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educed by avoiding situations in which the denominator is close to zero one of the other three methods looks as follows 6 7 q i 1 2 1 a 11 a 22 a 33 q j 1 4 q i a 12 a 21 q k 1 4 q i a 13 a 31 q r 1 4 q i a 32 a 23 displaystyle begin aligned q_ i frac 1 2 sqrt 1 a_ 11 a_ 22 a_ 33 q_ j frac 1 4q_ i left a_ 12 a_ 21 right q_ k frac 1 4q_ i left a_ 13 a_ 31 right q_ r frac 1 4q_ i left a_ 32 a_ 23 right end aligned the rotation matrix corresponding to the quaternion q can be computed as follows a q r 2 q ˇ t q ˇ i 3 2 q ˇ q ˇ t 2 q r q displaystyle mathbf a left q_ r 2 check mathbf q mathsf t check mathbf q right mathbf i _ 3 2 check mathbf q check mathbf q mathsf t 2q_ r mathbf mathcal q where q ˇ q i q j q k q 0 q k q j q k 0 q i q j q i 0 displaystyle check mathbf q begin bmatrix q_ i q_ j q_ k end bmatrix quad mathbf mathcal q begin bmatrix 0 q_ k q_ j q_ k 0 q_ i q_ j q_ i 0 end bmatrix which gives a 1 2 q j 2 2 q k 2 2 q i q j q k q r 2 q i q k q j q r 2 q i q j q k q r 1 2 q i 2 2 q k 2 2 q j q k q i q r 2 q i q k q j q r 2 q j q k q i q r 1 2 q i 2 2 q j 2 displaystyle mathbf a begin bmatrix 1 2q_ j 2 2q_ k 2 2 left q_ i q_ j q_ k q_ r right 2 left q_ i q_ k q_ j q_ r right 2 left q_ i q_ j q_ k q_ r right 1 2q_ i 2 2q_ k 2 2 left q_ j q_ k q_ i q_ r right 2 left q_ i q_ k q_ j q_ r right 2 left q_ j q_ k q_ i q_ r right 1 2q_ i 2 2q_ j 2 end bmatrix or equivalently a 1 2 q i 2 2 q r 2 2 q i q j q k q r 2 q i q k q j q r 2 q i q j q k q r 1 2 q j 2 2 q r 2 2 q j q k q i q r 2 q i q k q j q r 2 q j q k q i q r 1 2 q k 2 2 q r 2 displaystyle mathbf a begin bmatrix 1 2q_ i 2 2q_ r 2 2 left q_ i q_ j q_ k q_ r right 2 left q_ i q_ k q_ j q_ r right 2 left q_ i q_ j q_ k q_ r right 1 2q_ j 2 2q_ r 2 2 left q_ j q_ k q_ i q_ r right 2 left q_ i q_ k q_ j q_ r right 2 left q_ j q_ k q_ i q_ r right 1 2q_ k 2 2q_ r 2 end bmatrix this is called the euler rodrigues formula for the transformation matrix a displaystyle mathbf a euler angles quaternion edit main article conversion between quaternions and euler angles euler angles z x z extrinsic quaternion edit we will consider the x convention 3 1 3 extrinsic euler angles for the following algorithm the terms of the algorithm depend on the convention used we can compute the quaternion q q i q j q k q r q i i q j j q k k q r displaystyle mathbf q begin bmatrix q_ i q_ j q_ k q_ r end bmatrix q_ i mathbf i q_ j mathbf j q_ k mathbf k q_ r from the euler angles ϕ θ ψ as follows q i cos ϕ ψ 2 sin θ 2 q j sin ϕ ψ 2 sin θ 2 q k sin ϕ ψ 2 cos θ 2 q r cos ϕ ψ 2 cos θ 2 displaystyle begin aligned q_ i cos frac phi psi 2 sin frac theta 2 q_ j sin frac phi psi 2 sin frac theta 2 q_ k sin frac phi psi 2 cos frac theta 2 q_ r cos frac phi psi 2 cos frac theta 2 end aligned euler angles z y x intrinsic quaternion edit a quaternion equivalent to yaw ψ pitch θ and roll ϕ angles or intrinsic tait bryan angles following the z y x convention can be computed by q i sin ϕ 2 cos θ 2 