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Working With 3D Rotations Stan Melax Graphics Software Engineer, Intel Human Brain is wired for Spatial Computation Which shape is the same: a) b) c) “I don’t need to ask A childhood IQ test question for directions” Translations Rotations Agenda ● Rotations and Matrices (hopefully review) ● Combining Rotations ● Matrix and Axis Angle ● Challenges of deep Space (of Rotations) ● Quaternions ● Applications Terminology Clarification Preferred usages of various terms: Linear Angular Object Pose Position (point) Orientation A change in Pose Translation (vector) Rotation Rate of change Linear Velocity Spin also: Direction specifies 2 DOF, Orientation specifies all 3 angular DOF. Rotations Trickier than Translations Translations Rotations a then b == b then a x then y != y then x (non-commutative) ● Programming with rotations also more challenging! 2D Rotation θ Rotate [1 0] by θ about origin 1,1 [ cos(θ) sin(θ) ] θ θ sin cos θ 1,0 2D Rotation θ Rotate [0 1] by θ about origin -1,1 0,1 sin θ [-sin(θ) cos(θ)] θ θ cos 2D Rotation of an arbitrary point Rotate about origin by θ = cos θ + sin θ 2D Rotation of an arbitrary point 푥 Rotate 푦 about origin by θ 푥′, 푦′ 푥′ = 푥 cos θ − 푦 sin θ 푦′ = 푥 sin θ + 푦 cos θ 푦 푥 2D Rotation Matrix 푥 Rotate 푦 about origin by θ 푥′, 푦′ 푥′ = 푥 cos θ − 푦 sin θ 푦′ = 푥 sin θ + 푦 cos θ 푦 푥′ cos θ − sin θ 푥 푥 = 푦′ sin θ cos θ 푦 cos θ − sin θ Matrix is rotation by θ sin θ cos θ 2D Orientation 풚 Yellow grid placed over first grid but at angle of θ cos θ − sin θ sin θ cos θ 풙 Columns of the matrix are the directions of the axes. Matrix is yellow grid’s Orientation 2D Passive Transformation cos θ − sin θ 풙′, 풚′ Basis: 풙, 풚 sin θ cos θ 푥′ cos θ − sin θ 푥 = 푦′ sin θ cos θ 푦 (note: exact same math as before) 푥′ 푥 , both same point but 푦′ 푦 In different reference frames 3D Rotation around Z axis 푍 푎푥푠 푥′ cos θ − sin θ 0 푥 푥′, 푦′ 푦′ = sin θ cos θ 0 푦 푦 푧′ 0 0 1 푧 푥 Can Rotate around X and Y too 푧 푥′ 1 0 0 푥 푦 푦′ = 0 cos θ −sin θ 푦 X 푎푥푠 푥 푧′ 0 sin θ cos θ 푧 풙′, 풚′, 풛′ 푥′ cos θ 0 sin θ 푥 푦 푧 푥 풙′, 풚′, 풛′ 푦′ = 0 1 0 푦 푧′ − sin θ 0 cos θ 푧 Rotating Objects (changing orientation) 0 0 1 1 0 0 0 −1 0 0 1 0 0 0 −1 1 0 0 Rotations: −1 0 0 0 1 0 0 0 1 90˚ on Y 90˚ on X 90˚ on Z Orientations: 1 0 0 0 0 1 0 0 1 −1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0 0 1 −1 0 0 0 1 0 0 1 0 Matrices used for both rotations and orientations Quat rotations: 0 .7 0 .7 .7 0 0 .7 0 0 .7 .7 orientations: 0 0 0 1 0 0 .7 .7 [.5 .5 .5 .5] 0 .7 .7 0 Row vs Column Conventions OpenGL and most math books use column vectors: C O v’ = M v = B A v L U M N Some engines, APIs (DirectX) use row convention: v’ = v MT = v AT BT All the same. Combining Rotations Combine a sequence of Rotations A,B,… Rotate v by A, then B, then C... = C ( B ( Av )) Mathematically we know C ( B ( A v )) == ( C B A ) v So with matrix-matrix multiplication let: R = C B A R is a single rotation that is the same as rotating by A, then by B then C. Multiplication Order: top of page W Y 90˚ on 90˚ on O right side World Y “World” X X R of page L Z D 푨푤표푟푙푑 푩푤표푟푙푑 World Coordinate Frame Y L 푨푙표푐푎푙 푩푙표푐푎푙 O X 90˚ on 90˚ on C Local Y “Local” Z A Z (dice side 2) (dice side 3) L Dice Coordinate Frame Multiplication Order: Math Equations W 90˚ on 90˚ on O World Y “World” X R = * * L D 푨푤표푟푙푑 푩푤표푟푙푑 푑푐푒푛푒푤 = 푩푤표푟푙푑 ∗ 푨푤표푟푙푑 ∗ 푑푐푒 = * * L 푨푙표푐푎푙 푩푙표푐푎푙 O 90˚ on 90˚ on C 푑푐푒 = 푑푐푒 ∗ 푨 ∗ 푩 Local Y “Local” Z 푛푒푤 푙표푐푎푙 푙표푐푎푙 A (dice side 2) (dice side 3) L Both produce: 120˚ on [1 1 1] Example When to use Local frame ● Player “pulls up” on flight stick. ● Pitch upward about object wing (x) axis. ● World x irrelevant ● Multiply rotation (about x) on the right hand Math: 1 0 0 0 cos θ −sin θ side 0 sin θ cos θ = * Sidenote: a point doesn’t 푐푙푚푏푛푔 푐푟푢푠푒푛푔 푝푡푐ℎ_푢푝 have an orientation, so never = ∗ do this for points. 표푟푒푛푡푎푡표푛 표푟푒푛푡푎푡표푛 푟표푡푎푡표푛 Find Rotation R Between Orientations A and B need to be more specific ● Have an object with orientation A, what rotation R will change it to have orientation B? 푹 = 푩푨−ퟏ ● Given a direction v in reference frame A, what rotation R will show how v points according to B? 푹 = 푩−ퟏ푨 Be aware of all the details of the problem to be solved. Rotating (Reorienting) a Rotation 푟표푡 Machine that rotates an object by 푟표푡 : Apply 45˚ Tilt to the Machine: Rotating a Rotation – Its Different Neither of these 180 푟표푡 45 푡푙푡 multiplication sequences work 45 푡푙푡 180 푟표푡 Rotating a Rotation: Decompose Steps Tilted How to calculate Machine: what this new rotation will be? Rotate duck into and back out of the machine’s reference frame: Same Result! Rotating a Rotation: The Mathematics U P Initial R equation I : = * G H 풏풆풘 푑푢푐푘′ 푑푢푐푘 T = 푟표푡 ∗ 표푟푒푛푡푎푡표푛 표푟푒푛푡푎푡표푛 equation w tilt T I L : = * * * T E D 풏풆풘 푑푢푐푘′′ = 푡푙푡 ∗ 푟표푡 ∗ 푡푙푡−1 ∗ 푑푢푐푘 Rotating a Rotation: The Mathematics Now drop the duck… = * * −1 푟표푡푡푖푙푡푒푑 = 푡푙푡 ∗ 푟표푡 ∗ 