3D Mathematics

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3D Mathematics

• CIS 488/588
• Bruce R. Maxim
• UM-Dearborn

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2D Polar Coordinates

• Converting p(r,?) to p(x,y)
• x r cos(?)
• y r sin(?)
• Converting p(x,y) to p(r,?)
• x2 y2 r2
• r sqrt(x2 y2)
• ? tan-1(y/x)

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3D Coordinates

• Left-handed system
• Negative z axis is points toward viewer
• LaMothe uses LHS for most examples
• Right-handed system
• Positive z axis point toward viewer

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3D Space
Y
Y
Z
Z
X
X
Right-handed system
Left-handed system
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3D Cylindrical Coordinates

• Converting p(r,?,z) to p(x,y,z)
• x r cos(?)
• y r sin(?)
• z z
• Converting p(z,x,y) to p(r,?,z)
• r sqrt(x2 y2)
• ? tan-1(y/x)
• z z

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3D Spherical Coordinates

• Converting p(?,?,?) to p(x,y,z)
• x ? sin(?) cos(?)
• y ? sin(?) sin(?)
• z ? cos(?)
• Converting p(x,y,z) to p(?,?,?)
• r sqrt(x2 y2 )
• ? sqrt(x2 y2 z2)
• ? sin-1(r/?) or ? cos-1(z/?)
• ? tan-1(y/x)

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Radians

• C math library uses radians rather than degrees
  for the arguments to the trig functions
• 1 radian 57.296 degrees
• 1 degree 0.0175 radians
• 2? radians 360 degrees
• ? radians 180 degrees

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Trigonometric Functions

• hypotenuse2 adjacent2 opposite2
• r2 x2 y2
• cos(?) adj/hyp x/r 0 lt ? lt 2 ?
• sin(?) opp/hyp y/r 0 lt ? lt 2 ?
• tan(?) opp/adj y/x - ?/2 lt ? lt ?/2
• ? cos-1(x) -1 lt x lt 1
• ? sin-1(x) -1 lt x lt 1
• ? tan-1(x) -? lt x lt ?

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Use Trig Identities

• csc(?) 1/sin(?)
• sec(?) 1/cos(?)
• cot(?) 1/tan(?)
• sin2(?) cos2(?) 1
• sin(?) cos(? - ?/2)
• sin(-?) -sin(?)
• cos(-?) cos(?)

• sin(? ?)
• sin(?)cos(?) cos(?)sin(?)
• cos(? ?)
• cos(?)cos(?) - sin(?)sin(?)
• sin(? - ?)
• sin(?)cos(?) - cos(?)sin(?)
• cos(? - ?)
• cos(?)cos(?) sin(?)sin(?)

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Vectors - 1

• Vector u from p1 (initial point) to p2 (terminal
  point) also written p1p2
• u p2 p1 (x2 x1, y2 y1) ltux, uygt
• In 3D vector space the initial point is always
  (0, 0, 0)
• Length of 2D vector u ltux, uygt
• u sqrt(ux2 uy2)
• Length of 3D vector u ltux, uy, uzgt
• u sqrt(ux2 uy2 uz2)

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Vectors - 2

• Vector u normalization
• n n / n
• Scalar multiplication vector
• u ltux, uygt and k is real constant
• u k ltux, uygt ltk ux, k uygt
• Vector addition
• u v ltux, uygt ltvx, vygt
• ltux vx , uy vygt

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Vectors - 3

• Vector subtraction
• u - v ltux, uygt - ltvx, vygt
• ltux - vx , uy - vygt
• Vector multiplication
• u ? v ltux vx , uy vygt
• Dot product (can compute vector angle)
• u ? v ux vx uy vy
• u ? v u v cos(?)

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Vectors - 4

• Length of Parallel Projection of u onto v
• Projv u (u ? v) v / u v
• Dot Product Multiplicative Laws
• u ? v v ? u
• u ? (v w) (u ? v u ? w)
• k (u ? v) (k u) ? v (u ? (k v))

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Vectors - 5

• Cross Product
• u x v u v sin(?) n
• Cross Product Multiplication Laws
• u x v -(v x u)
• u x (v w) (u x v) (u x w)
• (u v) x w (u x w) (v x w)
• k (u x v) (k u) x v u x (k v)

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Vectors - 6

• Normal Vector Between u and v
• n (uyvz vyuz, -uxvz vxuz, uxvy
  vxuy)
• Displacement Vector p1 to p2
• p p1 t v
• v is unit vector in direction of p1 to p2

