1

Plücker Coordinate of a Line in 3-Space

Spring 2013

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!

2

Q

P

L

O

RttQPt ,1

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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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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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Mason’s Version

tqptx )(

qpq 0

Line in parametric form

Define

Plucker coordinate of the line (q, q0)

Six coordinate 4 DOFs:

]by [scale ,,

0

00

0

kqqqkqq

qq

p q

q0

•(q, q0): q0, q00: general line

•(q, q0): q0, q0=0: line through origin

•(q, q0): q=0, (q0=0): [not allowed]

O

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The following are from Shoemake’s note…

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Summary 1/3

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Summary 2/3

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Summary 3/3

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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: PQ, 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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Determinant DefinitionQ

P

L

O

1z

y

x

p

p

p

1z

y

x

q

q

q … row x

… row y

… row z

… row w

Make all possible determinants of pairs of rows

11xx qp

11yy qp

11zz qp

yy

xx

qp

qp

zz

yy

qp

qp

xx

zz

qp

qp

P–Q PQ

L={P-Q:PQ}L={P-Q:PQ}

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ExampleQ

P

L

O

P=(2,3,7), Q=(2,1,0). L = {U:V} = {0:2:7:-7:14:-4}.

Order does not matter

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

identical

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Tangent-Normal Definition

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

Question: any pair of points P,Q gives the same {U:V}? Yes

{p.14}

L={U:UQ}L={U:UQ}

PQ

U

L

O

U×Q

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Example

x

y

z y

U=(1,0,-1)Q=(0,0,1)UQ = (0,-1,0)L={1:0:-1:0:-1:0}

If we reverse the tangent:U=(-1,0,1)Q=(0,0,1)UQ = (0,1,0)L={-1:0:1:0:1:0}… still get the same line(but different orientation)Spring 2013

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Remark

P’=Q+kU

U’ = P’– Q = kUV’ = P’×Q = (Q+kU) ×Q = kU×Q

P’Q {kU:kV}

Q

P

P’

U

L

O

Moving P and/or Q scales U & V together!Similar to homogeneous coordinates

Moving P and/or Q scales U & V together!Similar to homogeneous coordinates

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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 (U0, 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 differentSpring 2013

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Exercise

x

y

zP

Q P=(1,0,0)Q=(0,1,0)

P=(0,1,0)Q=(1,0,0)

L={P-Q:PQ}L={U:UQ}

L={P-Q:PQ}L={U:UQ}

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 OriginT: 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)

Q

T

U

L

O Squared distance:

Closest point:

Vector triple product

L={U:V}

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Example

x

y

z y

U=(1,0,-1)Q=(0,0,1)UQ = (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

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Line as Intersection of Two Planes1

L

[E:e] [F:f]

P

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 = 0fE-eF defines the normal of an origin plane through Ldirection U = EF

L = {EF: fE – eF}

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Example

x

z

y

z = 0[0:0:1:0]

x = 1[1:0:0:-1]

E = [1:0:0:-1]F = [0:0:1:0]L = {EF:fE-eF} = {0:-1:0:0:0:1}

CheckP = (1,1,0), Q = (1,0,0)L = {P-Q:PQ} = {0:1:0:0:0:-1}

Q

P

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Line as Intersection of Two Planes2

• If both planes do not pass through origin, e0 and f0, we can normalize both planes to [E:1] and [F:1].

• The intersecting line then becomes {EF:E-F}

L

[E:1] [F:1]Q

P

L

O

{P-Q:PQ}

{EF:E-F}

Duality!Spring 2013

• [V:0] origin plane thru L (V0)

• [UV:V.V] plane thru L [V:0]

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Other Duality

• (U:0) direction of L• T=(VU:U.U) point

of L (U:0)

Q

T

U

L

O

O[V:0]

[UV:V.V]

Verify!(next page)P.18

L={U:V}

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Verify P.23R

O[V:0]

[UV:V.V]

L

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}

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Line-Plane Intersection1

• L and plane [N:0] Points on {VN:0} = (VN:w)Intersection: the point on [UV:V.V]!

