Type& absTol, const Type& relTol)
454{
455 // First check to see if we are inside the absolute tolerance
456 // Necessary for numbers close to 0
457 if (!(Abs(a - b) > absTol)) return true;
458
459 // Next check to see if we are inside the relative tolerance
460 // to handle large numbers that aren't within the abs tolerance
461 // but could be the closest floating point representation
462 double relError;
463 if (Abs(b) > Abs(a)) {
464 relError = Abs((a - b) / b);
465 } else {
466 relError = Abs((a - b) / a);
467 }
468 return (relError <= relTol);
469}
470
471 template<>
472 inline bool
473 isRelOrApproxEqual(const bool& a, const bool& b, const bool&, const bool&)
474{
475 return (a == b);
476}
477
478 inline int32_t
479 floatToInt32(const float f)
480{
481 // switch to std:bit_cast in C++20
482 static_assert(sizeof(int32_t) == sizeof f, "`float` has an unexpected size.");
483 int32_t ret;
484 std::memcpy(&ret, &f, sizeof(int32_t));
485 return ret;
486}
487
488 inline int64_t
489 doubleToInt64(const double d)
490{
491 // switch to std:bit_cast in C++20
492 static_assert(sizeof(int64_t) == sizeof d, "`double` has an unexpected size.");
493 int64_t ret;
494 std::memcpy(&ret, &d, sizeof(int64_t));
495 return ret;
496}
497
498// aUnitsInLastPlace is the allowed difference between the least significant digits
499// of the numbers' floating point representation
500// Please read the reference paper before trying to use isUlpsEqual
501// http://www.cygnus-software.com/papers/comparingfloats/comparingfloats.htm
502 inline bool
503 isUlpsEqual(const double aLeft, const double aRight, const int64_t aUnitsInLastPlace)
504{
505 int64_t longLeft = doubleToInt64(aLeft);
506 // Because of 2's complement, must restore lexicographical order
507 if (longLeft < 0) {
508 longLeft = INT64_C(0x8000000000000000) - longLeft;
509 }
510
511 int64_t longRight = doubleToInt64(aRight);
512 // Because of 2's complement, must restore lexicographical order
513 if (longRight < 0) {
514 longRight = INT64_C(0x8000000000000000) - longRight;
515 }
516
517 int64_t difference = Abs(longLeft - longRight);
518 return (difference <= aUnitsInLastPlace);
519}
520
521 inline bool
522 isUlpsEqual(const float aLeft, const float aRight, const int32_t aUnitsInLastPlace)
523{
524 int32_t intLeft = floatToInt32(aLeft);
525 // Because of 2's complement, must restore lexicographical order
526 if (intLeft < 0) {
527 intLeft = 0x80000000 - intLeft;
528 }
529
530 int32_t intRight = floatToInt32(aRight);
531 // Because of 2's complement, must restore lexicographical order
532 if (intRight < 0) {
533 intRight = 0x80000000 - intRight;
534 }
535
536 int32_t difference = Abs(intLeft - intRight);
537 return (difference <= aUnitsInLastPlace);
538}
539
540
541////////////////////////////////////////
542
543
544// ==========> Pow <==================
545
546/// Return @a x<sup>2</sup>.
547 template<typename Type>
548 inline Type Pow2(Type x) { return x*x; }
549
550/// Return @a x<sup>3</sup>.
551 template<typename Type>
552 inline Type Pow3(Type x) { return x*x*x; }
553
554/// Return @a x<sup>4</sup>.
555 template<typename Type>
556 inline Type Pow4(Type x) { return Pow2(Pow2(x)); }
557
558/// Return @a x<sup>n</sup>.
559 template<typename Type>
560Type
561 Pow(Type x, int n)
562{
563 Type ans = 1;
564 if (n < 0) {
565 n = -n;
566 x = Type(1)/x;
567 }
568 while (n--) ans *= x;
569 return ans;
570}
571
572//@{
573/// Return @a b<sup>e</sup>.
