Arbitrary Axis Rotations with Vector Algebra CS418 Computer Graphics - - PowerPoint PPT Presentation

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arbitrary axis rotations with vector algebra
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Arbitrary Axis Rotations with Vector Algebra CS418 Computer Graphics - - PowerPoint PPT Presentation

Mar 20, 2023 488 likes •625 views

Arbitrary Axis Rotations with Vector Algebra CS418 Computer Graphics John C. Hart Vector Algebra Forget homogenous coordinates for the moment Simple vectors, e.g. a = ( a x , a y , a z ), b = ( b x , b y , b z ) Vector Algebra

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SLIDE 1

Arbitrary Axis Rotations with Vector Algebra

CS418 Computer Graphics John C. Hart

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SLIDE 2

Vector Algebra

  • Forget homogenous coordinates for the moment
  • Simple vectors, e.g. a = (ax,ay,az), b = (bx,by,bz)
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SLIDE 3

Vector Algebra

  • Forget homogenous coordinates for the moment
  • Simple vectors, e.g. a = (ax,ay,az), b = (bx,by,bz)
  • Length: ||a|| = sqrt(ax

2 + ay 2 + az 2)

  • Normalizing a vector (a/||a||) makes it unit length
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SLIDE 4

Vector Algebra

  • Forget homogenous coordinates for the moment
  • Simple vectors, e.g. a = (ax,ay,az), b = (bx,by,bz)
  • Length: ||a|| = sqrt(ax

2 + ay 2 + az 2)

  • Normalizing a vector (a/||a||) makes it unit length
  • Dot product

a⋅b = axbx + ayby + azbz = ||a|| ||b|| cos θ a⋅a = ||a||2

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SLIDE 5

Vector Algebra

  • Forget homogenous coordinates for the moment
  • Simple vectors, e.g. a = (ax,ay,az), b = (bx,by,bz)
  • Length: ||a|| = sqrt(ax

2 + ay 2 + az 2)

  • Normalizing a vector (a/||a||) makes it unit length
  • Dot product

a⋅b = axbx + ayby + azbz = ||a|| ||b|| cos θ a⋅a = ||a||2

  • Cross product

a × b = (aybz – azby, azbx – axbz, axby – aybx) ||a × b|| = ||a|| ||b|| sin θ

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SLIDE 6

Arbitrary Axis Rotation

  • Rotations about x, y and z axes
  • Rotation * rotation = rotation
  • Can rotate about any axis direction

x y z v

θ

p p’

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SLIDE 7

Arbitrary Axis Rotation

  • Rotations about x, y and z axes
  • Rotation * rotation = rotation
  • Can rotate about any axis direction
  • Can do simply with vector algebra

– Ensure ||v|| = 1 x y z v

θ

p p’

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SLIDE 8

Arbitrary Axis Rotation

  • Rotations about x, y and z axes
  • Rotation * rotation = rotation
  • Can rotate about any axis direction
  • Can do simply with vector algebra

– Ensure ||v|| = 1 – Let o = (p⋅v)v x y z v

θ

p p’

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SLIDE 9

Arbitrary Axis Rotation

  • Rotations about x, y and z axes
  • Rotation * rotation = rotation
  • Can rotate about any axis direction
  • Can do simply with vector algebra

– Ensure ||v|| = 1 – Let o = (p⋅v)v – Let a = p – o x y z v

θ

p p’ a

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SLIDE 10

Arbitrary Axis Rotation

  • Rotations about x, y and z axes
  • Rotation * rotation = rotation
  • Can rotate about any axis direction
  • Can do simply with vector algebra

– Ensure ||v|| = 1 – Let o = (p⋅v)v – Let a = p – o – Let b = v × a, (note that ||b||=||a||) x y z v

θ

p p’ a b

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SLIDE 11

Arbitrary Axis Rotation

  • Rotations about x, y and z axes
  • Rotation * rotation = rotation
  • Can rotate about any axis direction
  • Can do simply with vector algebra

– Ensure ||v|| = 1 – Let o = (p⋅v)v – Let a = p – o – Let b = v × a, (note that ||b||=||a||) – Then p’ = o + a cos q + b sin q x y z v

θ

p p’ a b a b

θ

p p’

||a||cosθ ||a||sinθ

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SLIDE 12

Arbitrary Axis Rotation

  • Rotations about x, y and z axes
  • Rotation * rotation = rotation
  • Can rotate about any axis direction
  • Can do simply with vector algebra

– Ensure ||v|| = 1 – Let o = (p⋅v)v – Let a = p – o – Let b = v × a, (note that ||b||=||a||) – Then p’ = o + a cos q + b sin q

  • Simple solution to rotate a single point
  • Difficult to generate a rotation matrix

to rotate all vertices in a meshed model x y z v

θ

p p’ a b a b

θ

p p’

||a||cosθ ||a||sinθ