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Quaternions - Finding the most natural and compact way to present rotation and orientations ... Length Preserving (Isometric) Reflection Preserving. Orthonormal ...

Finding the most natural and compact way to present rotation and orientations ... Length Preserving (Isometric) Reflection Preserving. Orthonormal ...

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Quaternions - Quaternions. An easy approach to determine the rotation matrix around arbitrary axis ... Quaternion is Complex. Consist of a 3D vector and one scalar ...

Quaternions. An easy approach to determine the rotation matrix around arbitrary axis ... Quaternion is Complex. Consist of a 3D vector and one scalar ...

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Quaternions - This implies that there is possibly extra or redundant information in the axis/angle format ... Those extra 6 DOFs manifest themselves as 3 scales (x, y, and z) ...

This implies that there is possibly extra or redundant information in the axis/angle format ... Those extra 6 DOFs manifest themselves as 3 scales (x, y, and z) ...

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Quaternions - Quaternions can be converted to / from Euler angles and. rotation matrices ... An alternative to the correction term is to re-normalize. the quaternion periodically ...

Quaternions can be converted to / from Euler angles and. rotation matrices ... An alternative to the correction term is to re-normalize. the quaternion periodically ...

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Quaternions - Rotations are defined as rotations about the x, y and z axes. ... Useful in animation because it is easy to interpolate rotations ...

Rotations are defined as rotations about the x, y and z axes. ... Useful in animation because it is easy to interpolate rotations ...

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Quaternions - Quaternion Multiplication ... We need this when we want to apply a quaternion rotation to a vector ... vector, V with the quaternion, Q, we take the following ...

Quaternion Multiplication ... We need this when we want to apply a quaternion rotation to a vector ... vector, V with the quaternion, Q, we take the following ...

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Quaternionen PowerPoint PPT Presentation

Quaternionen - Quaternionen Eugenia Schwamberger Inhalt : Komplexe Zahlen Quaternionen Einheitsquaternionen Rotation mit Quaternionen Beispiel Matrix Conversion Anhang A: K rper ...

Quaternionen Eugenia Schwamberger Inhalt : Komplexe Zahlen Quaternionen Einheitsquaternionen Rotation mit Quaternionen Beispiel Matrix Conversion Anhang A: K rper ...

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Quaternionic%20Splines%20of%20Camera%20Paths PowerPoint PPT Presentation

Quaternionic%20Splines%20of%20Camera%20Paths - Quaternionic Splines of Paths Robert Shuttleworth Youngstown State University Professor George Francis, Director illiMath2001 Order of Events History of the ...

Quaternionic Splines of Paths Robert Shuttleworth Youngstown State University Professor George Francis, Director illiMath2001 Order of Events History of the ...

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Quaternions Chapter 11 PowerPoint PPT Presentation

Quaternions Chapter 11 - Quaternions (Chapter 11) A quaternion is an extension of the complex plane to a four ... The multiplicative inverse of a quaternion q is denoted q-1. ...

Quaternions (Chapter 11) A quaternion is an extension of the complex plane to a four ... The multiplicative inverse of a quaternion q is denoted q-1. ...

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Quaternions - Cartesian coordinates (x,y,z) are an easy and natural means of representing a ... of any fixed poles or longitude/latitude, because no matter which direction they ...

Cartesian coordinates (x,y,z) are an easy and natural means of representing a ... of any fixed poles or longitude/latitude, because no matter which direction they ...

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Quaternions Multivariate Vectors PowerPoint PPT Presentation

Quaternions Multivariate Vectors - ... are easy to see from diagrams. Wave theory: What is the speed ... Bohr is said to have developed his quantum atomic theory in 1913 immediately upon seeing it. ...

... are easy to see from diagrams. Wave theory: What is the speed ... Bohr is said to have developed his quantum atomic theory in 1913 immediately upon seeing it. ...

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Animating Rotations and Using Quaternions PowerPoint PPT Presentation

Animating Rotations and Using Quaternions - Euler Angles ... Euler Angles. . ?1. ?2. Can you do it? Line 1: P1 = (0,1,0) P2 = (1,0,1) ... Alternative to Euler angles for specifying orientation ...

Euler Angles ... Euler Angles. . ?1. ?2. Can you do it? Line 1: P1 = (0,1,0) P2 = (1,0,1) ... Alternative to Euler angles for specifying orientation ...