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Matrices - 1

• Matrix Addition and Subtraction
• A B 1 5 13 7 14 12
• -2 0 5 10 3 -10
• A - B 1 5 - 13 7 -12 -2
• -2 0 5 10 -7 10
• Transpose
• 5 6
• A 2 3 AT 5 2 1
• 1 4 6 3 4

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Matrices - 2

• Scalar Multiplication
• A 1 5 2A 21 25
• 3 0 23 20
• Matrix Multiplication
• A 2 3 B 5 2 1
• 1 4 6 3 4
• A B 2536 2233 2134
• 1546 1243 1144
• B A is not defined

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Matrices - 3

• Matrix Arithmetic Laws
• A B B A
• A (B C) (A B) C
• A (B C) (A B) C
• A (B C) A B A C
• k (A B) k A k B
• (A B) C A C B C
• A I I A A

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Matrices - 4

• 2x2 Determinant
• A a b Det(A) (ab bc)
• c d
• If Det(A) 0 system is not solvable
• Cramers Rule for Inverse
• A-1 1/Det(A) d b
• -c a

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Matrices - 4

• Solving 2D Linear Systems
• 3x 5y 6
• -2x 2y 4
• A 3 5 X x B 6
• -2 2 y 4
• AX B so X A-1B

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Matrices - 5

• Solving 3D Linear Systems
• A X B
• a b c j
• A d e f B k
• g h i l
• Det(A) (aei bfg cdh)
• (ceg bdi afh)

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Matrices - 5

• Solving 3D Linear Systems (cont)
• j b c
• x Det(k e f) / Det(A)
• l h i
• a j c
• y Det(d k f) / Det(A)
• g l i
• a b j
• z Det(d e k) / Det(A)
• g h l

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Homogeneous Coordinates

• In graphics work transformation as accomplished
  by matrix multiplication
• Customary to add an extra term w to convert
  point coordinates to homogeneous coordinates (w
  1 is common)
• To convert from homogeneous coordinates to point
  coordinates
• x y z w (x/w, y/w, z/w)

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2D Matrix Transformations - 1

• Translation
• 1 0 0
• x y 1 0 1 0
• dx dy 1
• Scaling
• sx 0 0
• x y 1 0 sy 0
• 0 0 1

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2D Matrix Transformations - 2

• Rotation
• cos(a) -sin(a) 0
• x y 1 sin(a) cos(a) 0
• 0 0 1
• (S R) T
• sxcos(a) sxsin(a) 0
• x y 1 sysin(a) sycos(a) 0
• dx dy 1

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3D Transformations - 1

• Translation
• x y z 1 x y z 1 1 0 0 0
• 0 1 0 0
• 0 0 1 0
• dx dy dz 1
• Constant Axis Scaling
• x y z 1 x y z 1 sx 0 0 0
• 0 sy 0 0
• 0 0 sz 0
• 0 0 0 1

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3D Transformations - 2

• Inverse Translation
• x y z 1 x y z 1 1 0 0 0
• 0 1 0 0
• 0 0 1 0
• -dx -dy -dz 1
• Inverse Constant Axis Scaling
• x y z 1 x y z 1 1/sx 0 0 0
• 0 1/sy 0 0
• 0 0 1/sz 0
• 0 0 0 1

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3D Transformations - 3

• Rotation Parallel to x-axis
• x y z 1 x y z 1 1 0 0 0
• 0 cos r sin r 0
• 0 -sin r cos r 0
• 0 0 0 1
• Rotation Parallel to y-axis
• x y z 1 x y z 1 cos r 0 -sin r 0
• 0 1 0 0
• sin r 0 cos r 0
• 0 0 0 1

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3D Transformations - 3

• Rotation Parallel to z-axis
• x y z 1 x y z 1 cos r sin r 0 0
• -sin r cos r 0 0
• 0 0 1 0
• 0 0 0 1

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Orthogonal Matrices

• Each row column of the matrix is orthogonal to
  the previous for a given basis
• Two vectors are orthogonal if
• u v 0
• An interesting property of orthogonal matrics is
  that MT M-1
• Inverse rotations is simply a matter of making
  use of the transpose of the rotation matrix

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Points and Lines

• Points
• x y w
• Lines - Slope Intercept Form
• y m x b
• m slope
• b y-intercepts
• Lines General Form
• ax by c 0
• c (mx0 y0) for any line point (x0, y0)