cos ψ 2 cos ϕ 2 sin θ 2 sin ψ 2 q j cos ϕ 2 sin θ 2 cos ψ 2 sin ϕ 2 cos θ 2 sin ψ 2 q k cos ϕ 2 cos θ 2 sin ψ 2 sin ϕ 2 sin θ 2 cos ψ 2 q r cos ϕ 2 cos θ 2 cos ψ 2 sin ϕ 2 sin θ 2 sin ψ 2 displaystyle begin aligned q_ i sin frac phi 2 cos frac theta 2 cos frac psi 2 cos frac phi 2 sin frac theta 2 sin frac psi 2 q_ j cos frac phi 2 sin frac theta 2 cos frac psi 2 sin frac phi 2 cos frac theta 2 sin frac psi 2 q_ k cos frac phi 2 cos frac theta 2 sin frac psi 2 sin frac phi 2 sin frac theta 2 cos frac psi 2 q_ r cos frac phi 2 cos frac theta 2 cos frac psi 2 sin frac phi 2 sin frac theta 2 sin frac psi 2 end aligned quaternion euler angles z x z extrinsic edit given the rotation quaternion q q i q j q k q r q i i q j j q k k q r displaystyle mathbf q begin bmatrix q_ i q_ j q_ k q_ r end bmatrix q_ i mathbf i q_ j mathbf j q_ k mathbf k q_ r the x convention 3 1 3 extrinsic euler angles φ θ ψ can be computed by ϕ atan2 q i q k q j q r q j q k q i q r θ arccos q i 2 q j 2 q k 2 q r 2 ψ atan2 q i q k q j q r q j q k q i q r displaystyle begin aligned phi operatorname atan2 left left q_ i q_ k q_ j q_ r right left q_ j q_ k q_ i q_ r right right theta arccos left q_ i 2 q_ j 2 q_ k 2 q_ r 2 right psi operatorname atan2 left left q_ i q_ k q_ j q_ r right left q_ j q_ k q_ i q_ r right right end aligned quaternion euler angles z y x intrinsic edit given the rotation quaternion q q i q j q k q r q i i q j j q k k q r displaystyle mathbf q begin bmatrix q_ i q_ j q_ k q_ r end bmatrix q_ i mathbf i q_ j mathbf j q_ k mathbf k q_ r yaw pitch and roll angles or intrinsic tait bryan angles following the z y x convention can be computed by roll atan2 2 q r q i q j q k 1 2 q i 2 q j 2 pitch arcsin 2 q r q j q k q i yaw atan2 2 q r q k q i q j 1 2 q j 2 q k 2 displaystyle begin aligned text roll operatorname atan2 left 2 left q_ r q_ i q_ j q_ k right 1 2 left q_ i 2 q_ j 2 right right text pitch arcsin left 2 left q_ r q_ j q_ k q_ i right right text yaw operatorname atan2 left 2 left q_ r q_ k q_ i q_ j right 1 2 left q_ j 2 q_ k 2 right right end aligned euler axis angle quaternion edit given the euler axis ê and angle θ the quaternion q q i q j q k q r q i i q j j q k k q r displaystyle mathbf q begin bmatrix q_ i q_ j q_ k q_ r end bmatrix q_ i mathbf i q_ j mathbf j q_ k mathbf k q_ r can be computed by q i e 1 sin θ 2 q j e 2 sin θ 2 q k e 3 sin θ 2 q r cos θ 2 displaystyle begin aligned q_ i hat e _ 1 sin frac theta 2 q_ j hat e _ 2 sin frac theta 2 q_ k hat e _ 3 sin frac theta 2 q_ r cos frac theta 2 end aligned given the rotation quaternion q define q ˇ q i q j q k displaystyle check mathbf q begin bmatrix q_ i q_ j q_ k end bmatrix then the euler axis ê and angle θ can be computed by e q ˇ q ˇ θ 2 arccos q r displaystyle begin aligned hat mathbf e frac check mathbf q left check mathbf q right theta 2 arccos q_ r end aligned rotation matrix rodrigues vector edit rodrigues vector rotation matrix edit since the definition of the rodrigues vector can be related