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Quaternions Multivariate Vectors - And we derive a potential energy relation, not a kinetic one. Kinetic theory of gases ... The potential energy equation applies to an orbit already in existence. ...

And we derive a potential energy relation, not a kinetic one. Kinetic theory of gases ... The potential energy equation applies to an orbit already in existence. ...

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Quaternions and Interpolation of Rotations PowerPoint PPT Presentation

Quaternions and Interpolation of Rotations - ... the quaternion multiplication is ... is a quaternion with zero real part (like v) ... unit quaternion multiplication corresponds to composition of rotations ...

... the quaternion multiplication is ... is a quaternion with zero real part (like v) ... unit quaternion multiplication corresponds to composition of rotations ...

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Quaternions Multivariate Vectors - Fundamental physics only concerned with fermions and their interactions via gauge bosons. ... The fundamental equation for the fermion is the Dirac equation, ...

Fundamental physics only concerned with fermions and their interactions via gauge bosons. ... The fundamental equation for the fermion is the Dirac equation, ...

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Quaternions Design and Creation of Virtual Environments CISE 6930/4930 Section 6589/3146 PowerPoint PPT Presentation

Quaternions Design and Creation of Virtual Environments CISE 6930/4930 Section 6589/3146 - To create a unit quaternion that represents a rotation of degrees about an ... Remember this is quaternion multiplication not matrix multiplication! ...

To create a unit quaternion that represents a rotation of degrees about an ... Remember this is quaternion multiplication not matrix multiplication! ...

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Attitude Determination PowerPoint PPT Presentation

Attitude Determination - Quaternions. A quaternion consists of four composants. Where i,j ... Quaternions. A quaternion can be thought of as a 4 dimensional. vector with unit length: ...

Quaternions. A quaternion consists of four composants. Where i,j ... Quaternions. A quaternion can be thought of as a 4 dimensional. vector with unit length: ...

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INCIDENCE GEOMETRIES PowerPoint PPT Presentation

INCIDENCE GEOMETRIES - Geometry from Quaternions. Example: Q = { 1,-1, i,-i, j,-j, k,-k} ... Quaternions Examle of Rank 4 Geometry. Levi graph was an octahedron. Notation: ...

Geometry from Quaternions. Example: Q = { 1,-1, i,-i, j,-j, k,-k} ... Quaternions Examle of Rank 4 Geometry. Levi graph was an octahedron. Notation: ...

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The Topological Structure of the Unit Octonions and Games with Quantum Communication PowerPoint PPT Presentation

The Topological Structure of the Unit Octonions and Games with Quantum Communication - The quaternions are spanned by the identity element 1 and ... A unit quaternion has length 1. Set of unit quaternions is S3 C2. Quaternionic multiplication: ...

The quaternions are spanned by the identity element 1 and ... A unit quaternion has length 1. Set of unit quaternions is S3 C2. Quaternionic multiplication: ...

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Find the Best Trademark Registration Cost PowerPoint PPT Presentation

Find the Best Trademark Registration Cost - Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in ... then promptly carved this equation into the side of the nearby Brougham Bridge ...

Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in ... then promptly carved this equation into the side of the nearby Brougham Bridge ...

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3D Mechanics PowerPoint PPT Presentation

3D Mechanics - Quaternions ... Quaternions are a four-dimensional extension to complex numbers: ... The complement of a quaternion is being defined as: ...

Quaternions ... Quaternions are a four-dimensional extension to complex numbers: ... The complement of a quaternion is being defined as: ...

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3D Mechanics II PowerPoint PPT Presentation

3D Mechanics II - quaternions), and in comparing the efficiency of the multi-bond graph solution ... The Cardan angles or the quaternion vector respectively are integrated from the ...

quaternions), and in comparing the efficiency of the multi-bond graph solution ... The Cardan angles or the quaternion vector respectively are integrated from the ...

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Technical Background PowerPoint PPT Presentation

Technical Background - Image space. Screen space. transformations. transformations. perspective ... Rw (g) Rv (b)Ru (a) = Rx (a)Ry (b) Rz (g) Rick Parent - CIS681. Quaternions ...

