1. Plücker Coordinate of a Line in 3-Space
Spring 2013

2. Motivation
The other way of representing lines in 3-space is parametric equation
We are interested in learning the aspects/features of Plucker coordinates that make life easier! Q P L O
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3. References Plucker coordinate tutorial, K. Shoemake [rtnews]
Plucker coordinates for the rest of us, L. Brits [flipcode]
Plucker line coordinate, J. Erickson [cgafaq]
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4. 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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5. Mason’s Version Line in parametric form Defineq q0
Plucker coordinate of the line (q, q0)
Six coordinate 4 DOFs: O
(q, q0): q0, q00: general line
(q, q0): q0, q0=0: line through origin
(q, q0): q=0, (q0=0): [not allowed]
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6. The following are from Shoemake’s note…
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7. Summary 1/3
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8. Summary 2/3
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9. Summary 3/3
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10. Notations Upper case letter: a 3-vector U = (ux,uy,uz)
Vector U; homogeneous version (U:0)
Point P; homo version (P:1), (P:w)
Cross and dot product: PQ, U.V
Plane equation: ax+by+cz+dw=0
[a:b:c:d] or [D:d] with D=(a,b,c)
[D:0] origin plane: plane containing origin
Plucker coordinate: {U:V}
Colon “:” proclaims homogeneity
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11. Determinant Definition
L={P-Q:PQ}
Determinant Definition Q P L O
… row x
… row y
… row z
… row w
Make all possible determinants of pairs of rows
P–Q
PQ
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12. Example Q P L O
P=(2,3,7), Q=(2,1,0).
L = {U:V} = {0:2:7:-7:14:-4}.
Order does not matter
identical
Q=(2,3,7), P=(2,1,0).
L = {U:V} = {0:-2:-7:7:-14:4}
Identical lines: two lines are distinct IFF their Plucker coordinates are linearly independent
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13. Tangent-Normal Definition
L={U:UQ}
Tangent-Normal Definition P Q U L O
U×Q
PQ: {U:V}
U = P–Q
V = P×Q = (U+Q)×Q = U×Q
(U:0)  direction of line
[V:0]  origin plane through L
V=0 implies?!
Question: any pair of points P,Q gives the same {U:V}? Yes {p.14}
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14. Example y U=(1,0,-1) Q=(0,0,1) UQ = (0,-1,0) L={1:0:-1:0:-1:0} x y z
If we reverse the tangent:
U=(-1,0,1)
Q=(0,0,1)
UQ = (0,1,0)
L={-1:0:1:0:1:0}
… still get the same line
(but different orientation)
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15. Remark L P’=Q+kU Q U’ = P’– Q = kU U V’ = P’×Q = (Q+kU) ×Q = kU×QO
P’=Q+kU
U’ = P’– Q = kU
V’ = P’×Q = (Q+kU) ×Q = kU×Q
P’Q {kU:kV}
Moving P and/or Q scales U & V together!
Similar to homogeneous coordinates
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16. Remarks Six numbers in Plucker coordinate {U:V} are not independent.
Line in R3 has 4 dof. : six variables, two equations: one from homogeneity; one from U.V = 0
Geometric interpretation {U:V}
U: line tangent (U0, by definition)
V: the normal of origin plane containing L (V=0  L through origin)
Identical lines: two lines are distinct IFF their Plucker coordinates are linearly independent
Ex: {0:-2:-7:7:-14:4} and {0:4:14:-14:28:-8} are the same (but different orientation); {2:1:0:0:0:0} is different
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17. Exercise L={P-Q:PQ} L={U:UQ} x y z P=(1,0,0) Q=(0,1,0) P=(0,1,0)
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18. Distance to Origin L Q T: closest to origin Any Q on L: Q = T + sU U
Vector triple product
Distance to Origin Q T U L O
T: closest to origin
Any Q on L: Q = T + sU
V = U×Q = U×(T+sU) =U×T
||V|| = ||U|| ||T|| sin90 = ||U|| ||T||
T.T = (V.V) / (U.U)
V×U=(U×T)×U = (U.U)T
T=(V×U:U.U)
Squared distance:
L={U:V}
Closest point:
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19. Example y U=(1,0,-1) Q=(0,0,1) UQ = (0,-1,0) L={1:0:-1:0:-1:0}
T=(V×U:U.U) = (1:0:1:2)= (1/2,0,1/2)
Squared distance = (V.V)/(U.U) = 1/2 x y z
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20. 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.P+e) – e(F.P+f) = 0
(fE – eF).P = 0
fE-eF defines the normal of an origin plane through L
direction U = EF L
[E:e]
[F:f] P
L = {EF: fE – eF}
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21. Example y E = [1:0:0:-1] F = [0:0:1:0]
L = {EF:fE-eF} = {0:-1:0:0:0:1}
Check
P = (1,1,0), Q = (1,0,0)
L = {P-Q:PQ} = {0:1:0:0:0:-1} P x Q z
z = 0
[0:0:1:0]
x = 1
[1:0:0:-1]
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22. 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 [E:1] and [F:1].