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3D Lines

• Consider a line passing through p0 and p1
• p0(x0, y0, z0)
• p1(x1, y1, z1)
• v ltx1-x0, y1-y0, z1-z0gt lta, b, cgt
• v v / v where v ltp1 p0gt
• Line - Parametric Vector Form
• p(x, y, z) p0 v t
• Line Explicit Form
• (x x0) / a (y y0) / b (z z0) / c

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Figure 4.38
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Planes

• Plane - Point-Normal Form
• n lta, b, cgt is the plane normal vector
• n ? p0-gtp
• lta, b, cgt ? ltx x0, y y0, z z0gt
• a (x x0) b (y y0) c (z z0) 0
• Plane - General Form
• a x b y c z d 0

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Testing for Half-Space Containing Point

• Use the following equation of the plane
• hs a (x x0) b (y y0) c (z z0)
• These are the outcomes
• If hs 0 then point lies in plane
• If hs gt 0 then point lies in positive half-space
• If hs lt 0 then point lies in negative half-space
• This can also be used for a polygon point
  containment test

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Parametric Lines - 1

• Line Segment
• p p0 v t for t in a, b
• v ltvx, vygt p0-gtp1 (x1-x0, y1-y0)
• Segment with Standard Direction Vector v
• p p0 v t for t in 0, 1
• Segment with Unit Direction Vector v 1
• p p0 v t for t in 0, v

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Parametric Lines - 2

• Line Segment
• p p0 v t for t in a, b
• v ltvx, vy, vzgt (x1-x0, y1-y0, z1 z0)
• Intersection of Two Parametric Lines (same
  technique used for 2D and 3D)
• Ship1 traveling p0(x0,y0) to p1(x1,y1)
• Ship2 traveling p2(x2,y2) to p3(x3,y3)
• v1 ltx1-x0, y1-y0gt
• v2 ltx3-x2, y3-y2gt

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Parametric Lines - 3

• Intersection (continued)
• p01 p0 v1 t1 ? 0 , 1
• p23 p2 v2 t2 ? 0 , 1
• For Ship1 (1,1)-gt(8,5) and Ship2 (3,6)-gt(8,3)
• v1 lt8 1, 5 1gt lt7, 4gt
• v2 lt8 3, 3 6gt lt5, -3gt
• p01(x,y) (1,1) lt7,4gt t1
• p23(x,y) (3,6) lt5, -3gt t2

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Parametric Lines - 4

• Intersection (continued)
• Rewriting the equations
• x 1 7 t1 and y 1 4 t1
• x 3 5 t2 and y 6 3 t2
• Eliminating x and y, then solving this system
• 7t1 5t2 2
• 4t1 3t2 5
• t1 0.756 t2 0.658
• Lines intersect if t1 ? 0,1 and t2 ? 0,1

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Intersection Plane and Parametric Line

• Parametric Line with t ? 0, 1
• p(x, y, z) p0(x0,y0,z0) vltvx,vy,vzgt t
• x x0 vxt
• y y0 vyt
• z z0 vzt
• 3D Plane
• n ? (p p0) 0
• nx (x x0) ny (y y0) nz (z z0) 0
• nxx nyy nzz (-nxx0 nyy0 - nzz0) 0

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Intersection Plane and Parametric Line

• Solving for t
• t -(ax0 by0 cz0 d)/
• (avx bvy cvz)
• a nx
• b ny
• c nz
• d (-nxx0 nyy0 - nzz0)
• All this was so that we can do 3D collision
  detection

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Complex Numbers - 1

• Let i sqrt(-1) then sqrt(-9) 3i
• Complex numbers have the form
• z (a b i)
• a is the real part and b is the imaginary part
• Complex numbers of the form (a,b) can be graphed
  on the complex plane, a is the real axis (x) and
  b is the imaginary axis (y)

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Complex Numbers - 2

• Scalar Multiplication
• k z k (a bi) ka kbi
• Addition
• z1 a b i
• z2 c d i
• z1 z2 (ac) (b d) i
• Additive Indentity
• (0 0i)
• Additive Inverse
• (a b i) (-a b i) (0 0i)

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Complex Numbers - 3

• Multiplication
• z1 z2 (ac bd) (ad bc) i
• Division
• z1/z2 (a b i) / (c d i)
• Need to multiply by conjugate of denominator
• (a b i) / (c d i) (c -
  d i) / (c - d i)
• (ac bd)/( a2 b2) (bc
  ad)/( a2 b2) i
• Multiplicative Inverse
• 1/z 1/(a b i) a/( a2 b2) b/( a2 b2)
  i