푡푙푡 Matrix & Axis Angle 3D Orientation / Rotation Matrix R 풛 푅푥푥 푅푦푥 푅푧푥 푹 = 푅푥푦 푅푦푦 푅푧푦 푅푥푧 푅푦푧 푅푧푧 푹풚 풚 푹풙 General form of Rotation Matrix: • Orthonormal basis: 푹풙 푹풚 푹풛 풙 • 푹풛 = 푹풙 × 푹풚 etc. • Determinant(R)==1 푹풛 • Inverse(R) == Transpose(R) • Has a corresponding axis of rotation Rotation Matrix – Finding its Axis Angle 풛 풛 푹풚 푹풙 휽 풚 풚 푹풛 풙 풙 푅푥푥 푅푦푥 푅푧푥 푹 = 푅푥푦 푅푦푦 푅푧푦 풂풙풊풔, 휽 푅푥푧 푅푦푧 푅푧푧 풂풙풊풔 will be an eigenvector of R Example of corresponding Matrix and Axis Angle 풛 풛 푹풚 휽 푹풙 풚 풚 푹풛 풙 풙 0 0 1 푹 = 1 0 0 풂풙풊풔, 휽 = ퟏ ퟏ ퟏ , ퟏퟐퟎ° 0 1 0 To check, verify: 풂풙풊풔 == 푹 ∗ 풂풙풊풔 Matrix from general axis a, angle θ 풂 푎푥푠 Matrix for a,θ ? Matrix from general axis a, angle θ 풂 푎푥푠 ? [푥, 푦, 푧] How would axis/angle rotate a point [푥, 푦, 푧]? Matrix from general axis a, angle θ • Find b,c unit vecs a,b,c orthonormal 풂 푎푥푠 풂 = 풃 × 풄 , 풄 = 풂 × 풃, 풃 = 풄 × 풂 [푥, 푦, 푧] Matrix from general axis a, angle θ • Find b,c unit vecs a,b,c orthonormal 풂 푎푥푠 풂 = 풃 × 풄 , 풄 = 풂 × 풃, 풃 = 풄 × 풂 • Get [xyz] as weighted sum of a,b,c [푥, 푦, 푧] Matrix from general axis a, angle θ • Find b,c unit vecs a,b,c orthonormal 풂 푎푥푠 풂 = 풃 × 풄 , 풄 = 풂 × 풃, 풃 = 풄 × 풂 • Get [xyz] as weighted sum of a,b,c [푥, 푦, 푧] • Stuff along a stays the same, • Results along b & c based on sinθ and cosθ portions along b & c 푥′ 푥 푥 푥 푥 푥 푦′ = 풂 풂 ∙ 푦 + 풃 풃 ∙ 푦 cos θ − 풄 ∙ 푦 sin θ + 풄 풃 ∙ 푦 sin θ + 풄 ∙ 푦 cos θ 푧′ 푧 푧 푧 푧 푧 Matrix from general axis a, angle θ • Find b,c unit vecs a,b,c orthonormal 풂 푎푥푠 풂 = 풃 × 풄 , 풄 = 풂 × 풃, 풃 = 풄 × 풂 • Get [xyz] as weighted sum of a,b,c [푥, 푦, 푧] • Stuff along a stays the same, • Results along b & c based on sinθ and cosθ portions along b & c 푥′ 푥 푥 푥 푥 푥 푦′ = 풂 풂 ∙ 푦 + 풃 풃 ∙ 푦 cos θ − 풄 ∙ 푦 sin θ + 풄 풃 ∙ 푦 sin θ + 풄 ∙ 푦 cos θ 푧′ 푧 푧 푧 푧 푧 푥′ 푥 “still need method 푇 푇 푇 푇 푇 푦′ = 풂풂 + 풃풃 cos θ − 풃풄 sin θ + 풄풃 sin θ + 풄풄 cos θ 푦 for finding b,c” 푧′ 푧 Matrix from general axis a, angle θ Alternatively (Equivalently): 풂 푎푥푠 [푥, 푦, 푧] Think of [b,c,a] as a 3x3 basis. 풄 ● Move/rotate into abc’s 푥′, 푦′, 푧′ reference frame. 풃 ● Do spin on ‘local’ z axis ● Rotate back out 3x3 Rotation Matrix 푥′ cos θ − sin θ 0 풃 푥 푏푥 푐푥 푎푥 cos θ − sin θ 0 푏푥 푏푦 푏푧 푥 푏 푐 푎 푐 푐 푐 푦 푦′ = 풃 풄 풂 sin θ cos θ 0 풄 푦 = 푦 푦 푦 sin θ cos θ 0 푥 푦 푧 푏푧 푐푧 푎푧 0 0 1 푎푥 푎푦 푎푧 푧 푧′ 0 0 1 풂 푧 “ok, but this math is still not concise.” Challenges with the Space of Rotations Matrix Disadvantages Great for some systems (batch rendering), but not ideal for animation, gameplay, or physics code.
