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Simbody
3.3
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Simbody
3.3
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A Quaternion is a Vec4 with the following behavior: More...
#include <Quaternion.h>
Inheritance diagram for SimTK::Quaternion_< P >:Public Member Functions | |
| Quaternion_ () | |
| Default constructor produces the ZeroRotation quaternion [1 0 0 0] (not NaN - even in debug mode). More... | |
| Quaternion_ (const Quaternion_ &q) | |
| Zero-cost copy constructor just copies the source without conversion to canonical form or normalization. More... | |
| Quaternion_ & | operator= (const Quaternion_ &q) |
| Zero-cost copy assignment just copies the source without conversion to canonical form or normalization. More... | |
| Quaternion_ (RealP e0, RealP e1, RealP e2, RealP e3) | |
| Construct a quaternion from four scalars and normalize the result, which costs about 40 flops. More... | |
| Quaternion_ (const Vec4P &q) | |
| Construct a quaternion from a 4-vector and normalize the result, which costs about 40 flops. More... | |
| Quaternion_ (const Rotation_< P > &) | |
| Constructs a canonical quaternion from a rotation matrix (cost is about 60 flops). More... | |
| void | setQuaternionToZeroRotation () |
| The ZeroRotation quaternion is [1 0 0 0]. More... | |
| void | setQuaternionToNaN () |
| Set quaternion to all-NaN. More... | |
| void | setQuaternionFromAngleAxis (const Vec4P &av) |
| Set this quaternion from an angle-axis rotation packed into a 4-vector as [a vx vy vz]. More... | |
| void | setQuaternionFromAngleAxis (const RealP &a, const UnitVec< P, 1 > &v) |
| Set this quaternion from an angle-axis rotation provided as an angle a and a separate unit vector [vx vy vz]. More... | |
| Vec4P | convertQuaternionToAngleAxis () const |
| Returns [ a vx vy vz ] with (a,v) in canonical form, i.e., -180 < a <= 180 and |v|=1. More... | |
| const Vec4P & | asVec4 () const |
| Zero-cost cast of a Quaternion_ to its underlying Vec4; this is not converted to axis-angle form. More... | |
| Quaternion_ & | normalizeThis () |
| Normalize an already constructed quaternion in place; but do you really need to do this? Quaternions should be kept normalized at all times. More... | |
| Quaternion_ | normalize () const |
| Return a normalized copy of this quaternion; but do you really need to do this? Quaternions should be kept normalized at all times. More... | |
| Quaternion_ (const Vec4P &v, bool) | |
| Use this constructor only if you are *sure* v is normalized to 1.0. More... | |
| Public Member Functions inherited from SimTK::Vec< 4, P > | |
| ScalarNormSq | scalarNormSqr () const |
| Scalar norm square is sum( conjugate squares of all underlying scalars ), where conjugate square of scalar s is conj(s)*s. More... | |
| TSqrt | sqrt () const |
| Elementwise square root; that is, the return value has the same length as this Vec but with each element replaced by whatever it thinks its square root is. More... | |
| TAbs | abs () const |
| Elementwise absolute value; that is, the return value has the same dimension as this Vec but with each element replaced by whatever it thinks its absolute value is. More... | |
| TStandard | standardize () const |
| Return a copy of this Vec but with the underlying scalar type converted (if necessary) to one of the C++ standard real or complex floating point types. More... | |
| EStandard | sum () const |
| Sum just adds up all the elements into a single return element that is the same type as this Vec's elements except standardized to use one of the C++ built-in real or complex types as its underlying scalars. More... | |
| Vec () | |
| Default construction initializes Vec's elements to NaN when debugging but leaves them uninitialized garbage otherwise, so declarations have zero cost in Release builds. More... | |
| Vec (const Vec &src) | |
| Copy constructor copies the logically-included elements from the source Vec; gaps due to stride are not accessed in either source or destination. More... | |
| Vec (const Vec< M, E, SS > &src) | |
| This is an implicit conversion from a Vec of the same length and element type but with a different stride. More... | |
| Vec (const Vec< M, ENeg, SS > &src) | |
| This is an implicit conversion from a Vec of the same length and negated element type (possibly with a different stride). More... | |
| Vec (const Vec< M, EE, SS > &src) | |
| Construct a Vec from a Vec of the same length, with any stride. More... | |
| Vec (const E &e) | |
