 # Triangle Mesh Completer

RHINOCEROS® plug-in & standalone program Triangle Mesh Completer Downloads  Plug-in for RHINOCEROS®  Standalone version  Theory in PDF file  Manual in CHM file TMC Rhinoceros plugin Manual  General information  Supposed input  System requirements  Working with the plugin  Examples TMC Standalone version Manual  General information  Input and output  System requirements  Interface  Examples Contact information Frequently Asked Questions  F.A.Q. Advanced Mesh Repairing Theory  Abstract  1.Introduction  2. Formalization  3. Concept of bridges  4. Missing surface field concept  5. Implemented missing s.fields  6. The implemented method  7. Tests and comparison  8. Conclusion  9. References

Alexander Emelyanov,
Institute of Computing for Physics and Technology
Protvino, Moscow region, Russia
ae@ae3d.ru
(Updated version of the paper presented on Graphicon 2010)

## 5. Implemented missing surface fields

### 5.1 General background

Described in this section implementations of AF are developed in a “physical” manner, so they have several common traits adduced below.

For each field the following two kinds of objects are defined: an elementary source and an object of the field action. The both kinds of objects are represented by single points supplemented by the corresponding sets of extra features (such set can be empty). An elementary source acts in the corresponding way on an object of action.

The action of an elementary source decreases with increasing the distance between it and a considered object of action. In each of the AF implementations described below this property is implemented by the corresponding distance function of the following kind: A has a shielding effect on AF. This effect currently implemented in the following way: an elementary source doesn’t act at a specified point if the segment between this point and the point of the source crosses A.

### 5.2 Boundary interpolation missing surface field

To determine the AF indices the implementation of AF described here uses an interpolation of boundaries of A. Because of that it is called the boundary interpolation missing surface field (BIAF).

Initially, let’s consider the basic geometric issues. Consider a boundary of A, a point (O) on it, the normal vector to A at the point ( nO), and the tangent line to the boundary at the point (F5.2.1). The tangent line splits the plane defined by O and nO to the “occupied” and the “empty” half-planes. Define the unit vector ( τO) on the tangent line such that the cross product nO X τO is in the “occupied” half-plane. Let’s call such vector the tangent vector of a specified boundary point. Consider connection of some point outside A( X) with A at O by an arbitrary narrow planar strip. It is obvious, that the plane of this strip should pass through X , O and the boundary tangent line at O.

From the vector cross-product properties it follows that the normal vector (nXO) of this strip is defined by the following equation: Now, assuming that a normal vector at X ( nX) is specified, let’s define the quality ( ηXO) of the considered strip connection in the following way: In this formulation the first multiplier indicates the “passability” of the strip; the second one expresses the degree of extrapolation of A at O by the strip; the third expresses the degree of extrapolation of an arbitrary small surface element defined by X and nX by the strip.

Now, let’s define the introduced above field. As an elementary source of the field the aggregate of a boundary point of A(O), the normal (nO) and the tangent vectors (τO) at the point ({ O,nO, τO}) is assumed. As an object of the field action let’s assume the last traced point of a force line with the origin at the corresponding boundary point ( B). That is the aggregate of a space point ( X), a specified normal vector at it (nX) and the tangent vector at the corresponding force line origin (τB): {X,nX, τB}. With these assumptions define the force (FX) of action of { O,nO, τO} on {X,nX, τB} in the following way: where constant c expresses the length of the boundary segment represented by O (in other words it is a “charge value” of the elementary source); λBI is a distance function of the kind introduced by (E5.1.1).

It also can be written in the following matrix form: where When the field at a point is created by a number of sources, this formulation allows obtaining the force vector for various nX without recalculation of the matrix.

The AF normal vector (NX), the potential (ΨX) and the attraction index (ΩX) created at X by action of { O,nO, τO} on {X,nX, τB} are defined in the following way: The defined field is a complete AF because it provides obtaining all the AF indices at a point; vectors nX and τB are considered as parameters of the state context of the field.

### 5.3 Point radial missing surface field

This implementation of AF is called the point radial missing surface field (PRAF). Its elementary source is a free point of a specified ICADM. An object of the field action is just a space point. This field is incomplete, because it provides obtaining only the potential value at a specified point. This potential, created by a free point (O) at a specified space point ( X) is defined by the following equation: where c∈(0,1] is the confidence value of the free point coordinates; λPR is a distance function of the kind introduced by (E5.1.1).

This AF can increase the adequateness of force line behavior of a composite AF with its participation, if a processed ICADM contains a sufficient number of free points. 