to rotation quaternions g i q i q r e x tan θ 2 g j q j q r e y tan θ 2 g k q k q r e z tan θ 2 displaystyle begin cases g_ i dfrac q_ i q_ r e_ x tan left dfrac theta 2 right g_ j dfrac q_ j q_ r e_ y tan left dfrac theta 2 right g_ k dfrac q_ k q_ r e_ z tan left dfrac theta 2 right end cases by making use of the following property 1 q r 2 q i 2 q j 2 q k 2 q r 2 1 q i 2 q r 2 q j 2 q r 2 q k 2 q r 2 q r 2 1 g i 2 g j 2 g k 2 displaystyle 1 q_ r 2 q_ i 2 q_ j 2 q_ k 2 q_ r 2 left 1 frac q_ i 2 q_ r 2 frac q_ j 2 q_ r 2 frac q_ k 2 q_ r 2 right q_ r 2 left 1 g_ i 2 g_ j 2 g_ k 2 right the formula can be obtained by factoring q 2 r from the final expression obtained for quaternions a q r 2 1 q r 2 2 q j 2 q r 2 2 q k 2 q r 2 2 q i q r q j q r q k q r 2 q i q r q k q r q j q r 2 q i q r q j q r q k q r 1 q r 2 2 q i 2 q r 2 2 q k 2 q r 2 2 q j q r q k q r q i q r 2 q i q r q k q r q j q r 2 q j q r q k q r q i q r 1 q r 2 2 q i 2 q r 2 2 q j 2 q r 2 displaystyle mathbf a q_ r 2 begin bmatrix frac 1 q_ r 2 2 frac q_ j 2 q_ r 2 2 frac q_ k 2 q_ r 2 2 left frac q_ i q_ r frac q_ j q_ r frac q_ k q_ r right 2 left frac q_ i q_ r frac q_ k q_ r frac q_ j q_ r right 2 left frac q_ i q_ r frac q_ j q_ r frac q_ k q_ r right frac 1 q_ r 2 2 frac q_ i 2 q_ r 2 2 frac q_ k 2 q_ r 2 2 left frac q_ j q_ r frac q_ k q_ r frac q_ i q_ r right 2 left frac q_ i q_ r frac q_ k q_ r frac q_ j q_ r right 2 left frac q_ j q_ r frac q_ k q_ r frac q_ i q_ r right frac 1 q_ r 2 2 frac q_ i 2 q_ r 2 2 frac q_ j 2 q_ r 2 end bmatrix leading to the final formula a 1 1 g i 2 g j 2 g k 2 1 g i 2 g j 2 g k 2 2 g i g j g k 2 g i g k g j 2 g i g j g k 1 g i 2 g j 2 g k 2 2 g j g k g i 2 g i g k g j 2 g j g k g i 1 g i 2 g j 2 g k 2 displaystyle mathbf a frac 1 1 g_ i 2 g_ j 2 g_ k 2 begin bmatrix 1 g_ i 2 g_ j 2 g_ k 2 2 left g_ i g_ j g_ k right 2 left g_ i g_ k g_ j right 2 left g_ i g_ j g_ k right 1 g_ i 2 g_ j 2 g_ k 2 2 left g_ j g_ k g_ i right 2 left g_ i g_ k g_ j right 2 left g_ j g_ k g_ i right 1 g_ i 2 g_ j 2 g_ k 2 end bmatrix conversion formulae for derivatives edit rotation matrix angular velocities edit the angular velocity vector ω ω x ω y ω z displaystyle boldsymbol omega begin bmatrix omega _ x omega _ y omega _ z end bmatrix can be extracted from the time derivative of the rotation matrix d a d t by the following relation ω 0 ω z ω y ω z 0 ω x ω y ω x 0 d a d t a t displaystyle boldsymbol omega _ times begin bmatrix 0 omega _ z omega _ y omega _ z 0 omega _ x omega _ y omega _ x 0 end bmatrix frac mathrm d mathbf a mathrm d t mathbf a mathsf t the derivation is adapted from ioffe 8 as follows for any vector r 0 consider r t a t r 0 and differentiate it d r d t d a d t r 0 d a d t a t t r t displaystyle frac mathrm d mathbf r mathrm d t frac mathrm d mathbf a mathrm d t mathbf r _ 0 frac mathrm d mathbf a mathrm d t mathbf a mathsf t t mathrm r t the derivative of a vector is the linear velocity of its tip since a is a rotation matrix by definition the length of r t is