O

[V:0]

[N:0]

L{U:V} [N:0] = (VN:U.N)

[UV:V.V]

{VN:0}

(U V).(V N)+w(V.V) = 0w(V.V) = (V U).(V N) = N.((V U) V)=N.(-(U.V)V+(V.V)U)=(V.V)(U.N)w = U.N

Triple product

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Line-Plane Intersection2

• L and plane [N:n]

O

[V:0]

[N:0]

L

{U:V} [N:n] = (VN–nU:U.N)

O

[V:0]

[N:0]

L

Derivation pending

[N:n]

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Example

x

y

z y

U=(1,0,-1)Q=(0,0,1)UQ = (0,-1,0)L={1:0:-1:0:-1:0}

VN–nU = (-1,0,0) – (-2)(1,0,-1) = (1,0,-2)U.N = (1,0,-1).(-1,0,0) = -1Intersection at (-1,0,2)!

Z = 2[0:0:1:-2]

Intersect with y = 0, [0:1:0:0](VN:U.N) = (0:0:0:0), overlap

Intersect with y = 1, [0:1:0:-1](VN–nU:U.N) = (1:0:-1:0)Intersect at infinity

{U:V} [N:n] = (VN–nU:U.N)

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Common Plane1

{U:V} and (P:w) [UP-wV:V.P]

U = (0,0,1)V = UQ = (0,0,1) (0,1,0) = (-1,0,0)

(P:w) = (1:1:0:1)

[UP-wV:V.P] = [0:1:0:-1]

Derivation pending

x

y

z

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Common Plane2

{U:V} and (N:0) [UN:V.N]

U = (0,0,1)V = UQ = (-1,0,0)

N = (1,0,0) …(-1,0,0) get the same … (1,0,1) also get the same (N need not U)⊥

[UN:V.N] = [0:1:0:-1]

Derivation pending

x

y

z

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Generate Points on Line1

Useful for:

• Computing transformed Plucker coordinate (p.50)

• Line-in-plane testO

[V:0]

[N:0]

LUse {U:V} [N:0] = (VN:U.N)

U

NAny N will do, as long as U.N0{Take non-zero component of U}N

O

Also related: p. 18, 35L

Does not work for line with V=0(line through origin)Spring 2013

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Example

As before: L = {U:V} = {0:0:1:-1:0:0}

Take N = (0,1,1)

{U:V} [N:0] = (VN:U.N) = (0:1:-1:1)x

y

z

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Line in Plane Test

Point-on-Plane Test [N:n] contain (P:w) IFF N.P+nw = 0

Is L in [1:1:0:0]? No

(1,1,0).(0,1,-1) + 0 0

Is L in [1:0:0:0]? Yes

(1,0,0).(0,1,-1) + 0 = 0

x

y

z

1. Generate two points on the line 2. Do point-on-plane test

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Point-on-Line Test

x

y

z

N

N1

N2

N,N1,N2: three base vectorsChoose N according to nonzero component of UN1 and N2 are the other two axes

Check point-in-plane with [UN1:V.N1] and [UN2:V.N2](common plane, p.29)

U

1. Generate two independent planes

containing the line.

2. Perform point-on-plane tests twice

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Example

x

y

z

N

N1

N2

U

L = {0:0:1:-1:0:0}, P = (0:1:-2:1)N = (0,0,1), N1 = (0,1,0), N2 = (1,0,0)Plane1 [-1:0:0:0] (-1,0,0).(0,1,-2)+0 = 0Plane2 [0:1:0:-1] (0,1,0).(0,1,-2) - 1 = 0

P

Spring 2013

• Parametric equation of L• Weighted sum of (U:0)

and T=(VU:U.U)

Pnt(t) = (VU+tU:U.U)

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Duality

• Parametric form of planes through L– Generate two planes as

page 33…

L = {0:0:1:-1:0:0}Pnt(t) = (0:-1:t:1)

x

y

z Spring 2013

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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, closest points

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Coplanarity Test(intersect)

• Two lines L1 {U1:V1}, L2 {U2:V2} are coplanar if

U1.V2+V1.U2 = 0

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L1&U2: [U1U2:V1.U2]

L2 &U1: [U2U1:V2.U1]

Same plane!