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    GG303 Lecture 11 9/4/01 1 ROTATIONS (II) I Main Topics A Rotations using a stereonet B Comments on rotations using stereonets and matrices C Uses of rotation in geology (and engineering) II II Rotations using a stereonet A Best uses 1 Rotation axis is vertical 2 Rotation axis is horizontal (e.g., to restore tilted beds) B Construction technique 1 Find orientation of rotation axis 2 Find angle of rotation and rotate pole to plane (or a linear feature) along a small circle perpendicular to the rotation axis. a For a horizontal rotation axis the small circle is vertical b For a vertical rotation axis the small circle is horizontal 3 WARNING: DON'T rotate a plane by rotating its dip direction vector; this doesn't work (see handout) IIIComments on rotations using stereonets and matrices Advantages Disadvantages Stereonets Good for visualization Relatively slow Can bring into field Need good stereonets, paper Matrices Speed and flexibility Computer really required to Good for multiple rotations cut down on errors A General comments about stereonets vs. rotation matrices 1 With stereonets, an object is usually considered to be rotated and the coordinate axes are held fixed. 2 With rotation matrices, an object is usually considered to be held fixed and the coordinate axes are rotated. B To return tilted bedding to horizontal choose a rotation axis that coincides with the direction of strike. The angle of rotation is the negative of the dip of the bedding (right-hand rule! ) if the bedding is to be rotated back to horizontal and positive if the axes are to be rotated to the plane of the tilted bedding.
  • Hypercomplex Numbers

    Hypercomplex Numbers

    Hypercomplex numbers Johanna R¨am¨o Queen Mary, University of London [email protected] We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers. It has not always been easy to accept new numbers. Negative numbers were rejected for cen- turies, and complex numbers, the square roots of negative numbers, were considered impossible. Complex numbers behave like ordinary numbers. You can add, subtract, multiply and divide them, and on top of that, do some nice things which you cannot do with real numbers. Complex numbers are now accepted, and have many important applications in mathematics and physics. Scientists could not live without complex numbers. What if we take the next step? What comes after the complex numbers? Is there a bigger set of numbers that has the same nice properties as the real numbers and the complex numbers? The answer is yes. In fact, there are two (and only two) bigger number systems that resemble real and complex numbers, and their discovery has been almost as dramatic as the discovery of complex numbers was. 1 Complex numbers Complex numbers where discovered in the 15th century when Italian math- ematicians tried to find a general solution to the cubic equation x3 + ax2 + bx + c = 0: At that time, mathematicians did not publish their results but kept them secret. They made their living by challenging each other to public contests of 1 problem solving in which the winner got money and fame.
  • Euler Quaternions

    Euler Quaternions

    Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1.
  • Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space

    Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space

    Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space Murat Babaarslan and Yusuf Yayli Abstract. A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternions and spacelike constant slope surfaces in Minkowski 3-space. We show that spacelike constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit timelike quaternions with the spacelike vector parts and homothetic motions. Subsequently we give some examples to illustrate our main results. Mathematics Subject Classification (2010). Primary 53A05; Secondary 53A17, 53A35. Key words: Spacelike constant slope surface, split quaternion, homothetic motion. 1. Introduction Quaternions were discovered by Sir William Rowan Hamilton as an extension to the complex number in 1843. The most important property of quaternions is that every unit quaternion represents a rotation and this plays a special role in the study of rotations in three- dimensional spaces. Also quaternions are an efficient way understanding many aspects of physics and kinematics. Many physical laws in classical, relativistic and quantum mechanics can be written nicely using them. Today they are used especially in the area of computer vision, computer graphics, animations, aerospace applications, flight simulators, navigation systems and to solve optimization problems involving the estimation of rigid body transformations. Ozdemir and Ergin [9] showed that a unit timelike quaternion represents a rotation in Minkowski 3-space.