574 inline float
575 Pow(float b, float e)
576{
577 assert( b >= 0.0f && "Pow(float,float): base is negative" );
578 return powf(b,e);
579}
580
581 inline double
582 Pow(double b, double e)
583{
584 assert( b >= 0.0 && "Pow(double,double): base is negative" );
585 return std::pow(b,e);
586}
587//@}
588
589
590// ==========> Max <==================
591
592/// Return the maximum of two values
593 template<typename Type>
594 inline const Type&
595 Max(const Type& a, const Type& b)
596{
597 return std::max(a,b);
598}
599
600/// Return the maximum of three values
601 template<typename Type>
602 inline const Type&
603 Max(const Type& a, const Type& b, const Type& c)
604{
605 return std::max(std::max(a,b), c);
606}
607
608/// Return the maximum of four values
609 template<typename Type>
610 inline const Type&
611 Max(const Type& a, const Type& b, const Type& c, const Type& d)
612{
613 return std::max(std::max(a,b), std::max(c,d));
614}
615
616/// Return the maximum of five values
617 template<typename Type>
618 inline const Type&
619 Max(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e)
620{
621 return std::max(std::max(a,b), Max(c,d,e));
622}
623
624/// Return the maximum of six values
625 template<typename Type>
626 inline const Type&
627 Max(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e, const Type& f)
628{
629 return std::max(Max(a,b,c), Max(d,e,f));
630}
631
632/// Return the maximum of seven values
633 template<typename Type>
634 inline const Type&
635 Max(const Type& a, const Type& b, const Type& c, const Type& d,
636 const Type& e, const Type& f, const Type& g)
637{
638 return std::max(Max(a,b,c,d), Max(e,f,g));
639}
640
641/// Return the maximum of eight values
642 template<typename Type>
643 inline const Type&
644 Max(const Type& a, const Type& b, const Type& c, const Type& d,
645 const Type& e, const Type& f, const Type& g, const Type& h)
646{
647 return std::max(Max(a,b,c,d), Max(e,f,g,h));
648}
649
650
651// ==========> Min <==================
652
653/// Return the minimum of two values
654 template<typename Type>
655 inline const Type&
656 Min(const Type& a, const Type& b) { return std::min(a, b); }
657
658/// Return the minimum of three values
659 template<typename Type>
660 inline const Type&
661 Min(const Type& a, const Type& b, const Type& c) { return std::min(std::min(a, b), c); }
662
663/// Return the minimum of four values
664 template<typename Type>
665 inline const Type&
666 Min(const Type& a, const Type& b, const Type& c, const Type& d)
667{
668 return std::min(std::min(a, b), std::min(c, d));
669}
670
671/// Return the minimum of five values
672 template<typename Type>
673 inline const Type&
674 Min(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e)
675{
676 return std::min(std::min(a,b), Min(c,d,e));
677}
678
679/// Return the minimum of six values
680 template<typename Type>
681 inline const Type&
682 Min(const Type& a, const Type& b, const Type& c, const Type& d, const Type& e, const Type& f)
683{
684 return std::min(Min(a,b,c), Min(d,e,f));
685}
686
687/// Return the minimum of seven values
688 template<typename Type>
689 inline const Type&
690 Min(const Type& a, const Type& b, const Type& c, const Type& d,
691 const Type& e, const Type& f, const Type& g)
692{
693 return std::min(Min(a,b,c,d), Min(e,f,g));
694}
695
696/// Return the minimum of eight values
697 template<typename Type>
698 inline const Type&
699 Min(const Type& a, const Type& b, const Type& c, const Type& d,
700 const Type& e, const Type& f, const Type& g, const Type& h)
701{
702 return std::min(Min(a,b,c,d), Min(e,f,g,h));
703}
704
705
706// ============> Exp <==================
707
708/// Return @a e<sup>x</sup>.
709 template<typename Type>
710 inline Type Exp(const Type& x) { return std::exp(x); }
711
712// ============> Sin <==================
713
714//@{
715/// Return sin @a x.
716 inline float Sin(const float& x) { return std::sin(x); }
717
718 inline double Sin(const double& x) { return std::sin(x); }
719//@}
720
721// ============> Cos <==================
722
723//@{
724/// Return cos @a x.
725 inline float Cos(const float& x) { return std::cos(x); }
726
727 inline double Cos(const double& x) { return std::cos(x); }
728//@}
729
730
731////////////////////////////////////////
732
733
734/// Return the sign of the given value as an integer (either -1, 0 or 1).
735 template <typename Type>
736 inline int Sign(const Type &x) { return (zeroVal<Type>() < x) - (x < zeroVal<Type>()); }
737
738
739/// @brief Return @c true if @a a and @a b have different signs.
740/// @note Zero is considered a positive number.
