Geometric Transformation

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Geometric Transformation

• Pradondet Nilagupta

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Coordinate Systems
Left-handed
Right-handed
coordinate
coordinate
system
system
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3-D Vectors

• Have length and direction V vx, vy, vz
• Length is given by the Euclidean Norm V
• Dot Product VU vx, vy, vz ux, uy, uz
  vxux vyuy vzuz
• V U cos ß
• result is a scalar value
• Cross Product V x U vyuz - vzuy, vzux - vxuz,
  vxuy - vyux
• V x U - ( U x V)
• Direction of resulting vector depends on
  coordinate system and order of vectors but
  always perpendicular to other two vectors.

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Parametric Equation of a Line
t gt 1
Given two points P1 (x1, y1, z1), P2 ( x2,
y2, z2) x x1 t (x2 - x1) y y1 t (y2 -
y1) z z1 t (z2 - z1)
t 1
t 0
P2
t lt 0
P1

• Given a point P1 and a vector V vx, vy, vz
• x x1 t vx, y y1 t vy , z
  z1 t vz
• COMPACT FORM L P1 tP2 - P1
  or L P1 Vt

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Equation of a plane

• Ax By Cz D 0
• Given Ax By Cz D 0 Then A, B, C is a
  normal vector
• Pf Given two points P1 and P2 in the plane, the
  vector P2 - P1 is in the plane and A,B, C
  P2 - P1 (Ax2 By2 Cz2) - (Ax1 B y1
  Cz1) ( - D )
  - ( - D )
  0
• Alternate Form A'x B'y C'z D' 0
• where A' A/d, B' B/d, C' C/d, D'
  D/d
• d
• Distance between a point and the plane is given
  by A'x B'y C'z D' (sign indicates which
  side)

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Derivation of Plane Equation

• To derive equation of the plane given three
  points P1, P2, P3
• P3 - P1 x P2 - P1 N, orthogonal vector
• Given a general point P (x,y,z)
• N P - P1 0 if P is in the plane.
• Or given a point (x,y,z) in the plane and normal
  vector N then N x,y,z -D

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Plane derivation example

• Find the plane containing the points (0,0,0),
  (1,2,3), and (4,5,6).

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Plane derivation example

• Find the plane containing the points (0,0,0),
  (1,2,3), and (4,5,6).
• 1,2,3 x 4,5,6 -3,6,-3 Normal vector
• -3x 6y - 3z D 0
• substitute known point (0,0,0)
• 000D 0
• D0
• Plane equation -3x 6y - 3z 0

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Basic 3D Transformations

• Translation
• Scale
• Rotation
• Shear
• As in 2D, we use homogeneous coordinates
  (x,y,z,w), so that transformations may be
  composited together via matrix multiplication.

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3D Translation in Homogeneous Coordinates

• TP (x tx, y ty, z tz)

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3D Scale

• SP (sxx, syy, szz)

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3D Rotations

• Positive Rotations are defined as follows
• Axis of rotation is Direction of positive
  rotation is
• x y to z
• y z to x
• z x to y

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3D Rotations
About y-axis Ry(ß)P About x-axis Rx(ß)P

• About z-axis Rz(ß)P

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3D Shears

• xy Shear
• SHxyP

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Mapping Homogeneous Coordinates to R3

• A transformation Tr can cause the w-component to
  become non-zero
• TrP Tr(x,y,z,1) (x', y', z',w) P
• PR3 (x'/w, y'/w, z'/w)

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Rotation About An Arbitrary Axis

• Situation Given an object whose vertices are
  defined in the world coordinate system, rotate it
  by R about an axis defined by two points, P1 and
  P2.
• Example moon rotating about a planet
• Strategy transform this arbitrary situation into
  something specific we know how to handle

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1. Translate one end of the axis to the origin
P2
Z
P2-P1 u1, u2, u3
U
P1
c
u3

• a
• b
• c
• cosß u3/a
• sinß u1/a

Y
a
ß
u2
b
u1
X
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2. Rotate about the y-axis an angle -ß
Z
U
Z
a
U
c
u3
a
Y
ß
Y
u2
u2
b
u1
X
After Ry(-ß), U lies in the y-z plane
X
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3. Rotate about the x-axis through an angle µ
U

• This step aligns U with the z-axis.
• 4. When U is aligned with the z-axis, apply the
  original rotation, R, about the z-axis.
• 5. Apply the inverses of the transformations in
  reverse order.

Rx (µ) cos µ a / u sinµ u2 / u
Z
a
µ
Y
u2
X
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Rotation About an Arbitrary Axis

• T-1 Ry(ß) Rx(-µ) R Rx(µ) Ry(-ß) T P