Plcker Coordinate of a Line in 3Space

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Plücker Coordinate of a Line in 3-Space
2
References

• Plucker coordinate tutorial, K. Shoemake rtnews
• Plucker coordinates for the rest of us, L. Brits
  flipcode
• Plucker line coordinate, J. Erickson cgafaq

3
Introduction

• A line in 3-space has four degree-of-freedom (why
  so?!)
• Plucker coordinates are concise and efficient for
  numerous chores
• One special case of Grassmann coordinates
• Uniformly manage points, lines, planes and flats
  in spaces of any dimension.
• Can generate, intersect, with simple equations.

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Masons Version
Line in parametric form
Define
Plucker coordinate of the line (q, q0)
Six coordinate 4 DOFs
O

• (q, q0) q?0, q0?0 general line
• (q, q0) q?0, q00 line through origin
• (q, q0) q0, (q00) not allowed

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The following from Shoemakes note
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Summary 1/3
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Summary 2/3
8
Summary 3/3
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Notations

• Upper case letter a 3-vector U (ux,uy,uz)
• Vector U homogeneous version (U0)
• Point P homo version (P1), (Pw)
• Cross and dot product P?Q, U.V
• Plane equation axbyczdw0
• abcd or Dd with D(a,b,c)
• D0 origin plane plane containing origin
• Plucker coordinate UV
• Colon proclaims homogeneity

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Determinant Definition
LP-QP?Q
row x
row y
row z
row w
Make all possible determinants of pairs of rows
PQ
P?Q
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Example
P(2,3,7), Q(2,1,0). L UV
027-714-4.
Order does not matter
identical
Q(2,3,7), P(2,1,0). L UV
0-2-77-144
Identical lines two lines are distinct IFF their
Plucker coordinates are linearly independent
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Tangent-Normal Definition
LUU?Q
PQ UV U PQ V PQ (UQ)Q UQ
(U0) ? direction of line V0 ? origin plane
through L
Question any pair of points P,Q gives the same
UV? p.14
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Example
y
U(1,0,-1) Q(0,0,1) U?Q (0,-1,0) L10-10-1
0
x
y
z
If we reverse the tangent U(-1,0,1) Q(0,0,1) U?
Q (0,1,0) L-101010 still get the same
line
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Remark
PQkU U P Q kU V PQ (QkU) Q
kUQ PQ kUkV
Moving P and/or Q scales U V together! Similar
to homogeneous coordinates
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Remarks

• Six numbers in Plucker coordinate UV are not
  independent.
• Line in R3 has 4 dof. six variables, two
  equations one from homogeneity one from U.V 0
• Geometric interpretation UV
• U line tangent (U?0, by definition)
• V the normal of origin plane containing L (V0 ?
  L through origin)
• Identical lines two lines are distinct IFF their
  Plucker coordinates are linearly independent

Ex 0-2-77-144 and 0414-1428-8 are
the same (but different orientation)
210000 is different
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Example
LP-QP?Q LUU?Q
P(0,1,0) Q(1,0,0)
P(1,0,0) Q(0,1,0)
Q
P
U(1,-1,0) Q(0,1,0)
U(2,-2,0) Q(0,1,0)
U(-1,1,0) Q(0,1,0)
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Distance to Origin
T closest to origin Any Q on L Q T sU V
UQ U(TsU) UT V U T sin90
U T T.T (V.V) / (U.U) VU(UT)U
(U.U)T T(VUU.U)
Squared distance
Closest point
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Example
y
U(1,0,-1) Q(0,0,1) U?Q (0,-1,0) L10-10-1
0 T(VUU.U) (1012) (1/2,0,1/2) Squared
distance (V.V)/(U.U) 1/2
x
y
z
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Line as Intersection of Two Planes1
Plane equation ax by cz d 0 P
(x,y,z), point on L E.P e 0 F.P f 0
f(E.Pe) e(F.Pf) 0 (fE eF).P 0 fE-eF
defines the normal of an origin plane through
L direction U E?F
P
L E?F fE eF
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Example
y
E 100-1 F 0010 L E?FfE-eF
0-10001 Check P (1,1,0), Q (1,0,0) L
P-QP?Q 01000-1
P
x
Q
z
z 0 0010
x 1 100-1
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Line as Intersection of Two Planes2