741 template <typename Type>
742 inline bool
743 SignChange(const Type& a, const Type& b)
744{
745 return ( (a<zeroVal<Type>()) ^ (b<zeroVal<Type>()) );
746}
747
748
749/// @brief Return @c true if the interval [@a a, @a b] includes zero,
750/// i.e., if either @a a or @a b is zero or if they have different signs.
751 template <typename Type>
752 inline bool
753 ZeroCrossing(const Type& a, const Type& b)
754{
755 return a * b <= zeroVal<Type>();
756}
757
758
759//@{
760/// Return the square root of a floating-point value.
761 inline float Sqrt(float x) { return std::sqrt(x); }
762 inline double Sqrt(double x) { return std::sqrt(x); }
763 inline long double Sqrt(long double x) { return std::sqrt(x); }
764//@}
765
766
767//@{
768/// Return the cube root of a floating-point value.
769 inline float Cbrt(float x) { return std::cbrt(x); }
770 inline double Cbrt(double x) { return std::cbrt(x); }
771 inline long double Cbrt(long double x) { return std::cbrt(x); }
772//@}
773
774
775//@{
776/// Return the remainder of @a x / @a y.
777 inline int Mod(int x, int y) { return (x % y); }
778 inline float Mod(float x, float y) { return std::fmod(x, y); }
779 inline double Mod(double x, double y) { return std::fmod(x, y); }
780 inline long double Mod(long double x, long double y) { return std::fmod(x, y); }
781 template<typename Type> inline Type Remainder(Type x, Type y) { return Mod(x, y); }
782//@}
783
784
785//@{
786/// Return @a x rounded up to the nearest integer.
787 inline float RoundUp(float x) { return std::ceil(x); }
788 inline double RoundUp(double x) { return std::ceil(x); }
789 inline long double RoundUp(long double x) { return std::ceil(x); }
790//@}
791/// Return @a x rounded up to the nearest multiple of @a base.
792 template<typename Type>
793 inline Type
794 RoundUp(Type x, Type base)
795{
796 Type remainder = Remainder(x, base);
797 return remainder ? x-remainder+base : x;
798}
799
800
801//@{
802/// Return @a x rounded down to the nearest integer.
803 inline float RoundDown(float x) { return std::floor(x); }
804 inline double RoundDown(double x) { return std::floor(x); }
805 inline long double RoundDown(long double x) { return std::floor(x); }
806//@}
807/// Return @a x rounded down to the nearest multiple of @a base.
808 template<typename Type>
809 inline Type
810 RoundDown(Type x, Type base)
811{
812 Type remainder = Remainder(x, base);
813 return remainder ? x-remainder : x;
814}
815
816
817//@{
818/// Return @a x rounded to the nearest integer.
819 inline float Round(float x) { return RoundDown(x + 0.5f); }
820 inline double Round(double x) { return RoundDown(x + 0.5); }
821 inline long double Round(long double x) { return RoundDown(x + 0.5l); }
822//@}
823
824
825/// Return the euclidean remainder of @a x.
826/// Note unlike % operator this will always return a positive result
827 template<typename Type>
828 inline Type
829 EuclideanRemainder(Type x) { return x - RoundDown(x); }
830
831
832/// Return the integer part of @a x.
833 template<typename Type>
834 inline Type
835 IntegerPart(Type x)
836{
837 return (x > 0 ? RoundDown(x) : RoundUp(x));
838}
839
840/// Return the fractional part of @a x.
841 template<typename Type>
842 inline Type
843 FractionalPart(Type x) { return Mod(x,Type(1)); }
844
845
846//@{
847/// Return the floor of @a x.
848 inline int Floor(float x) { return int(RoundDown(x)); }
849 inline int Floor(double x) { return int(RoundDown(x)); }
850 inline int Floor(long double x) { return int(RoundDown(x)); }
851//@}
852
853
854//@{
855/// Return the ceiling of @a x.
856 inline int Ceil(float x) { return int(RoundUp(x)); }
857 inline int Ceil(double x) { return int(RoundUp(x)); }
858 inline int Ceil(long double x) { return int(RoundUp(x)); }
859//@}
860
861
862/// Return @a x if it is greater or equal in magnitude than @a delta. Otherwise, return zero.
863 template<typename Type>
864 inline Type Chop(Type x, Type delta) { return (Abs(x) < delta ? zeroVal<Type>() : x); }
865
866
867/// Return @a x truncated to the given number of decimal digits.
