Page 288 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 23: Developable Surface Fitting to Point Clouds

23.4.1 Estimation of Tangent Planes Notes

We are given data points p , i = 1, . . . , N, with Cartesian coordinates x , y , z in  and a
3
i
i
i
i
triangulation of the data with triangles t . The triangulation gives topological information
j
about the point cloud, and we are able to define adjacent points q for any data point p.
k
The estimated tangent plane T at p shall be a plane best fitting the data points q . T can be
k
computed as minimizer (in the l or l -sense) of the vector of distances dist(q , T) between the
1
2
k
data points q and the plane T. This leads to a set of N estimated tangent planes T corresponding
k
i
to the data points p . For more information concerning reverse engineering.
i
Assuming that the original surface with measurement points p is a developable surface D, the
i
image points b(T ) of the estimated tangent planes T will form a curve-like region on B. To check
i
i
the property ’curvelike’, neighborhoods with respect to a metric on B will be defined. Later, we
will fit a curve c(t) to the curve-like set of image points b(T ), and this fitting is implemented
i
according to the chosen metric.
23.4.2 A Euclidean Metric in the Set of Planes
Now we show that the simplest choice, namely the canonical Euclidean metric in the surrounding
space R of the Blaschke cylinder B, is a quite useful metric for data analysis and fitting. This says
4
that the distance dist(E, F) between two planes E, F
E : e x + e x + e x + e = 0, F : f x + f x + f x + f = 0,
3 3
1 1
1 1
4
2 2
4
2 2
3 3
with normalized normal vectors e = (e , e , e ) and f = (f , f , f ) ( e = f = 1) is defined to be the
3
2
3
2
1
1
Euclidean distance of their image points b(E) and b(F). Thus, the squared distance between E and
F is defined by
dist(E, F) = (e – f ) + (e – f ) + (e – f ) + (e – f ) . (1)
2
2
2
2
2
3
2
2
4
1
4
3
1
To illustrate the geometric meaning of dist(E, F) between two planes E and F we choose a fixed
2
plane M(= F) in  as x – m = 0. Its Blaschke image is b(M) = (1, 0, 0, –m). All points of the Blaschke
3
cylinder, whose Euclidean distance to b(M) equals r, form the intersection surface S of B with the
2
2
2
2
2
three-dimensional sphere (u – 1) + u  u +(u + m) = r . Thus, S is an algebraic surface of
3
2
4
1
order 4 in general. Its points are Blaschke images b(E) of planes E in  which have constant
3
distance r from M and their coordinates ei satisfy
(e – 1) + e + e + (e + m) – r = 0. (2)
2
2
2
2
2
3
2
1
4
2
2
2
The coefficients e satisfy the normalization e  e  e  1. If we consider a general homogeneous
2
3
1
i
equation E : w x + w x + w x + w = 0 of E, these coefficients wi are related to e by
2 2
i
3 3
4
1 1
w
e = i , i  1, 2, 3, 4.
i
2
2
w  w  w 2 3
2
1
We plug this into (2) and obtain the following homogeneous relation of degree four in plane
coordinates w , i
2
2
2
2 2
2
2
2
2
2
2


[(2 r  m )(w  w  w ) w ]  4(w  mw ) (w  w  w ). (3)
2
3
1
1
4
3
2
1
4
Hence, all planes E, having constant distance dist(E, M) = r from a fixed plane M, form the
tangent planes of an algebraic surface b (S) = U of class 4, and U bounds the tolerance region of
–1
the plane M. If a plane E deviates from a plane M in the sense, that b(E) and b(M) have at most
distance r, then the plane E lies in a region of  , which is bounded by the surface U (3).
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