| Construction from a single value of this Vec's element type assigns that value to each element. More... | |
| Vec (const ENeg &ne) | |
| Construction from a single value of this Vec's negated element type assigns that value to each element, requiring floating point negation to be performed once to compute the type-E representation of the type negator<E> value provided. More... | |
| Vec (int i) | |
| Given an int value, turn it into a suitable floating point number, convert that to element type E and then feed that to the above single-element constructor. More... | |
| Vec (const E &e0, const E &e1) | |
| Construct a Vec<2,E> from two elements of type E, etc. More... | |
| Vec (const E &e0, const E &e1, const E &e2) | |
| Vec (const E &e0, const E &e1, const E &e2, const E &e3) | |
| Vec (const E &e0, const E &e1, const E &e2, const E &e3, const E &e4) | |
| Vec (const E &e0, const E &e1, const E &e2, const E &e3, const E &e4, const E &e5) | |
| Vec (const E &e0, const E &e1, const E &e2, const E &e3, const E &e4, const E &e5, const E &e6) | |
| Vec (const E &e0, const E &e1, const E &e2, const E &e3, const E &e4, const E &e5, const E &e6, const E &e7) | |
| Vec (const E &e0, const E &e1, const E &e2, const E &e3, const E &e4, const E &e5, const E &e6, const E &e7, const E &e8) | |
| Vec (const EE *p) | |
| Construction from a pointer to elements of any type EE assumes we're pointing at a C++ array of EE's of the right length, and that EE is assignment compatible with this Vec's element type E. More... | |
| Vec & | operator= (const Vec &src) |
| Copy assignment operator copies the logically-included elements from the source Vec; gaps due to stride are not accessed in either source or destination. More... | |
| Vec & | operator= (const EE *p) |
| Assignment to a pointer to elements of any type EE assumes we're pointing at a C++ array of EE's of the right length, and that EE is assignment compatible with this Vec's element type E. More... | |
| Vec & | operator= (const Vec< M, EE, SS > &vv) |
| Assignment to a conforming Vec, of any element type and stride, provided that the element types are assignment-compatible. More... | |
| Vec & | operator= (const EE &e) |
| Vec & | operator+= (const Vec< M, EE, SS > &r) |
| Add in a conforming Vec, of any element type and stride, provided that the element types are addition-compatible. More... | |
| Vec & | operator+= (const Vec< M, negator< EE >, SS > &r) |
| Add in a conforming Vec, of any negated element type and stride, provided that the element types are addition-compatible. More... | |
| Vec & | operator+= (const EE &e) |
| Vec & | operator-= (const Vec< M, EE, SS > &r) |
| Subtract off a conforming Vec, of any element type and stride, provided that the element types are addition-compatible. More... | |
| Vec & | operator-= (const Vec< M, negator< EE >, SS > &r) |
| Subtract off a conforming Vec, of any negated element type and stride, provided that the element types are addition-compatible. More... | |
| Vec & | operator-= (const EE &e) |
| Vec< M, typename CNT< E > ::template Result< EE >::Add > | conformingAdd (const Vec< M, EE, SS > &r) const |
| Vector addition – use operator+ instead. More... | |
| Vec< M, typename CNT< E > ::template Result< EE >::Sub > | conformingSubtract (const Vec< M, EE, SS > &r) const |
| Vector subtraction – use operator- instead. More... | |
| Mat< M, M, typename CNT< E > ::template Result< EE >::Mul > | conformingMultiply (const Row< M, EE, SS > &r) const |
| Same as outer product (m = col*row) – use operator* or outer() instead. More... | |
| Vec< M, typename CNT< E > ::template Result< EE >::Mul > | elementwiseMultiply (const Vec< M, EE, SS > &r) const |
| Elementwise multiply (Matlab . More... | |
| Vec< M, typename CNT< E > ::template Result< EE >::Dvd > | elementwiseDivide (const Vec< M, EE, SS > &r) const |
| Elementwise divide (Matlab . More... | |
| const E & | operator[] (int i) const |
| Select an element of this Vec and return a const reference to it. More... | |
| E & | operator[] (int i) |
| Select an element of this Vec and return a writable reference to it. More... | |
| const E & | operator() (int i) const |
| Same as const operator[] above. More... | |
| E & | operator() (int i) |
| Same as non-const operator[] above. More... | |
| ScalarNormSq | normSqr () const |
| CNT< ScalarNormSq >::TSqrt | norm () const |
| TNormalize | normalize () const |
| If the elements of this Vec are scalars, the result is what you get by dividing each element by the norm() calculated above. More... | |