868 template<typename Type>
869 inline Type
870 Truncate(Type x, unsigned int digits)
871{
872 Type tenth = static_cast<Type>(Pow(size_t(10), digits));
873 return RoundDown(x*tenth+0.5)/tenth;
874}
875
876////////////////////////////////////////
877
878
879/// @brief 8-bit integer values print to std::ostreams as characters.
880/// Cast them so that they print as integers instead.
881 template<typename T>
882 inline auto PrintCast(const T& val) -> typename std::enable_if<!std::is_same<T, int8_t>::value
883 && !std::is_same<T, uint8_t>::value, const T&>::type { return val; }
884 inline int32_t PrintCast(int8_t val) { return int32_t(val); }
885 inline uint32_t PrintCast(uint8_t val) { return uint32_t(val); }
886
887
888////////////////////////////////////////
889
890
891/// Return the inverse of @a x.
892 template<typename Type>
893 inline Type
894 Inv(Type x)
895{
896 assert(x);
897 return Type(1)/x;
898}
899
900
901 enum Axis {
902 X_AXIS = 0,
903 Y_AXIS = 1,
904 Z_AXIS = 2
905};
906
907// enum values are consistent with their historical mx analogs.
908 enum RotationOrder {
909 XYZ_ROTATION = 0,
910 XZY_ROTATION,
911 YXZ_ROTATION,
912 YZX_ROTATION,
913 ZXY_ROTATION,
914 ZYX_ROTATION,
915 XZX_ROTATION,
916 ZXZ_ROTATION
917};
918
919
920 template <typename S, typename T>
921 struct promote {
922 using type = typename boost::numeric::conversion_traits<S, T>::supertype;
923};
924
925/// @brief Return the index [0,1,2] of the smallest value in a 3D vector.
926/// @note This methods assumes operator[] exists.
927/// @details The return value corresponds to the largest index of the of
928/// the smallest vector components.
929 template<typename Vec3T>
930 size_t
931 MinIndex(const Vec3T& v)
932{
933 size_t r = 0;
934 for (size_t i = 1; i < 3; ++i) {
935 // largest index (backwards compatibility)
936 if (v[i] <= v[r]) r = i;
937 }
938 return r;
939}
940
941/// @brief Return the index [0,1,2] of the largest value in a 3D vector.
942/// @note This methods assumes operator[] exists.
943/// @details The return value corresponds to the largest index of the of
944/// the largest vector components.
945 template<typename Vec3T>
946 size_t
947 MaxIndex(const Vec3T& v)
948{
949 size_t r = 0;
950 for (size_t i = 1; i < 3; ++i) {
951 // largest index (backwards compatibility)
952 if (v[i] >= v[r]) r = i;
953 }
954 return r;
955}
956
957} // namespace math
958} // namespace OPENVDB_VERSION_NAME
959} // namespace openvdb
960
961#endif // OPENVDB_MATH_MATH_HAS_BEEN_INCLUDED
OPENVDB_EXACT_IS_APPROX_EQUAL
#define OPENVDB_EXACT_IS_APPROX_EQUAL(T)
Definition Math.h:421
OPENVDB_NO_FP_EQUALITY_WARNING_END
#define OPENVDB_NO_FP_EQUALITY_WARNING_END
Definition Math.h:49
OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
#define OPENVDB_NO_FP_EQUALITY_WARNING_BEGIN
Definition Math.h:48
Platform.h
OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN
#define OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN
Bracket code with OPENVDB_NO_TYPE_CONVERSION_WARNING_BEGIN/_END, to inhibit warnings about type conve...
Definition Platform.h:227
OPENVDB_NO_TYPE_CONVERSION_WARNING_END
#define OPENVDB_NO_TYPE_CONVERSION_WARNING_END
Definition Platform.h:228
openvdb::v11_0::math::Rand01
Simple generator of random numbers over the range [0, 1)
Definition Math.h:167
openvdb::v11_0::math::Rand01::setSeed
void setSeed(unsigned int seed)
Set the seed value for the random number generator.
Definition Math.h:184
openvdb::v11_0::math::Rand01::Rand01
Rand01(const EngineType &engine)
Initialize the generator.