Image space. Screen space. transformations. transformations. perspective ... Rw (g) Rv (b)Ru (a) = Rx (a)Ry (b) Rz (g) Rick Parent - CIS681. Quaternions ...

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2. From Groups to Surfaces PowerPoint PPT Presentation

2. From Groups to Surfaces - Quaternions H. - Exercises ... Exercise: Represent quaternions by complex matrices (matrix addition and matrix multiplication) ...

Quaternions H. - Exercises ... Exercise: Represent quaternions by complex matrices (matrix addition and matrix multiplication) ...

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Dr. Scott Schaefer PowerPoint PPT Presentation

Dr. Scott Schaefer - Quaternions History. Hamilton attempted to extend complex numbers from 2D to 3D... impossible ... The quaternion representing rotation about the unit axis v by is ...

Quaternions History. Hamilton attempted to extend complex numbers from 2D to 3D... impossible ... The quaternion representing rotation about the unit axis v by is ...

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CV - Affine transformation (rotation, scaling, etc.) is every transformation which is ... Non-affine (projection matrices). Quaternions. ...

Affine transformation (rotation, scaling, etc.) is every transformation which is ... Non-affine (projection matrices). Quaternions. ...

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The synthesis of motion capture data PowerPoint PPT Presentation

The synthesis of motion capture data - ... simple and dual quaternions efficiency. 3. Agenda. Overview ... bone, skeleton, animation, quaternion classes involved. Delphi import packages, manual included ...

... simple and dual quaternions efficiency. 3. Agenda. Overview ... bone, skeleton, animation, quaternion classes involved. Delphi import packages, manual included ...

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Early - Thomas Sowell, a senior fellow at the Hoover Institute, Stanford University ... Interlude 2: More Games!! PART 3: Advanced Early Number Concepts Quaternions! ...

Thomas Sowell, a senior fellow at the Hoover Institute, Stanford University ... Interlude 2: More Games!! PART 3: Advanced Early Number Concepts Quaternions! ...

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Fractal%20Research,%20Technology,%20Development,%20Design%20and%20Production%20of%20Artificial%20Fractal%20Products PowerPoint PPT Presentation

Fractal%20Research,%20Technology,%20Development,%20Design%20and%20Production%20of%20Artificial%20Fractal%20Products - Fractal Research, Technology, Development, Design and ... Fractal. Products. Fractal Geometry in Nature (Mandelbrot) Fractal Generator in quaternions (foxels) ...

Fractal Research, Technology, Development, Design and ... Fractal. Products. Fractal Geometry in Nature (Mandelbrot) Fractal Generator in quaternions (foxels) ...

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Rigid Body Dynamics I An Introduction PowerPoint PPT Presentation

Rigid Body Dynamics I An Introduction - Unit quaternions can do it with 4 ... A rotation of ? about a unit axis u can be represented by the unit quaternion: ... Unit Quaternion Usage ...

Unit quaternions can do it with 4 ... A rotation of ? about a unit axis u can be represented by the unit quaternion: ... Unit Quaternion Usage ...

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Quaternion - Certain special properties exist for quaternions numbers too. ... Quaternion Julia Fractals use the same type of generating function, f(Z) = Z ...

Certain special properties exist for quaternions numbers too. ... Quaternion Julia Fractals use the same type of generating function, f(Z) = Z ...

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COMP790072 Robotics: An Introduction PowerPoint PPT Presentation

COMP790072 Robotics: An Introduction - Rotation as quaternion. Rotation as Euler Angles. Homogeneous TXF: T=H(R,d) UNC Chapel Hill ... is composed of some matrices or quaternions. UNC Chapel Hill ...

Rotation as quaternion. Rotation as Euler Angles. Homogeneous TXF: T=H(R,d) UNC Chapel Hill ... is composed of some matrices or quaternions. UNC Chapel Hill ...

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Emerging Technologies for Games Review PowerPoint PPT Presentation

Emerging Technologies for Games Review - Euler angles, Matrices, Angle-axis form & Quaternions. Form of a quaternion with formulae. Using quaternions (formulae in grey): Conversions, addition & scaling ...

Euler angles, Matrices, Angle-axis form & Quaternions. Form of a quaternion with formulae. Using quaternions (formulae in grey): Conversions, addition & scaling ...