The intersecting line then becomes {EF:E-F} L Q
[E:1]
[F:1]
{P-Q:PQ} P
{EF:E-F} L
Duality! O
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23. Other Duality (U:0) direction of L T=(VU:U.U) point of L  (U:0)
L={U:V}
(U:0) direction of L
T=(VU:U.U) point of L  (U:0)
[V:0] origin plane thru L (V0)
[UV:V.V] plane thru L  [V:0] Q T U L O
[UV:V.V]
Verify!
(next page) O
[V:0]
P.18
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24. Verify P.23R L={E  F:fE-eF} = {(U  V)  V: -(V.V)V} (U  V)  VO
[V:0]
[UV:V.V]
L={E  F:fE-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)} = {U:V} L
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25. Line-Plane Intersection1
Triple product
Line-Plane Intersection1
L and plane [N:0]
Points on {VN:0} = (VN:w)
Intersection: the point on [UV:V.V]!
{VN:0} O
[V:0]
[N:0] L
(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V:V.V]
{U:V}  [N:0] = (VN:U.N)
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26. Line-Plane Intersection2
[N:n]
L and plane [N:n]
[N:0] O
[V:0]
[N:0] L O L
[V:0]
Derivation pending
{U:V}  [N:n] = (VN–nU:U.N)
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27. Example U=(1,0,-1) Q=(0,0,1) UQ = (0,-1,0) L={1:0:-1:0:-1:0}
VN–nU = (-1,0,0) – (-2)(1,0,-1) = (1,0,-2)
U.N = (1,0,-1).(-1,0,0) = -1
Intersection at (-1,0,2)!
{U:V}  [N:n] = (VN–nU:U.N)
Example y x
Intersect with y = 0, [0:1:0:0]
(VN:U.N) = (0:0:0:0), overlap
Intersect with y = 1, [0:1:0:-1]
(VN–nU:U.N) = (1:0:-1:0)
Intersect at infinity y z
Z = 2
[0:0:1:-2]
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28. Common Plane1 {U:V} and (P:w)  [UP-wV:V.P] Derivation pending y
V = UQ = (0,0,1) (0,1,0) = (-1,0,0)
(P:w) = (1:1:0:1)
[UP-wV:V.P] = [0:1:0:-1] x z
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29. Common Plane2 {U:V} and (N:0)  [UN:V.N] Derivation pending y
V = UQ = (-1,0,0)
N = (1,0,0) …(-1,0,0) get the same
… (1,0,1) also get the same
(N need not ⊥U)
[UN:V.N] = [0:1:0:-1] x z
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30. Generate Points on Line1
Also related: p. 18, 35L
Generate Points on Line1 O
[V:0]
[N:0] L
Useful for:
Computing transformed Plucker coordinate (p.50)
Line-in-plane test
Use {U:V}  [N:0] = (VN:U.N)
Any N will do, as long as U.N0
{Take non-zero component of U} N N U
Does not work for line with V=0
(line through origin) O
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31. Example y As before: L = {U:V} = {0:0:1:-1:0:0} Take N = (0,1,1)
{U:V}  [N:0] = (VN:U.N) = (0:1:-1:1) x z
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32. Line in Plane Test y Generate two points on the line
Do point-on-plane test
Is L in [1:1:0:0]? No
(1,1,0).(0,1,-1) + 0  0 x z
Point-on-Plane Test
[N:n] contain (P:w)
IFF N.P+nw = 0
Is L in [1:0:0:0]? Yes
(1,0,0).(0,1,-1) + 0 = 0
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33. Point-on-Line Test y N2 N1 U N,N1,N2: three base vectors
1. Generate two independent planes
containing the line.
2. Perform point-on-plane tests twice
Point-on-Line Test 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N1:V.N1] and [UN2:V.N2]
(common plane, p.29) x N z
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34. Example L = {0:0:1:-1:0:0}, P = (0:1:-2:1) y P N2
N = (0,0,1), N1 = (0,1,0), N2 = (1,0,0)
Plane1 [-1:0:0:0] (-1,0,0).(0,1,-2)+0 = 0
Plane2 [0:1:0:-1] (0,1,0).(0,1,-2) - 1 = 0 y P N2 N1 U N x z
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35. Duality Parametric equation of L Weighted sum of (U:0) and T=(VU:U.U)
Pnt(t) = (VU+tU:U.U)
Parametric form of planes through L
Generate two planes as page 33… x y z
L = {0:0:1:-1:0:0}
Pnt(t) = (0:-1:t:1)
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36. Two Lines Can Be … Identical Coplanar: find common plane
Linearly dependent Plucker coordinate
Coplanar: find common plane
Intersecting: find intersection
Parallel: find distance
Skewed: find distance, closest points
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37. Coplanarity Test (intersect)
L1&U2: [U1U2:V1.U2]
Coplanarity Test (intersect) L1
Two lines L1 {U1:V1}, L2 {U2:V2} are coplanar if
U1.V2+V1.U2 = 0
Same plane!