| TInvert | invert () const |
| This method is not supported for Vec objects. More... | |
| const Vec & | operator+ () const |
| Unary plus does nothing. More... | |
| const TNeg & | operator- () const |
| Unary minus recasts this Vec to a type that has the opposite interpretation of the sign but is otherwise identical, so no computation or copying is performed here. More... | |
| TNeg & | operator- () |
| Recast to negated type and return a writable reference; writing to this will cause the negated result to be placed in the original Vec. More... | |
| const THerm & | operator~ () const |
| The Hermitian transpose operator recasts this Vec to a type that specifies the opposite storage order (row vs. column) then returns a reference, so no computation or copying is performed here. More... | |
| THerm & | operator~ () |
| Recast to Hermitian transposed type and return a writable reference; the effect is that writing to elements of the result affects the transposed element of the original Vec. More... | |
| const TNeg & | negate () const |
| Non-operator version of unary negation; just a recast. More... | |
| TNeg & | updNegate () |
| Non-operator version of unary negation; recasts and returns a writable reference. More... | |
| const THerm & | transpose () const |
| Non-operator version of Hermitian transpose; just a recast. More... | |
| THerm & | updTranspose () |
| Non-operator version of Hermitian transpose; recasts and returns a writable reference. More... | |
| const TPosTrans & | positionalTranspose () const |
| Positional transpose turns this Vec into a Row but does not transpose the individual elements. More... | |
| TPosTrans & | updPositionalTranspose () |
| Positional transpose returning a writable reference. More... | |
| const TReal & | real () const |
| Return a reference to the real portion of this Vec if it has complex elements; otherwise the type doesn't change. More... | |
| TReal & | real () |
| Recast to show only the real portion of this Vec and return a writable reference. More... | |
| const TImag & | imag () const |
| Return a reference to the imaginary portion of this Vec if it has complex elements; otherwise the type doesn't change. More... | |
| TImag & | imag () |
| Recast to show only the imaginary portion of this Vec and return a writable reference. More... | |
| const TWithoutNegator & | castAwayNegatorIfAny () const |
| Recast to remove negators from this Vec's type if present; this is handy for simplifying operations where we know the sign can be ignored such as squaring. More... | |
| TWithoutNegator & | updCastAwayNegatorIfAny () |
| Recast to remove negators from this Vec's type if present and return a writable reference. More... | |
| Vec< M, typename CNT< E > ::template Result< EE >::Mul > | scalarMultiply (const EE &e) const |
| Vec< M, typename CNT< EE > ::template Result< E >::Mul > | scalarMultiplyFromLeft (const EE &e) const |
| Vec< M, typename CNT< E > ::template Result< EE >::Dvd > | scalarDivide (const EE &e) const |
| Vec< M, typename CNT< EE > ::template Result< E >::Dvd > | scalarDivideFromLeft (const EE &e) const |
| Vec< M, typename CNT< E > ::template Result< EE >::Add > | scalarAdd (const EE &e) const |
| Vec< M, typename CNT< E > ::template Result< EE >::Sub > | scalarSubtract (const EE &e) const |
| Vec< M, typename CNT< EE > ::template Result< E >::Sub > | scalarSubtractFromLeft (const EE &e) const |
| Vec & | operator*= (const EE &e) |
| Vec & | operator/= (const EE &e) |
| Vec & | scalarEq (const EE &ee) |
| Vec & | scalarEq (int ee) |
| Vec & | scalarPlusEq (const EE &ee) |
| Vec & | scalarPlusEq (int ee) |
| Vec & | scalarMinusEq (const EE &ee) |
| Vec & | scalarMinusEq (int ee) |
| Vec & | scalarMinusEqFromLeft (const EE &ee) |
| Vec & | scalarMinusEqFromLeft (int ee) |
| Vec & | scalarTimesEq (const EE &ee) |
| Vec & | scalarTimesEq (int ee) |
| Vec & | scalarTimesEqFromLeft (const EE &ee) |
| Vec & | scalarTimesEqFromLeft (int ee) |
| Vec & | scalarDivideEq (const EE &ee) |
| Vec & | scalarDivideEq (int ee) |
| Vec & | scalarDivideEqFromLeft (const EE &ee) |
| Vec & | scalarDivideEqFromLeft (int ee) |
| void | setToNaN () |
| Set every scalar in this Vec to NaN; this is the default initial value in Debug builds, but not in Release. More... | |
| void | setToZero () |
| Set every scalar in this Vec to zero. More... | |
| const Vec< MM, P, STRIDE > & | getSubVec (int i) const |
| Extract a const reference to a sub-Vec with size known at compile time. More... | |
| Vec< MM, P, STRIDE > & | updSubVec (int i) |