Definition Math.h:177
openvdb::v11_0::math::Rand01< double, std::mt19937 >::engine
const std::mt19937 & engine() const
Definition Math.h:190
openvdb::v11_0::math::Rand01::operator()
FloatType operator()()
Return a uniformly distributed random number in the range [0, 1).
Definition Math.h:193
openvdb::v11_0::math::Rand01::Rand01
Rand01(unsigned int seed)
Initialize the generator.
Definition Math.h:181
openvdb::v11_0::math::Rand01::ValueType
FloatType ValueType
Definition Math.h:173
openvdb::v11_0::math::RandInt
Simple random integer generator.
Definition Math.h:203
openvdb::v11_0::math::RandInt::setSeed
void setSeed(unsigned int seed)
Set the seed value for the random number generator.
Definition Math.h:233
openvdb::v11_0::math::RandInt::RandInt
RandInt(unsigned int seed, IntType imin, IntType imax)
Initialize the generator.
Definition Math.h:221
openvdb::v11_0::math::RandInt::setRange
void setRange(IntType imin, IntType imax)
Change the range over which integers are distributed to [imin, imax].
Definition Math.h:227
openvdb::v11_0::math::RandInt::operator()
IntType operator()()
Return a randomly-generated integer in the current range.
Definition Math.h:242
openvdb::v11_0::math::RandInt< int, std::mt19937 >::engine
const std::mt19937 & engine() const
Definition Math.h:239
openvdb::v11_0::math::RandInt::RandInt
RandInt(const EngineType &engine, IntType imin, IntType imax)
Initialize the generator.
Definition Math.h:213
openvdb::v11_0::math::RandInt::operator()
IntType operator()(IntType imin, IntType imax)
Return a randomly-generated integer in the new range [imin, imax], without changing the current range...
Definition Math.h:246
openvdb::v11_0::math
Definition Types.h:28
openvdb::v11_0::math::Sign
int Sign(const Type &x)
Return the sign of the given value as an integer (either -1, 0 or 1).
Definition Math.h:736
openvdb::v11_0::math::isApproxLarger
bool isApproxLarger(const Type &a, const Type &b, const Type &tolerance)
Return true if a is larger than b to within the given tolerance, i.e., if b - a < tolerance.
Definition Math.h:434
openvdb::v11_0::math::MaxIndex
size_t MaxIndex(const Vec3T &v)
Return the index [0,1,2] of the largest value in a 3D vector.
Definition Math.h:947
openvdb::v11_0::math::RoundUp
float RoundUp(float x)
Return x rounded up to the nearest integer.
Definition Math.h:787
openvdb::v11_0::math::SmoothUnitStep
Type SmoothUnitStep(Type x)
Return 0 if x < 0, 1 if x > 1 or else (3 − 2 x) x².
Definition Math.h:287
openvdb::v11_0::math::Ceil
int Ceil(float x)
Return the ceiling of x.
Definition Math.h:856
openvdb::v11_0::math::isApproxZero
bool isApproxZero(const Type &x)
Return true if x is equal to zero to within the default floating-point comparison tolerance.
Definition Math.h:349
openvdb::v11_0::math::isInfinite
bool isInfinite(const float x)
Return true if x is an infinity value (either positive infinity or negative infinity).
Definition Math.h:385
openvdb::v11_0::math::IntegerPart
Type IntegerPart(Type x)
Return the integer part of x.
Definition Math.h:835
openvdb::v11_0::math::cwiseLessThan
bool cwiseLessThan(const Mat< SIZE, T > &m0, const Mat< SIZE, T > &m1)
Definition Mat.h:1015
openvdb::v11_0::math::pi
constexpr T pi()
Pi constant taken from Boost to match old behaviour.
Definition Math.h:119
openvdb::v11_0::math::Pow
Type Pow(Type x, int n)
Return xn.
Definition Math.h:561
openvdb::v11_0::math::Sqrt
float Sqrt(float x)
Return the square root of a floating-point value.
Definition Math.h:761
openvdb::v11_0::math::isApproxEqual
bool isApproxEqual(const Type &a, const Type &b, const Type &tolerance)
Return true if a is equal to b to within the given tolerance.
Definition Math.h:406
openvdb::v11_0::math::isRelOrApproxEqual
bool isRelOrApproxEqual(const Type &a, const Type &b, const Type &absTol, const Type &relTol)
Definition Math.h:453
openvdb::v11_0::math::Max
const Type & Max(const Type &a, const Type &b)
Return the maximum of two values.