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Complex Numbers - 4

• Vector Representation
• z a b 1
• z lta, bgt
• Norm of Complex Number
• z sqrt(z z) sqrt(a2 b2)

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Quaterions - 1

• Hyper-Complex Numbers of the Form
• q q0 q1 i q2 j q3 k
• or
• q q0 qv, where qv q1 i q2 j q3 k
• i lt1,0,0gt, jlt0,1,0gt, klt0,0,1gt
• ltq0,q1,q2,q3gt are all real numbers
• i,j,k are imaginary numbers forming the basis of
  the quaterion
• i2 j2 k2 -1 ijk

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Quaterions - 2

• Quaternion Addition
• q q0 qv
• p p0 pv
• p q (q0 p0) (qv pv)
• lt3,4,5,6gt lt-5,2,2,-3gt lt-2,6,7,3gt
• Additive Inverse
• - q -q0 - qv
• q - q lt0,0,0,0gt

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Quaternions - 3

• Quaternion Multiplication
• p p0 p1 i p2 j p3 k
• q q0 q1 i q2 j q3 k
• p q (p0q0 - (pv qv))
• (p0qv q0pv pv x qv)
• Multiplicative Identity
• q1 1 0 i 0 j 0 k
• q q1 q1 q q

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Quaternions - 4

• Quaternion Multiplication
• q q0 q1 i q2 j q3 k q0 qv
• q q0 q1 i q2 j q3 k q0 (- qv)
• q q q02 q12 q22 q32
• Norm of a Quaternion
• q sqrt(q02 q12 q22 q32) sqrt(q q)
• q2 (q q)
• Quaternion Inverse
• q-1 q / q2
• q q-1 1 q-1 q
• for unit Quaternion q-1 q

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Quaternion Uses

• Can be for ordinary 3D rotatiosn and
  interpolation of rotation
• LaMothe uses quaternions to interpolate between
  different camera angles
• Quaternions with their 4D character can handle
  rotation of a 3D object around an arbitrary axis
  more easily than vectors
• So if you have lots of matrix multiplication in a
  loop switch to quaternions

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Figure 4.44
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Quaternion Rotation

• v ltx,y,zgt
• vq lt0,x,y,zgt note q0 0
• q is unit quaternion
• Right-handed System
• vq q vq q clockwise rotation
• vq q vq q counter-clockwise rotation
• Left-handed System
• vq q vq q clockwise rotation
• vq q vq q counter-clockwise rotation

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Figure 4.45
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Conversion of Axis and Angle

• For a unit quaternion
• vq is the vector being rotated
• q q0 qv cos(?/2) sin(?/2) vq
• qv is the axis of rotation sin(?/2) vq
• q0 contains angle of rotation cos(?/2)
• If you dont have an vector to rotate around you
  can compute a product like
• qfinal qz0 qy0 qx0
• roll yaw pitch

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Euler Angles for Quaternion - 1

• qx0 cos(x0 / 2) sin(x0 / 2) i 0 j 0
  k
• q0 cos(x0 / 2) , qv ltsin(x0 / 2) , 0, 0gt
• qy0 cos(y0 / 2) 0 i sin(y0 / 2 j 0
  k
• q0 cos(y0 / 2) , qv lt0, sin(y0 / 2) , 0gt
• qz0 cos(z0 / 2) 0 i 0 j sin(z0 / 2)
  k
• q0 cos(z0 / 2) , qv lt0, 0, sin(z0 / 2)gt
• Rotation
• vq(0,x,y,z) (qz0 qy0 qx0) vq (qz0
  qy0 qx0)
• vq lt0,y,y,zgt

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Euler Angles for Quaternion - 2

• Each of the factors qx? qy? qz?
• q(i,j,k) cos(? / 2) sin(x0 / 2) (i , j , k)
• The product qz? qy? qx? ends up as
• q0 cos(z?/2) cos(y?/2) cos(x?/2)
• sin(z?/2) sin(y?/2) sin(x?/2)
• q1 cos(z?/2) cos(y?/2) sin(x?/2) -
• sin(z?/2) sin(y?/2) cos(x?/2)
• q2 cos(z?/2) sin(y?/2) cos(x?/2)
• sin(z?/2) cos(y?/2) sin(x?/2)
• q0 sin(z?/2) cos(y?/2) cos(x?/2) -
• cos(z?/2) sin(y?/2) sin(x?/2)

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Calculus Rdeview

• Pages 337 to 361