always equal to the length of r 0 and hence it does not change with time thus when r t rotates its tip moves along a circle and the linear velocity of its tip is tangential to the circle i e always perpendicular to r t in this specific case the relationship between the linear velocity vector and the angular velocity vector is d r d t ω t r t ω r t displaystyle frac mathrm d mathbf r mathrm d t boldsymbol omega t times mathbf r t boldsymbol omega _ times mathbf r t see circular motion and cross product by the transitivity of the abovementioned equations d a d t a t t r t ω r t displaystyle frac mathrm d mathbf a mathrm d t mathbf a mathsf t t mathbf r t boldsymbol omega _ times mathbf r t which implies d a d t a t t ω displaystyle frac mathrm d mathbf a mathrm d t mathbf a mathsf t t boldsymbol omega _ times quaternion angular velocities edit the angular velocity vector ω ω x ω y ω z displaystyle boldsymbol omega begin bmatrix omega _ x omega _ y omega _ z end bmatrix can be obtained from the derivative of the quaternion d q d t as follows 9 0 ω x ω y ω z 2 d q d t q displaystyle begin bmatrix 0 omega _ x omega _ y omega _ z end bmatrix 2 frac mathrm d mathbf q mathrm d t tilde mathbf q where q̃ is the conjugate inverse of q conversely the derivative of the quaternion is d q d t 1 2 0 ω x ω y ω z q displaystyle frac mathrm d mathbf q mathrm d t frac 1 2 begin bmatrix 0 omega _ x omega _ y omega _ z end bmatrix mathbf q rotors in a geometric algebra edit the formulations of geometric algebra ga provides an extension and interpretation of the quaternion method central to ga is the geometric product of vectors an extension of the traditional inner and cross products given by a b a b a b displaystyle mathbf ab mathbf a cdot mathbf b mathbf a wedge mathbf b where the symbol denotes the exterior product or wedge product this product of vectors a and b produces two terms a scalar part from the inner product and a bivector part from the wedge product this bivector describes the plane perpendicular to what the cross product of the vectors would return bivectors in ga have some unusual properties compared to vectors under the geometric product bivectors have a negative square the bivector x̂ŷ describes the xy plane its square is x̂ŷ 2 x̂ŷx̂ŷ because the unit basis vectors are orthogonal to each other the geometric product reduces to the antisymmetric outer product so x̂ and ŷ can be swapped freely at the cost of a factor of 1 the square reduces to x̂x̂ŷŷ 1 since the basis vectors themselves square to 1 this result holds generally for all bivectors and as a result the bivector plays a role similar to the imaginary unit geometric algebra uses bivectors in its analogue to the quaternion the rotor given by r exp b θ 2 cos θ 2 b sin θ 2 displaystyle mathbf r exp left frac hat mathbf b theta 2 right cos frac theta 2 hat mathbf b sin frac theta 2 where b̂ is a unit bivector that describes the plane of rotation because b̂ squares to 1 the power series expansion of r generates