Definition Math.h:595
openvdb::v11_0::math::MinIndex
size_t MinIndex(const Vec3T &v)
Return the index [0,1,2] of the smallest value in a 3D vector.
Definition Math.h:931
openvdb::v11_0::math::cwiseAdd
Mat3< Type1 > cwiseAdd(const Mat3< Type1 > &m, const Type2 s)
Definition Mat3.h:806
openvdb::v11_0::math::isFinite
bool isFinite(const float x)
Return true if x is finite.
Definition Math.h:375
openvdb::v11_0::math::PrintCast
auto PrintCast(const T &val) -> typename std::enable_if<!std::is_same< T, int8_t >::value &&!std::is_same< T, uint8_t >::value, const T & >::type
8-bit integer values print to std::ostreams as characters. Cast them so that they print as integers i...
Definition Math.h:882
openvdb::v11_0::math::operator+
Vec3< typename promote< T, typename Coord::ValueType >::type > operator+(const Vec3< T > &v0, const Coord &v1)
Allow a Coord to be added to or subtracted from a Vec3.
Definition Coord.h:527
openvdb::v11_0::math::Pow4
Type Pow4(Type x)
Return x4.
Definition Math.h:556
openvdb::v11_0::math::EuclideanRemainder
Type EuclideanRemainder(Type x)
Definition Math.h:829
openvdb::v11_0::math::Remainder
Type Remainder(Type x, Type y)
Definition Math.h:781
openvdb::v11_0::math::Inv
Type Inv(Type x)
Return the inverse of x.
Definition Math.h:894
openvdb::v11_0::math::Exp
Type Exp(const Type &x)
Return ex.
Definition Math.h:710
openvdb::v11_0::math::isNegative
bool isNegative(const Type &x)
Return true if x is less than zero.
Definition Math.h:367
openvdb::v11_0::math::isNan
bool isNan(const float x)
Return true if x is a NaN (Not-A-Number) value.
Definition Math.h:395
openvdb::v11_0::math::doubleToInt64
int64_t doubleToInt64(const double d)
Definition Math.h:489
openvdb::v11_0::math::Clamp01
Type Clamp01(Type x)
Return x clamped to [0, 1].
Definition Math.h:271
openvdb::v11_0::math::Cos
float Cos(const float &x)
Return cos x.
Definition Math.h:725
openvdb::v11_0::math::isUlpsEqual
bool isUlpsEqual(const double aLeft, const double aRight, const int64_t aUnitsInLastPlace)
Definition Math.h:503
openvdb::v11_0::math::RandomInt
RandInt< int, std::mt19937 > RandomInt
Definition Math.h:253
openvdb::v11_0::math::RoundDown
float RoundDown(float x)
Return x rounded down to the nearest integer.
Definition Math.h:803
openvdb::v11_0::math::Min
const Type & Min(const Type &a, const Type &b)
Return the minimum of two values.
Definition Math.h:656
openvdb::v11_0::math::Abs
Coord Abs(const Coord &xyz)
Definition Coord.h:517
openvdb::v11_0::math::Mod
int Mod(int x, int y)
Return the remainder of x / y.
Definition Math.h:777
openvdb::v11_0::math::Truncate
Type Truncate(Type x, unsigned int digits)
Return x truncated to the given number of decimal digits.
Definition Math.h:870
openvdb::v11_0::math::negative
T negative(const T &val)
Return the unary negation of the given value.
Definition Math.h:128
openvdb::v11_0::math::isExactlyEqual
bool isExactlyEqual(const T0 &a, const T1 &b)
Return true if a is exactly equal to b.
Definition Math.h:443
openvdb::v11_0::math::Round
float Round(float x)
Return x rounded to the nearest integer.
Definition Math.h:819
openvdb::v11_0::math::Pow3
Type Pow3(Type x)
Return x3.
Definition Math.h:552
openvdb::v11_0::math::Clamp
Type Clamp(Type x, Type min, Type max)
Return x clamped to [min, max].
Definition Math.h:261
openvdb::v11_0::math::isZero
bool isZero(const Type &x)
Return true if x is exactly equal to zero.
Definition Math.h:337
openvdb::v11_0::math::Chop
Type Chop(Type x, Type delta)
Return x if it is greater or equal in magnitude than delta. Otherwise, return zero.