• If both planes do not pass through origin, e?0
  and f?0, we can normalize both planes to E1
  and F1.
• The intersecting line then becomes E?FE-F

L
Q
E1
F1
P-QP?Q
P
E?FE-F
L
Duality!
O
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Other Duality

• V0 origin plane thru L (V?0)
• T(V?UU.U) point of L ? (U0)

• (U0) direction of L
• U?VV.V plane thru L ? V0

U?VV.V
O
Verify!
V0
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Verify
LE ? FfE-eF (U ? V) ? V -(V.V)V (U ?
V) ? V -U(V.V)V(U.V) -U(V.V) L
-U(V.V)-V(V.V) UV
L
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Line-Plane Intersection1
Points on V?N0 (V?Nw) Intersection the
point on U?VV.V!

• L and plane N0

V?N0
(U ? V).(V ? N)w(V.V) 0 w(V.V) (V ? U).(V ?
N) N.((V ? U) ? V) N.(-(U.V)V(V.V)U)(V.V)(U.
N) w U.N
U?VV.V
UV ? N0 (V?NU.N)
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Line-Plane Intersection2
Nn

• L and plane Nn

N0
O
L
V0
Derivation pending
UV ? Nn (V?NnUU.N)
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Example
U(1,0,-1) Q(0,0,1) U?Q (0,-1,0) L10-10-1
0 V?NnU (-1,0,0) (-2)(1,0,-1)
(1,0,-2) U.N (1,0,-1).(-1,0,0)
-1 Intersection at (-1,0,2)!
UV ? Nn (V?NnUU.N)
y
x
Intersect with y 0, 0100 (V?NU.N)
(0000), overlap Intersect with y 1,
010-1 (V?NnUU.N) (10-10) Intersect at
infinity
y
z
Z 2 001-2
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Common Plane1
Derivation pending

• UV and (Pw) ? U?P-wVV.P

y
U (0,0,1) V U?Q (0,0,1) ?(0,1,0)
(-1,0,0) (Pw) (1101) U?P-wVV.P
010-1
x
z
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Common Plane2
Derivation pending

• UV and (N0) ? U?NV.N

y
U (0,0,1) V U?Q (-1,0,0) N (1,0,0)
(-1,0,0) get the same U?NV.N 010-1
x
z
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Generate Points on Line1

• Useful for
• Computing transformed Plucker coordinate
• Line-in-plane test

Use UV ? N0 (V?NU.N)
Any N will do, as long as U.N?0 Take non-zero
component of U
N
N
U
O
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Example
y
As before L UV 001-100 Take N
(0,1,1) UV ? N0 (V?NU.N) (01-11)
x
z
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Example (cont)
Point-on-Plane Test Nn contain (Pw) IFF
N.Pnw 0
y
Is L in 1100? No (1,1,0).(0,1,-1) 0 ? 0
x
Is L in 1000? Yes (1,0,0).(0,1,-1) 0 0
z
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Point-on-Line Test
1. Generate two independent planes
containing the line. 2. Perform point-in-plane
tests twice
y
N2
N1
U
N,N1,N2 three base vectors Choose N according to
nonzero component of U N1 and N2 are the other
two axes Check point-in-plane with U?N1V.N1
and U?N2V.N2
x
N
z
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Example
L 001-100, P (01-21) N (0,0,1),
N1 (0,1,0), N2 (1,0,0) Plane1 -1000
(-1,0,0).(0,1,-2)0 0 Plane2 010-1
(0,1,0).(0,1,-2) - 1 0
y
P
N2
N1
U
N
x
z
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Duality

• Parametric equation of L
• Weighted sum of (U0) and T(V?UU.U)
• Pnt(t) (V?UtUU.U)