the trigonometric functions the rotation formula that maps a vector a to a rotated vector b is then b r a r displaystyle mathbf b mathbf rar dagger where r exp 1 2 b θ cos θ 2 b sin θ 2 displaystyle mathbf r dagger exp left frac 1 2 hat mathbf b theta right cos frac theta 2 hat mathbf b sin frac theta 2 is the reverse of r displaystyle scriptstyle r reversing the order of the vectors in b displaystyle b is equivalent to changing its sign example a rotation about the axis v 1 3 x y z displaystyle hat mathbf v frac 1 sqrt 3 left hat mathbf x hat mathbf y hat mathbf z right can be accomplished by converting v̂ to its dual bivector b x y z v i v displaystyle hat mathbf b hat mathbf x hat mathbf y hat mathbf z hat mathbf v mathbf i hat mathbf v where i x̂ŷẑ is the unit volume element the only trivector pseudoscalar in three dimensional space the result is b 1 3 y z z x x y displaystyle hat mathbf b frac 1 sqrt 3 left hat mathbf y hat mathbf z hat mathbf z hat mathbf x hat mathbf x hat mathbf y right in three dimensional space however it is often simpler to leave the expression for b̂ iv̂ using the fact that i commutes with all objects in 3d and also squares to 1 a rotation of the x̂ vector in this plane by an angle θ is then x r x r e i v θ 2 x e i v θ 2 x cos 2 θ 2 i x v v x cos θ 2 sin θ 2 v x v sin 2 θ 2 displaystyle hat mathbf x mathbf r hat mathbf x mathbf r dagger e i hat mathbf v frac theta 2 hat mathbf x e i hat mathbf v frac theta 2 hat mathbf x cos 2 frac theta 2 mathbf i left hat mathbf x hat mathbf v hat mathbf v hat mathbf x right cos frac theta 2 sin frac theta 2 hat mathbf v hat mathbf x hat mathbf v sin 2 frac theta 2 recognizing that i x v v x 2 i x v displaystyle mathbf i hat mathbf x hat mathbf v hat mathbf v hat mathbf x 2 mathbf i hat mathbf x wedge hat mathbf v and that v̂x̂v̂ is the reflection of x̂ about the plane perpendicular to v̂ gives a geometric interpretation to the rotation operation the rotation preserves the components that are parallel to v̂ and changes only those that are perpendicular the terms are then computed v x v 1 3 x 2 y 2 z 2 i x v 2 i 1 3 x y x z 2 3 y z displaystyle begin aligned hat mathbf v hat mathbf x hat mathbf v frac 1 3 left hat mathbf x 2 hat mathbf y 2 hat mathbf z right 2 mathbf i hat mathbf x wedge hat mathbf v 2 mathbf i frac 1 sqrt 3 left hat mathbf x hat mathbf y hat mathbf x hat mathbf z right frac 2 sqrt 3 left hat mathbf y hat mathbf z right end aligned the result of the rotation is then x x cos 2 θ 2 1 3 sin 2 θ 2 2 3 y sin θ 2 sin θ 2 3 cos θ 2 2 3 z sin θ 2 sin θ 2 3 cos θ 2 displaystyle hat mathbf x hat mathbf x left cos 2 frac theta 2 frac 1 3 sin 2 frac theta 2 right frac 2 3 hat mathbf y sin frac theta 2 left sin frac theta 2 sqrt 3 cos frac theta 2 right frac 2 3 hat mathbf z sin frac theta 2 left sin frac theta 2 sqrt 3 cos frac theta 2 right a simple check on this result is the angle θ 2 3 π such a rotation should map x̂ to ŷ indeed the rotation reduces to x x 1 4 1 3 3 4 2 3 y 3 2 3 ...