L1

L2parallel lines

(U1U2=0) are (by definition) coplanar

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L1 & L2 Coplanar

• Intersecting point (non-parallel)– Find the common plane: [U1U2:V1.U2]

– ((V1.N)U2-(V2.N)U1-(V1.U2)N:(U1U2).N)

– Where N is unit basis vector, independent of U1 and U2, (U1U2).N ≠0)

• Parallel (distinct) lines (U1U2 = 0)Common plane:

– [(U1.N)V2-(U2.N)V1:(V1V2).N] with N.U10

L1: {U1:V1}L2: {U2:V2}

Spring 2013

Example

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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:(U1U2).N)=(1:2:0:1)

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:(V1V2).N]

=[0:0:-6:0]

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L1 & L2 Skewed

• Not coplanar IFF skewed

• Find distance

• Find pair of closet points

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Distance Computation in R3

221

221

221 zzyyxx 222

111

cba

dczbyax

222

21

cba

dd

  Point (x2:y2:z2:1)

Line {U:V} Plane [D:d]

Point (x1:y1:z1:1)

(1)

Line {U:V}  (2a): parallel

(2b): skewed

If no intersection,

generate a point on line & point-plane distance

Plane [D:d]   

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(1) Line-Point Distance

L

p

D

1. Generate 1 containing L & p as [D:d]2. Generate 2 containing L & D3. Compute distance from p to 2

1=[D:d]

2

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(2a) Parallel Line Distance

[D:d]

D U

1 2

L1 L2Find the common plane [D:d] Find 1 containing L1 and DFind 2 containing L2 and DFind distance between 1 & 2

D

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(2b) Skewed Line Distance

1

2

L1

L2

U1

U2

Generate 1 containing L1 and U2

Generate 2 containing L2 and U1

Find distance between 1 & 2

How to find the pair of points that are

closest?Spring 2013

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Application

• Ray-polygon and ray-convex volume intersection

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Relative Position Between 2 Lines

Looking from tail of L1 …

Here, the lines are “oriented”!!{orientation defined by U}

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Example

x

y

zL1

L2L3

R

L1={1:0:0:0:0:0}L2={-1:1:0:0:0:-1}

L3={0:-1:0:0:0:0}

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

P:(1/3,1/3,0)Q:(1/3,1/3,1)

R vs. L3: (0:0:-3).(0:0:0) + (1:-1:0).(0:-1:0) = 1 > 0

Note here the line is “oriented”; L and –L are not the same

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Example

x

y

zL1

L2L3

R

L1={1:0:0:0:0:0}L2={-1:1:0:0:0:-1}

L3={0:-1:0:0:0:0}

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

P:(1,1,0)Q:(1,1,1)

R vs. L3: (0:0:-1).(0:0:0) + (1:-1:0).(0:-1:0) = 1 > 0Spring 2013

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Discussion

• Plucker coordinate of transformed line– More efficient by computing the Plucker

coordinates of the transformed points (p.30)

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IndexConstructors,

two points 11

tangent-normal 13

two planes 20

Distance (closest pt) to origin 18

Line-plane intersect25,26

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

50

Line in plane30,32

Point in line 33

Point on plane 32

Line-line configuration37-40

Parallel (distance, common plane) 38

Intersect (point, common plane) 38

Skew 44

Distance (point-line-plane)41-44

Winding 45

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Vector triple product

Triple product

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[Example]

x

y

z

U=(1,0,-1)V=(0,-1,0)L={1:0:-1:0:-1:0}

Different normal gives different lineL’ = {1:0:-1: 0:-2:0}

Reverse normal gives different lineU=(1,0,-1)V=(0,1,0)L’={1:0:-1:0:1:0}

x

y

z Spring 2013