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Groups, Group Theory, Symmetry PowerPoint PPT Presentation

Groups, Group Theory, Symmetry - Homomorphism: two groups are homomorphic if a unidirectional correspondence exists. ... represented by quaternions are homomorphic with but not the same ...

Homomorphism: two groups are homomorphic if a unidirectional correspondence exists. ... represented by quaternions are homomorphic with but not the same ...

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Quaternionization of Maximally Entangled Two Player Quantum Games via Quantum Prisoners Dilemma PowerPoint PPT Presentation

Quaternionization of Maximally Entangled Two Player Quantum Games via Quantum Prisoners Dilemma - Let be real numbers that are coefficients of 1, i,j,k in the unit quaternion pq. ... Quaternions are a non-commutative 4 dimensional real normed division algebra. ...

Let be real numbers that are coefficients of 1, i,j,k in the unit quaternion pq. ... Quaternions are a non-commutative 4 dimensional real normed division algebra. ...

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Emerging Technologies for Games Animation: Interpolation PowerPoint PPT Presentation

Emerging Technologies for Games Animation: Interpolation - Rotation: Euler angles, a matrix or a quaternion. Scaling: vector / scalars or a matrix ... Normalising quaternions is trivial ...

Rotation: Euler angles, a matrix or a quaternion. Scaling: vector / scalars or a matrix ... Normalising quaternions is trivial ...

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U - Angle between two quaternions is half the angle between the corresponding ... neighbourhood of a reference rotation form a circle on the quaternion sphere. ...

Angle between two quaternions is half the angle between the corresponding ... neighbourhood of a reference rotation form a circle on the quaternion sphere. ...

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6.837 Fall 2001 PowerPoint PPT Presentation

6.837 Fall 2001 - Quaternion Interpolation. Linear interpolation (lerp) of quaternion representation of orientations gives ... two general quaternions are multiplied by a special ...

Quaternion Interpolation. Linear interpolation (lerp) of quaternion representation of orientations gives ... two general quaternions are multiplied by a special ...

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681 Introduction to Computer Graphics PowerPoint PPT Presentation

681 Introduction to Computer Graphics - Quaternion. Rick Parent - CIS681. Interpolation. O1. O2. O 1.5. Rick Parent - CIS681. Concatenation ... Quaternions - composite transformations. Rotq(Rotp(v) ...

Quaternion. Rick Parent - CIS681. Interpolation. O1. O2. O 1.5. Rick Parent - CIS681. Concatenation ... Quaternions - composite transformations. Rotq(Rotp(v) ...

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Vector , like the word vehicle, is derived from the Latin PowerPoint PPT Presentation

Vector , like the word vehicle, is derived from the Latin - Radius' was dropped and vector' was given a place in the algebra of quaternions. ... A Quaternion is a way of representing every vector in the form (ix jy kz) ...

Radius' was dropped and vector' was given a place in the algebra of quaternions. ... A Quaternion is a way of representing every vector in the form (ix jy kz) ...

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More TRANSF, VR Programming PowerPoint PPT Presentation

More TRANSF, VR Programming - Quaternions are of the form a bi cj dk, for a, b, c, d real numbers. ... We can think of the i, j, k part of a quaternion as a 3-D vector. ...

Quaternions are of the form a bi cj dk, for a, b, c, d real numbers. ... We can think of the i, j, k part of a quaternion as a 3-D vector. ...

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681 - Introduction to Computer Graphics - ... Review of Quaternion Rotations Rotq(v) = v = q * [0,v] * q-1 Rotq(v ... Interpolation Background De Casteljau construction of Bezier curve ...

... Review of Quaternion Rotations Rotq(v) = v = q * [0,v] * q-1 Rotq(v ... Interpolation Background De Casteljau construction of Bezier curve ...

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Pr - On recommence un nombre suffisant de fois pour avoir des d tails de taille assez petite afin que l'image soit ... lectroniques Compression d images ...

On recommence un nombre suffisant de fois pour avoir des d tails de taille assez petite afin que l'image soit ... lectroniques Compression d images ...

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681 - Introduction to Computer Graphics - Double the arc. Bisect the span. Rick Parent - CIS681. Quaternion Interpolation. P n-1 ... Double(q1,q2) = q2q1-1q2. Q n =Double(P n-1, P n) Rick Parent ...