| Extract a writable reference to a sub-Vec with size known at compile time. More... | |
| Vec< M-1, P, 1 > | drop1 (int p) const |
| Return a vector one smaller than this one by dropping the element at the indicated position p. More... | |
| Vec< M+1, P, 1 > | append1 (const EE &v) const |
| Return a vector one larger than this one by adding an element to the end. More... | |
| Vec< M+1, P, 1 > | insert1 (int p, const EE &v) const |
| Return a vector one larger than this one by inserting an element before the indicated one. More... | |
| bool | isNaN () const |
| Return true if any element of this Vec contains a NaN anywhere. More... | |
| bool | isInf () const |
| Return true if any element of this Vec contains a +Infinity or -Infinity somewhere but no element contains a NaN anywhere. More... | |
| bool | isFinite () const |
| Return true if no element of this Vec contains an Infinity or a NaN anywhere. More... | |
| bool | isNumericallyEqual (const Vec< M, E2, RS2 > &v, double tol) const |
| Test whether this vector is numerically equal to some other vector with the same shape, using a specified tolerance. More... | |
| bool | isNumericallyEqual (const Vec< M, E2, RS2 > &v) const |
| Test whether this vector is numerically equal to some other vector with the same shape, using a default tolerance which is the looser of the default tolerances of the two objects being compared. More... | |
| bool | isNumericallyEqual (const P &e, double tol=getDefaultTolerance()) const |
| Test whether every element of this vector is numerically equal to the given element, using either a specified tolerance or the vector's default tolerance (which is always the same or looser than the default tolerance for one of its elements). More... | |
| std::string | toString () const |
| Print Vec into a string and return it. More... | |
| void | set (int i, const E &value) |
| Variant of operator[] that's scripting friendly to set ith entry. More... | |
| const E & | get (int i) const |
| Variant of operator[] that's scripting friendly to get const reference to ith entry. More... | |
Additional Inherited Members | |
| Public Types inherited from SimTK::Vec< 4, P > | |
| enum | |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef P | E |
| Element type of this Vec. More... | |
| typedef CNT< E >::TNeg | ENeg |
| Negated version of this Vec's element type; ENeg==negator< E >. More... | |
| typedef CNT< E >::TWithoutNegator | EWithoutNegator |
| Element type, stripped of negator<> if it has one. More... | |
| typedef CNT< E >::TReal | EReal |
| Type showing just the real part of an element of this Vec if elements are complex; otherwise just the element type. More... | |
| typedef CNT< E >::TImag | EImag |
| Type showing the imaginary part of an element of this Vec as real, if elements are complex; otherwise a type that can hold a zero of the element type. More... | |
| typedef CNT< E >::TComplex | EComplex |
| Type that elements would have if complex, if E is currently real; otherwise just the element type E. More... | |
| typedef CNT< E >::THerm | EHerm |
| Type of the Hermitian transpose of an element of this Vec. More... | |
| typedef CNT< E >::TPosTrans | EPosTrans |
| Type of a positional transpose of an element of this Vec. More... | |
| typedef CNT< E >::TSqHermT | ESqHermT |
| Type of the expression ~E*E (default vector and matrix square; symmetric). More... | |
| typedef CNT< E >::TSqTHerm | ESqTHerm |
| Type of the expression E*~E ("row square"; symmetric). More... | |
| typedef CNT< E >::TSqrt | ESqrt |
| Type required to hold the result of sqrt(E). More... | |
| typedef CNT< E >::TAbs | EAbs |
| Type required to hold the result of abs(E). More... | |
| typedef CNT< E >::TStandard | EStandard |
| Return type of standardize(E) method; a packed type that can hold the value of an element after eliminating negator and conjugate types. More... | |
| typedef CNT< E >::TInvert | EInvert |
| Packed type that can hold the value returned from invert(E), the inverse type of an element. More... | |
| typedef CNT< E >::TNormalize | ENormalize |
| Packed type that can hold the value returned from normalize(E). More... | |
| typedef CNT< E >::Scalar | EScalar |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef CNT< E >::ULessScalar | EULessScalar |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef CNT< E >::Number | ENumber |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef CNT< E >::StdNumber | EStdNumber |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef CNT< E >::Precision | EPrecision |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef CNT< E >::ScalarNormSq | EScalarNormSq |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef Vec< M, E, STRIDE > | T |