Definition Math.h:864
openvdb::v11_0::math::FractionalPart
Type FractionalPart(Type x)
Return the fractional part of x.
Definition Math.h:843
openvdb::v11_0::math::Axis
Axis
Definition Math.h:901
openvdb::v11_0::math::Z_AXIS
@ Z_AXIS
Definition Math.h:904
openvdb::v11_0::math::X_AXIS
@ X_AXIS
Definition Math.h:902
openvdb::v11_0::math::Y_AXIS
@ Y_AXIS
Definition Math.h:903
openvdb::v11_0::math::cwiseGreaterThan
bool cwiseGreaterThan(const Mat< SIZE, T > &m0, const Mat< SIZE, T > &m1)
Definition Mat.h:1029
openvdb::v11_0::math::ClampTest01
bool ClampTest01(Type &x)
Return true if x is outside [0,1].
Definition Math.h:277
openvdb::v11_0::math::Sin
float Sin(const float &x)
Return sin x.
Definition Math.h:716
openvdb::v11_0::math::floatToInt32
int32_t floatToInt32(const float f)
Definition Math.h:479
openvdb::v11_0::math::Pow2
Type Pow2(Type x)
Return x2.
Definition Math.h:548
openvdb::v11_0::math::RotationOrder
RotationOrder
Definition Math.h:908
openvdb::v11_0::math::YXZ_ROTATION
@ YXZ_ROTATION
Definition Math.h:911
openvdb::v11_0::math::ZXY_ROTATION
@ ZXY_ROTATION
Definition Math.h:913
openvdb::v11_0::math::YZX_ROTATION
@ YZX_ROTATION
Definition Math.h:912
openvdb::v11_0::math::ZXZ_ROTATION
@ ZXZ_ROTATION
Definition Math.h:916
openvdb::v11_0::math::XZX_ROTATION
@ XZX_ROTATION
Definition Math.h:915
openvdb::v11_0::math::XYZ_ROTATION
@ XYZ_ROTATION
Definition Math.h:909
openvdb::v11_0::math::ZYX_ROTATION
@ ZYX_ROTATION
Definition Math.h:914
openvdb::v11_0::math::XZY_ROTATION
@ XZY_ROTATION
Definition Math.h:910
openvdb::v11_0::math::Cbrt
float Cbrt(float x)
Return the cube root of a floating-point value.
Definition Math.h:769
openvdb::v11_0::math::Floor
int Floor(float x)
Return the floor of x.
Definition Math.h:848
openvdb::v11_0::math::SignChange
bool SignChange(const Type &a, const Type &b)
Return true if a and b have different signs.
Definition Math.h:743
openvdb::v11_0::math::ZeroCrossing
bool ZeroCrossing(const Type &a, const Type &b)
Return true if the interval [a, b] includes zero, i.e., if either a or b is zero or if they have diff...
Definition Math.h:753
openvdb::v11_0::math::Random01
Rand01< double, std::mt19937 > Random01
Definition Math.h:196
openvdb::v11_0::zeroVal
constexpr T zeroVal()
Return the value of type T that corresponds to zero.
Definition Math.h:70
openvdb
Definition Exceptions.h:13
std
Definition Coord.h:589
openvdb::v11_0::math::Delta< double >::value
static double value()
Definition Math.h:157
openvdb::v11_0::math::Delta< float >::value
static float value()
Definition Math.h:156
openvdb::v11_0::math::Delta
Delta for small floating-point offsets.
Definition Math.h:155
openvdb::v11_0::math::Delta::value
static T value()
Definition Math.h:155
openvdb::v11_0::math::Tolerance< double >::value
static double value()
Definition Math.h:150
openvdb::v11_0::math::Tolerance< float >::value
static float value()
Definition Math.h:149
openvdb::v11_0::math::Tolerance
Tolerance for floating-point comparison.
Definition Math.h:148
openvdb::v11_0::math::Tolerance::value
static T value()
Definition Math.h:148
openvdb::v11_0::math::promote
Definition Math.h:921
openvdb::v11_0::math::promote::type
typename boost::numeric::conversion_traits< S, T >::supertype type
Definition Math.h:922
OPENVDB_VERSION_NAME
#define OPENVDB_VERSION_NAME
The version namespace name for this library version.
Definition version.h.in:121
OPENVDB_USE_VERSION_NAMESPACE
#define OPENVDB_USE_VERSION_NAMESPACE
Definition version.h.in:212