L2 &U1: [U2U1:V2.U1]
parallel lines (U1U2=0) are (by definition) coplanar L2
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38. L1 & L2 Coplanar Intersecting point (non-parallel)
L1: {U1:V1}
L2: {U2:V2}
L1 & L2 Coplanar
Intersecting point (non-parallel)
Find the common plane: [U1U2:V1.U2]
((V1.N)U2-(V2.N)U1-(V1.U2)N:(U1U2).N)
Where N is unit basis vector, independent of U1 and U2, (U1U2).N ≠0)
Parallel (distinct) lines (U1U2 = 0)
Common plane:
[(U1.N)V2-(U2.N)V1:(V1V2).N] with N.U10
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39. Example L1={1:1:0:0:0:1} L2={2:2:0:0:0:-4} Pick N = (1,0,0)
[(U1.N)V2-(U2.N)V1:(V1V2).N]
=[0:0:-6:0]
L1={1:1:0:0:0:1}
L2={0:1:0:0:0:-1}
Pick N = (0,0,1)
((V1.N)U2-(V2.N)U1-(V1.U2)N:(U1U2).N)
=(1:2:0:1)
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40. L1 & L2 Skewed Not coplanar IFF skewed Find distance
Find pair of closet points
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41. Distance Computation in R3
Point (x2:y2:z2:1)
Line {U:V}
Plane [D:d]
Point (x1:y1:z1:1)
(1)
(2a): parallel
(2b): skewed
If no intersection, generate a point on line & point-plane distance
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42. (1) Line-Point Distance
Generate P1 containing L & p as [D:d]
Generate P2 containing L & D
Compute distance from p to P2 p
P1=[D:d]
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43. (2a) Parallel Line DistanceU
Find the common plane [D:d]
Find P1 containing L1 and D
Find P2 containing L2 and D
Find distance between P1 & P2
[D:d] L1 L2 P1 P2
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44. (2b) Skewed Line DistanceU2 P2 U1 P1
Generate P1 containing L1 and U2
Generate P2 containing L2 and U1
Find distance between P1 & P2 L1
How to find the pair of points that are closest?
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45. Application
Ray-polygon and ray-convex volume intersection
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46. Relative Position Between 2 Lines
Here, the lines are “oriented”!!
{orientation defined by U}
Relative Position Between 2 Lines
Looking from tail of L1 …
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47. Note here the line is “oriented”; L and –L are not the same
Example y R
L1={1:0:0:0:0:0}
P:(1/3,1/3,0)
Q:(1/3,1/3,1) L3 L2
L2={-1:1:0:0:0:-1} x
L3={0:-1:0:0:0:0} L1 z
R={0:0:-1:1/3:-1/3:0}
= {0:0:-3:1:-1:0}
R vs. L1: (0:0:-3).(0:0:0) + (1:-1:0).(1:0:0) = 1 > 0
R vs. L2: (0:0:-3).(0:0:-1) + (1:-1:0).(-1:1:0) = 1 > 0
R vs. L3: (0:0:-3).(0:0:0) + (1:-1:0).(0:-1:0) = 1 > 0
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48. Example R y L1={1:0:0:0:0:0} P:(1,1,0) Q:(1,1,1) L3 L2z
R={0:0:-1:1:-1:0}
R vs. L1: (0:0:-1).(0:0:0) + (1:-1:0).(1:0:0) = 1 > 0
R vs. L2: (0:0:-1).(0:0:-1) + (1:-1:0).(-1:1:0) = -1 < 0
R vs. L3: (0:0:-1).(0:0:0) + (1:-1:0).(0:-1:0) = 1 > 0
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49. Discussion Plucker coordinate of transformed line
More efficient by computing the Plucker coordinates of the transformed points (p.30)
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50. Index
Constructors, two points 11 tangent-normal 13 two planes 20 Distance (closest pt) to origin 18 Line-plane intersect 25,26 Line-line intersect 38 Common plane line, point 28 line, dir 29 Generate points on line 30, 35L Parametric equation of line 35L Parametric plane of line 35R
Line in plane 30,32
Point in line 33
Point on plane 32
Line-line configuration 37-40
Parallel (distance, common plane) 38
Intersect (point, common plane) 38
Skew 44
Distance (point-line-plane) 41-44
Winding 45
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51. Triple product
Vector triple product
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52. [Example] y U=(1,0,-1) V=(0,-1,0) L={1:0:-1:0:-1:0}
Different normal gives different line
L’ = {1:0:-1: 0:-2:0} x y z
Reverse normal gives different line
U=(1,0,-1)
V=(0,1,0)
L’={1:0:-1:0:1:0} x z
Spring 2013