| The type of this Vec. More... | |
| typedef Vec< M, ENeg, STRIDE > | TNeg |
| Type this Vec would have if its elements were interpreted as negated. More... | |
| typedef Vec< M, EWithoutNegator, STRIDE > | TWithoutNegator |
| Type of this Vec with negator removed from its element type, if the element is negated. More... | |
| typedef Vec< M, EReal, STRIDE *CNT< E >::RealStrideFactor > | TReal |
| Type of this Vec cast to show only the real part of its element; this might affect the stride. More... | |
| typedef Vec< M, EImag, STRIDE *CNT< E >::RealStrideFactor > | TImag |
| Type of this Vec cast to show only the imaginary part of its element; this might affect the stride. More... | |
| typedef Vec< M, EComplex, STRIDE > | TComplex |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef Row< M, EHerm, STRIDE > | THerm |
| Type of this Vec after casting to its Hermitian transpose; that is, the Vec turns into a Row and each element turns into its Hermitian transpose. More... | |
| typedef Row< M, E, STRIDE > | TPosTrans |
| Type of this Vec after casting to its positional transpose; that is, the Vec turns into a Row but the element type remains unchanged. More... | |
| typedef E | TElement |
| Element type of this Vec. More... | |
| typedef E | TRow |
| Type of a row of this CNT object (for a Vec, just its element type). More... | |
| typedef Vec | TCol |
| Type of a column of this CNT object (for a Vec, the whole thing). More... | |
| typedef Vec< M, ESqrt, 1 > | TSqrt |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef Vec< M, EAbs, 1 > | TAbs |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef Vec< M, EStandard, 1 > | TStandard |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef Row< M, EInvert, 1 > | TInvert |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef Vec< M, ENormalize, 1 > | TNormalize |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef ESqHermT | TSqHermT |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef SymMat< M, ESqTHerm > | TSqTHerm |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef EScalar | Scalar |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef EULessScalar | ULessScalar |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef ENumber | Number |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef EStdNumber | StdNumber |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef EPrecision | Precision |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| typedef EScalarNormSq | ScalarNormSq |
| These compile-time constants are required of every Composite Numerical Type (CNT). More... | |
| Static Public Member Functions inherited from SimTK::Vec< 4, P > | |
| static int | size () |
| The number of elements in this Vec (note that stride does not affect this number.) More... | |
| static int | nrow () |
| The number of rows in a Vec is the number of elements. More... | |
| static int | ncol () |
| The number of columns in a Vec is always 1. More... | |
| static const Vec & | getSubVec (const Vec< MM, P, STRIDE > &v, int i) |
| Extract a subvector of type Vec from a longer one that has the same element type and stride, and return a const reference to the selected subsequence. More... | |
| static Vec & | updSubVec (Vec< MM, P, STRIDE > &v, int i) |
| Extract a subvector of type Vec from a longer one that has the same element type and stride, and return a writable reference to the selected subsequence. More... | |
| static const Vec & | getAs (const P *p) |
| Recast an ordinary C++ array E[] to a const Vec<M,E,S>; assumes compatible length, stride, and packing. More... | |
| static Vec & | updAs (P *p) |
| Recast a writable ordinary C++ array E[] to a writable Vec<M,E,S>; assumes compatible length, stride, and packing. More... | |
| static Vec< M, P, 1 > | getNaN () |
| Return a Vec of the same length and element type as this one but with all elements set to NaN. More... | |
| static double | getDefaultTolerance () |
| For approximate comparisions, the default tolerance to use for a vector is the same as its elements' default tolerance. More... | |
A Quaternion is a Vec4 with the following behavior:
Conversion from quaternion to (angle,axis) form is handled here also. (angle,axis) is in canonical form when -180 < angle <= 180 and |axis|=1. However, (angle,axis) is meaningful for any angle and for any axis where |axis| > 0.
