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Figure 2: Description of a rectangle in 2D. Left: using 5 
parameters. Right: using 4 corner points. 
a total of 5 parameters (figure 2, left). From those 5 param- 
eters, all vertices and edges of the rectangle can be derived. 
The rectangle can also be described by its vertices (points), 
edges (lines), and constraints (figure 2, right). If the ver- 
tices are given, the edges are defined as well and vice versa. 
Using vertices as unknowns, one obtains the 8 unknowns 
(;,9:).1 € i < 4. Since two of the edges are perpen- 
dicular and one is parallel to the first edge, 3 constraints 
apply, so that again 5 parameters remain. Formulating 
constraints can be done in different ways. Especially in 
projective geometry, linear constraints can often be ob- 
tained (Heuel, 2002). However, in this case, the points 
as well as the lines are unknown which leads to bilinear 
equations, as used e.g. by (Brenner, 2000b) or (Griin and 
Wang, 2001) in the context of building models. The con- 
straint equations thus are linearized and have to be iterated. 
Four line equations given in Hesse normal form (HNF) 
GZ by +o.= 0,1 < 1 < A.vield 1? additional pa- 
rameters and 4 additional constraints for normalizing the 
normal vectors, a? -- b? — 1. Two points are on each line, 
leading to two equations of the form a;x; + b;y; + €; = 0. 
Three other equations hold for the normal vectors of the 
lines, enforcing parallel and perpendicular edges. Table 2 
summarizes unknowns and constraints for this case. 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
| * | Description [UTC | 
4 | Points: (z;, y) 8 
4 | Lines a;z + biy +6: = 0 12 
4 | Normal vector length = 1: a? + b7 = 1 4 
8 | Point on line: a;x; + biy; + € = 0 8 
2 | Perpendicular normals: a;a; + b;b; = 0 2 
| |. Parallel normal: apbs — boas = 0 l 
| Total unknowns | 5 | | 
  
Table 2: Unknowns (U) and constraints (C) for a simple 
rectangle in 2D. 
4.3 Packaging Constraints as Weak Primitives 
If one “packages” unknowns and constraints as weak prim- 
itives, some of the constraints will become invisible: en- 
forcing the normal vector length and points lying on their 
respective lines are internal constraints which a user won°t 
change. Parallel and perpendicular constraints will be- 
come properties of the object to be switched on and off. 
The point coordinates, being unknowns, will become fields 
which can be connected to fields of other primitives. What 
about the HNF line coefficients? In the above formulation, 
they are needed when regularization conditions should be 
met. However, after performing an adjustment, their esti- 
mated values are not used, since the object is defined by its 
points only, which are estimated as well. Thus, a line has 
1089 
       
  
    
      
   
  
  
  
  
  
   
   
    
    
   
   
   
    
   
    
    
   
    
    
   
   
    
   
   
   
   
   
    
    
    
   
  
   
    
     
       
   
    
     
    
     
    
    
     
  
   
   
   
   
    
8 and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
| 7 Point on line EN 
[CS 7 Aline] | \ [^58 | 
——— 
  
  
  
  
  
  
  
| Points[4] | J B.Poins(0] | 7 Points[4] | 
Lines[4] x E UAE ped / Lines[4] 
A >] Point on line 2 
B A-Lines[1] | 
[ B. Points[3} | 3] | 
  
  
  
Figure 3: Two rectangles, A and B, standing side by side. 
Left: geometry. Right: user view of constraints between A 
and B. 
e Fone 
s ug 08080 
Fag 
   
Figure 4: Internal representation of the two rectangles of 
figure 3 as a graph connecting unknowns by constraints. 
Unknowns are depicted as circles, constraints as boxes. 
The icons represent the “rectangular” and “point on line” 
constraints. 
only to take part in the estimation if i) regularization condi- 
tions are used which involve the line, or ii) the line is a field 
which is connected by a constraint to another primitive. 
Thus, one sees that fields offered by a primitive are inde- 
pendent from the parameters which represent its geome- 
try. If a field directly corresponds to a parameter, this pa- 
rameter will be introduced as unknown into the estimation. 
If not, the field will be introduced as unknown and equa- 
tions will be added relating the unknowns to the parame- 
ters of the primitive. In fact, both the “5-parameter” and 
the “4-point” representations can define the same interface 
in terms of the fields which are available for connection by 
constraints. 
Figure 3, left shows an example. Two rectangles are re- 
quired to stand side by side. In order to make them pre- 
cisely aligned, constraints are introduced, say in this case 
that the two left points of rectangle B have to lie on the 
right line of rectangle A. So from an operator’s point of 
view, the rather simple figure 3, right, reflects the scene 
structure: two objects, À and B, are present which are con- 
nected by constraints of type “Point on line”. 
Internally, however, if the rectangles are represented by 
four points, the structure of unknowns and constraints 
more complex. Figure 4 depicts the graph of unknowns 
and constraints which results. Each time an operator inter- 
action changes parts of the graph, the required graph nodes 
are determined, unknowns and constraints are set up, and 
an estimation is performed. Figure 5 shows how the two 
example primitives react to different user interactions. 
5 CONCLUSIONS AND OUTLOOK 
In this paper, a new modelling approach termed weak CSG 
modelling is proposed. Its main idea is to package, or hide,