• Parametric form of planes through L
• Generate two planes as previous page

L 001-100 Pnt(t) (0-1t1)
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Two Lines Can Be

• Identical
• Linearly dependent Plucker coordinate
• Coplanar find common plane
• Intersecting find intersection
• Parallel find distance
• Skewed find distance

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Coplanarity Test
L2 U2V2
L1 U1V1
U1.V2V1.U2 0
L1U2 U1?U2V1.U2
L2 U1 U2?U1V2.U1
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L1 L2 Coplanar
L1 U1V1 L2 U2V2

• Find the common plane
• Intersecting lines U1?U2V1.U2
• Parallel (distinct) lines (U1?U2 0)
• (U1.N)V2-(U2.N)V1(V1?V2).N with N.U1?0
• Intersection (coplanar and nonparallel)
• ((V1.N)U2-(V2.N)U1-(V1.U2)N(U1?U2).N)
• Where N is unit axis vector, independent of U1
  and U2

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Intersection Point

• Coplanar and non-parallel ? intersect
• ((V1.N)U2-(V2.N)U1-(V1.U2)N(U1?U2).N)
• Where N is unit axis vector, independent of U1
  and U2

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L1 L2 Skewed

• Not coplanar IFF skewed

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Distance Computation in R3
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Line-Point Distance
L
P2
D

• Generate P1 containing L p as Dd
• Generate P2 containing L D
• Compute distance from p to P2

p
P1Dd
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Parallel Line Distance
D
D
U
Find the common plane Dd Find P1 containing
L1 and D Find P2 containing L2 and D Find
distance between P1 P2
Dd
L2
L1
P1
P2
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Skewed Line Distance
L2
U2
P2
U1
P1
Generate P1 containing L1 and U2 Generate P2
containing L2 and U1 Find distance between P1 P2
L1
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Application

• Ray-polygon and ray-convex volume intersection

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Relative Position Between 2 Lines
Here, the lines are oriented!! orientation
defined by U
Looking from tail of L1
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Example
Note here the line is oriented L and L are
not the same
y
R
L1100000
P(1/3,1/3,0) Q(1/3,1/3,1)
L3
L2
L2-11000-1
x
L30-10000
L1
R00-11/3-1/30 00-31-10
z
R vs. L1 (00-3).(000) (1-10).(100) 1
gt 0
R vs. L2 (00-3).(00-1) (1-10).(-110)
1 gt 0
R vs. L3 (00-3).(000) (1-10).(0-10)
1 gt 0
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Example
R
y
L1100000
P(1,1,0) Q(1,1,1)
L3
L2
L2-11000-1
x
L30-10000
L1
R00-11-10
z
R vs. L1 (00-1).(000) (1-10).(100) 1
gt 0
R vs. L2 (00-1).(00-1) (1-10).(-110)
-1 lt 0
R vs. L3 (00-1).(000) (1-10).(0-10)
1 gt 0
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Summary
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Defining Lines

• L UV, UV 0 and U?0
• L PQ PQ, P,Q distinct points on L line is
  directed from Q?P
• L U UQ, U direction, Q a point on line
• L EFfE eF Ee Ff distinct planes
  containing line

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Basic Evaluations

• Minimum squared distance from origin
• (V.V)/(U.U)
• Point closest to origin
• (VUU.U)
• Plane/line intersection
• (VN-UnU.N)
• Plane defined by L and (Pw)
• UPVwV.P
• Plane containing L with direction N
• UNV.N

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The End
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Questions

• Plucker coordinate of transformed line
• Nonperspective transform P U
• Perspective transform P1 P2
• http//www.gamedev.net/community/forums/topic.asp?
  topic_id275350

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Example
y
U(1,0,-1) V(0,-1,0) L10-10-10 Different
normal gives different line L 10-1
0-20
x
y
z
Reverse normal gives different line U(1,0,-1) V(
0,1,0) L10-1010
x
z