|
Plücker Coordinates for the Rest of Us - Part 4 - Applications by Lionel Brits (15 November 2001) |
Return to The Archives |
Basic Properties of Plücker Coordinates
 |
Not all Plücker coordinates correspond to real lines in  .
In the first representation, a necessary condition for 
to be real is ,given that 
(a line must have a direction), assuming that 
(lines not passing through the origin; recall  ).
Since lines passing through the origin are just as real as other lines, this
test only rules out non-real lines, but doesn't confirm that a line is real. In
the second representation, a line is real when Furthermore, two Plücker coordinates
and 
describe the same line if for some  and
describe equal but oppositely directed lines for  .
Also lines
and
are perpendicular when
and parallel when  ,
although these two points are simple consequences from the geometry upon which
Plücker coordinates are based.
|
|
Plane-Line Intersection
|
Recall that the equation that defines a plane with normal  and
distance from the origin  ,
the set of points  such
that .Given a plane 
and a line  ,
the homogeneous coordinate for the intersection of the two is given by: which in Cartesian coordinates is equivalent to 
|
|
Plane-Plane Intersection
|
Given two planes
and  ,
the Plücker coordinate for their intersection is given by:
|
|
Ray-Polygon and Ray-Convex Volume Intersection
|
The feature of Plücker coordinates which attracted me most was the
simplicity of testing ray-polygon intersections [2]. If all the edges of a
polygon are converted to their Plücker representations (taking care to order
the vertices in the same direction) then taking the permuted inner product of
the ray with each edge we can determine whether or not the ray passes through
the polygon. For the ray to pass through the polygon, all the products must
agree in sign (or one product, max two, must be zero, and the others must
agree.) In the following diagram the edges move counterclockwise around
the intersecting ray. We can go further to compute the point of intersection with a triangle or a quadrilateral, if an intersection did occur. Consider a triangle with edges  ,
 ,  ,
where 
is the edge opposite to the nth vertex. Take the ray 
and its products with edges ,
, ,
being  ,
 , 
respectively. If their signs agree, we know that an intersection occurred. It
turns out that these are the unnormalized barycentric coordinates for the point
of intersection. If we take the normalization factor to be 
then the normalized barycentric coordinates are: ,  ,
 .Now the point of intersection in Cartesian coordinates is the linear combination of the vertices  ,
 , 
. The point of intersection,  ,
is then .The same method can be extended for testing for intersections of rays and convex volumes (e.g. a cube). The ray needs to intersect at least one face of the volume in order to intersect with the volume. The ray-polygon test can be optimized, however, because each edge is shared. Only one permuted inner product is therefore necessary per face. |
|
Conclusion
|
|
Sadly, it's time to say goodbye. If you've made it this far without excessive convulsion, I salute you.
If you've found you Zen, I salute you too! If you've found my Zen, please
tell it to come home immediately. Let me
know what you thought of my ramblings. - Lionel Brits (iam@cadvision.com) |
|
Greets and Thanks
|
|
Conor Stokes - for explaining a lot of concepts Per Vognsen - for explaining what Conor meant Grandpa Rogers - for super happy fun times with the hose #flipcode and muh homies - for keeping me awake |
|
References
|
|
Article Series:
|