Double the arc. Bisect the span. Rick Parent - CIS681. Quaternion Interpolation. P n-1 ... Double(q1,q2) = q2q1-1q2. Q n =Double(P n-1, P n) Rick Parent ...

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Basic Kinematics PowerPoint PPT Presentation

Basic Kinematics - Basic Kinematics Spatial Descriptions and Transformations Introduction to Motion Objectives of the Lecture Learn to represent position and orientation Be able to ...

Basic Kinematics Spatial Descriptions and Transformations Introduction to Motion Objectives of the Lecture Learn to represent position and orientation Be able to ...

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Computer Animation Algorithms and Techniques PowerPoint PPT Presentation

Computer Animation Algorithms and Techniques - Computer Animation Algorithms and Techniques Interpolating Values Bezier construction using quaternion operators Need quaternion-friendly operations to interpolate ...

Computer Animation Algorithms and Techniques Interpolating Values Bezier construction using quaternion operators Need quaternion-friendly operations to interpolate ...

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Advanced Game Engineering RealTime Rendering PowerPoint PPT Presentation

Advanced Game Engineering RealTime Rendering - Illustration of the rotation transform represented. by a unit quaternion, ... slerp could be used in a straightforward fashion. ...

Illustration of the rotation transform represented. by a unit quaternion, ... slerp could be used in a straightforward fashion. ...

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It - It rations de fonctions non lin aires 1. Introduction 2. It ration de fonctions quadratiques dans les complexes 2.1 f(z) = z2, z C 2.2 f(z) = z2 + c, z C et l ...

It rations de fonctions non lin aires 1. Introduction 2. It ration de fonctions quadratiques dans les complexes 2.1 f(z) = z2, z C 2.2 f(z) = z2 + c, z C et l ...

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Orientation - Matrix to quaternion is not too bad, I just don't have room for it here ... Another possibility is to also compute the world matrices as quaternion multiplications. ...

Matrix to quaternion is not too bad, I just don't have room for it here ... Another possibility is to also compute the world matrices as quaternion multiplications. ...

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Rigid motions PowerPoint PPT Presentation

Rigid motions - Representation of Rotation. Before we go further, we should consider a few different ways of representing orientation ... Possible rotation representations ...

Representation of Rotation. Before we go further, we should consider a few different ways of representing orientation ... Possible rotation representations ...

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Orientations PowerPoint PPT Presentation

Orientations - Orientations Goal: Convenient representation of orientation of objects & characters in a scene Applications to: inverse kinematics rigid body simulation

Orientations Goal: Convenient representation of orientation of objects & characters in a scene Applications to: inverse kinematics rigid body simulation

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Range data - Range data

Range data

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Basic Kinematics - Basic Kinematics Spatial Descriptions and Transformations Introduction to Motion Objectives of the Lecture Learn to represent position and orientation Be able to ...

Basic Kinematics Spatial Descriptions and Transformations Introduction to Motion Objectives of the Lecture Learn to represent position and orientation Be able to ...

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3D orientation PowerPoint PPT Presentation

3D orientation - Quaternion. Exponential map. Joints and rotations ... Rotation of P=(0,r) about the unit vector n by an angle ? using the unit quaternion q=(s,v) ...

Quaternion. Exponential map. Joints and rotations ... Rotation of P=(0,r) about the unit vector n by an angle ? using the unit quaternion q=(s,v) ...

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Deterministic Sampling Methods for Spheres and SO(3) PowerPoint PPT Presentation

Deterministic Sampling Methods for Spheres and SO(3) - Deterministic Sampling Methods for Spheres and SO(3) Anna Yershova Steven M. LaValle Dept. of Computer Science University of Illinois

Deterministic Sampling Methods for Spheres and SO(3) Anna Yershova Steven M. LaValle Dept. of Computer Science University of Illinois

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Plateforme de calcul massivement parall PowerPoint PPT Presentation

Plateforme de calcul massivement parall - Plateforme de calcul massivement parall le volutionnaire EASEA Pierre Collet, Nicolas Lachiche, Ogier Maitre, Fr d ric Kr ger, St phane Querry

Plateforme de calcul massivement parall le volutionnaire EASEA Pierre Collet, Nicolas Lachiche, Ogier Maitre, Fr d ric Kr ger, St phane Querry

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Quaternions PowerPoint PPT Presentations