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RollPitchYawAngles[r]

gives the roll-pitch-yaw angles {α,β,γ} corresponding to the rotation matrix r.

RollPitchYawAngles[r,{a,b,c}]

gives the roll-pitch-yaw angles {α,β,γ} corresponding to rotation order {a,b,c}.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
Possible Issues  
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RollPitchYawAngles

RollPitchYawAngles[r]

gives the roll-pitch-yaw angles {α,β,γ} corresponding to the rotation matrix r.

RollPitchYawAngles[r,{a,b,c}]

gives the roll-pitch-yaw angles {α,β,γ} corresponding to rotation order {a,b,c}.

Details

  • RollPitchYawAngles is used to decompose into fixed axis-oriented rotations.
  • RollPitchYawAngles[r,{a,b,c}] gives angles {α,β,γ} such that RollPitchYawMatrix[{α,β,γ},{a,b,c}]r.
  • RollPitchYawAngles[r] is equivalent to RollPitchYawAngles[r,{3,2,1}], the z-y-x rotation.
  • The default z-y-x angles RollPitchYawAngles[r,{3,2,1}] decompose rotation into three steps:
  • The rotation axes a, b, and c can be any integer 1, 2, or 3, but there are only twelve combinations that are general enough to be able to specify any 3D rotation.
  • Rotations with the first and last axis repeated:
  • {3,2,3}z-y-z rotation
    {3,1,3}z-x-z rotation
    {2,3,2}y-z-y rotation
    {2,1,2}y-x-y rotation
    {1,3,1}x-z-x rotation
    {1,2,1}x-y-x rotation
  • Rotations with all three axes different:
  • {1,2,3}x-y-z rotation
    {1,3,2}x-z-y rotation
    {2,1,3}y-x-z rotation
    {2,3,1}y-z-x rotation
    {3,1,2}z-x-y rotation
    {3,2,1}z-y-x rotation (default)
  • Rotations with subsequent axes repeated may not be invertible since these are not capable of representing all possible rotations in 3D.

Examples

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Basic Examples  (2)

Get roll-pitch-yaw angles from the rotation matrix:

Wolfram Language code: m = RollPitchYawMatrix[{Pi / 4, Pi / 2, 0}];
Wolfram Language code: RollPitchYawAngles[m]

Get roll-pitch-yaw angles from the rotation matrix with the given rotation order:

Wolfram Language code: m = RollPitchYawMatrix[{Pi / 2, Pi / 3, Pi / 4}, {1, 2, 3}];
Wolfram Language code: RollPitchYawAngles[m, {1, 2, 3}]

Scope  (2)

Get roll-pitch-yaw angles from a rotation matrix:

Wolfram Language code: m = RollPitchYawMatrix[{Pi / 7, Pi / 3, Pi / 3}];
Wolfram Language code: RollPitchYawAngles[m, {3, 2, 1}]

Get roll-pitch-yaw angles from a rotation matrix with the given rotation order:

Wolfram Language code: m = RollPitchYawMatrix[{Pi / 2, Pi / 3, Pi / 4}, {1, 2, 3}];
Wolfram Language code: RollPitchYawAngles[m, {1, 2, 3}]

Properties & Relations  (1)

RollPitchYawAngles returns angles for which RollPitchYawMatrix gives the same rotation matrix:

Wolfram Language code: m1 = RollPitchYawMatrix[{π / 3, π / 2, π / 4}]; m2 = RollPitchYawMatrix[RollPitchYawAngles[m1]];
Wolfram Language code: m1 == m2

The angles need not be the same:

Wolfram Language code: a1 = {π / 2, π, π / 3}; m1 = RollPitchYawMatrix[a1];
Wolfram Language code: a2 = RollPitchYawAngles[m1]

However, both sets of angles produce the same rotation matrix:

Wolfram Language code: RollPitchYawMatrix[a1] == RollPitchYawMatrix[a2]

Possible Issues  (1)

RollPitchYawMatrix allows equal consecutive axes, and this generates a rotation matrix:

Wolfram Language code: m = RollPitchYawMatrix[{Pi, Pi / 2, Pi / 8}, {1, 1, 2}]
Wolfram Language code: {OrthogonalMatrixQ[m], Det[m]}//Simplify

However, RollPitchYawAngles requires consecutive axes to be distinct:

Wolfram Language code: RollPitchYawAngles[m, {1, 1, 2}]

This is because with consecutive axes equal, some rotation matrices cannot be represented:

Wolfram Language code: m1 = RollPitchYawMatrix[{Pi / 3, 0, 0}, {3, 2, 1}]
Wolfram Language code: m2 = RollPitchYawMatrix[{α, β, γ}, {1, 1, 2}]
Wolfram Language code: FindInstance[And@@Thread[m1 == m2], {α, β, γ}]
Wolfram Research (2015), RollPitchYawAngles, Wolfram Language function, https://reference.wolfram.com/language/ref/RollPitchYawAngles.html.

Text

Wolfram Research (2015), RollPitchYawAngles, Wolfram Language function, https://reference.wolfram.com/language/ref/RollPitchYawAngles.html.

CMS

Wolfram Language. 2015. "RollPitchYawAngles." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RollPitchYawAngles.html.

APA

Wolfram Language. (2015). RollPitchYawAngles. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RollPitchYawAngles.html

BibTeX

@misc{reference.wolfram_2026_rollpitchyawangles, author="Wolfram Research", title="{RollPitchYawAngles}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/RollPitchYawAngles.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_rollpitchyawangles, organization={Wolfram Research}, title={RollPitchYawAngles}, year={2015}, url={https://reference.wolfram.com/language/ref/RollPitchYawAngles.html}, note=[Accessed: 13-June-2026]}