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Default constructor produces the ZeroRotation quaternion [1 0 0 0] (not NaN - even in debug mode).
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Zero-cost copy constructor just copies the source without conversion to canonical form or normalization.
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Construct a quaternion from four scalars and normalize the result, which costs about 40 flops.
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inlineexplicit |
Construct a quaternion from a 4-vector and normalize the result, which costs about 40 flops.
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explicit |
Constructs a canonical quaternion from a rotation matrix (cost is about 60 flops).
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Use this constructor only if you are *sure* v is normalized to 1.0.
This zero cost method is faster than the Quaternion(Vec4) constructor which normalizes the Vec4. The second argument forces the compiler to call the fast constructor; it is otherwise ignored. By convention, set the second argument to "true".
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Zero-cost copy assignment just copies the source without conversion to canonical form or normalization.
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The ZeroRotation quaternion is [1 0 0 0].
Note: Default constructor is ZeroRotation (unlike Vec4P which start as NaN in Debug mode).
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Set quaternion to all-NaN.
Note that this is not the same as produced by default construction, even in Debug mode – default construction always produces an identity rotation of [1 0 0 0].
| void SimTK::Quaternion_< P >::setQuaternionFromAngleAxis | ( | const Vec4P & | av | ) |
Set this quaternion from an angle-axis rotation packed into a 4-vector as [a vx vy vz].
The result will be put in canonical form, i.e., it will have a non-negative first element. If the "axis" portion of av is a zero vector on input, the quaternion is set to all-NaN.
| void SimTK::Quaternion_< P >::setQuaternionFromAngleAxis | ( | const RealP & | a, |
| const UnitVec< P, 1 > & | v | ||
| ) |
Set this quaternion from an angle-axis rotation provided as an angle a and a separate unit vector [vx vy vz].
The result will be put in canonical form, i.e., it will have a non-negative first element.
| Vec4P SimTK::Quaternion_< P >::convertQuaternionToAngleAxis | ( | ) | const |
Returns [ a vx vy vz ] with (a,v) in canonical form, i.e., -180 < a <= 180 and |v|=1.
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Zero-cost cast of a Quaternion_ to its underlying Vec4; this is not converted to axis-angle form.
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Normalize an already constructed quaternion in place; but do you really need to do this? Quaternions should be kept normalized at all times.
One of the advantages of using them is that you don't have to check if they are normalized or renormalize them. However, under some situations they do need renormalization, but it is costly if you don't actually need it. If the quaternion is exactly zero, set it to [1 0 0 0]. If its magnitude is 0 < magnitude < epsilon (epsilon is machine tolerance), set it to NaN (treated as an error). Otherwise, normalize the quaternion which costs about 40 flops. The quaternion is NOT put in canonical form.
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Return a normalized copy of this quaternion; but do you really need to do this? Quaternions should be kept normalized at all times.
One of the advantages of using them is that you don't have to check if they are normalized or renormalize them. However, under some situations they do need renormalization, but it is costly if you don't actually need it.
1.8.6 
