3D Mathematical Preliminaries

1
3-D Mathematical Preliminaries
2
Coordinate Systems
Left-handed
Right-handed
coordinate
coordinate
system
system
3
3-D Vectors

• Have length and direction V xv, yv, zv
• Length is given by the Euclidean Norm V ?(
  xv2 yv2 zv2 )
• Dot Product VU xv, yv, zv xu, yu, zu
  xvxu yvyu zvzu
• V U cos ß
• Cross Product V x U vyuz - vzuy, -vxuz
  vzux, vxuy - vyux
• V x U - ( U x V)
• Direction of resulting vector depends on
  coordinate system and order of vectors.

4
Parametric Definition of a Line
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 gt 1
t 1
t 0
P2
t lt 0
P1

• Given a point P1 and a vector V xv, yv, zv
• x x1 t xv, y y1 t yv , z
  z1 t zv
• COMPACT FORM L P1 tP2 - P1
  or L P1 Vt

5
Equation of a plane

• Ax By Cz 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 ?(A2 B2 C2)
• Distance between a point and the plane is given
  by
• A'x B'y C'z D' (sign indicates which side)
• Given Ax By Cz D 0 Then A, B, C is a
  normal vector
• Pf Given two point 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

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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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Basic Transformations

• Translation
• Scale
• Rotation
• Shear

8
Translation in Homogeneous Coordinates

• TP (x tx, y ty, z tz)

T
P
9
Scale

• SP (sxx, syy, szz)

Note A scale may also translate an object!
10
Rotations

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

11
Rotations

• About the z axis Rz(ß) P
• About the x axis Rx(ß) P
• About the y axis Ry(ß) P

12
Shears

• xy Shear
• SHxyP

Y
Y
X
X
Z
Z
13
Mapping a 4D point into R3

• If Tr is any transformation, then it is possible
  that the 4th component of the point will not be
  1.
• TrP Tr(x,y,z,1) (x', y', z',w)
• Pplotted (x'/w, y'/w, z'/w)
• Transformations may be appended together via
  matrix multiplication.

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Rotation About An Arbitrary Axis
P2
1. Translate one end of the axis to the
origin P2-P1 u1, u2, u3
Z

• a ? (u12 u32)
• b ? (u12 u22)
• c ? (u22 u32)
• cosß u3/a
• sinß u1/a

U
P1
c
u3
Y
a
ß
u2
b
u1
X
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2. Rotate the coordinate axes about the y-axis
an angle -ß
Z
Z
a
c
u3
a
Y
ß
µ
Y
u2
u2
b
u1
X
X
After Ry(-ß), µ lies in the y-z plane
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3. Rotate the coordinate axes about the x-axis
through an angle µ to align the z-axis with U
U
Z
Rx (µ) cos µ a/ u sinµ u2 / u

• 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.

µ
Y
X
17
Rotation About an Arbitrary Axis

• T-1 Ry(ß)Rx(-µ)RRx(µ)Ry(-ß)T P

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Transformation Types

• A transformation maps points in one coordinate
  system to points in another coordinate system.
• Rigid body transformations
• Do not distort shapes line lengths and angles
  are preserved
• Rotations, Translations, and combinations of both

• Affine transformations
• Keep parallel lines parallel, but do not preserve
  line lengths or angles
• Rotations, Translations, Scales, Shears, and
  combinations of these