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Advanced Mesh Repairing Theory

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**)

Described in this section implementations of **A**F 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 **A**F implementations described below this property is implemented by the corresponding distance function of the following kind:

**A** has a shielding effect on **A**F. 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**.

To determine the **A**F indices the implementation of **A**F described here uses an interpolation of boundaries of **A**. Because of that it is called the b*oundary interpolation missing surface field *(*BI***A***F*).

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 ( * n^{O}*), and the tangent line to the boundary at the point (F5.2.1). The tangent line splits the plane defined by

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 (* n^{XO}*) of this strip is defined by the following equation:

Now, assuming that a normal vector at *X* ( * n^{X}*) is specified, let’s define the

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 * n^{X}* 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 (* n^{O}*) and the tangent vectors (

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); *λ^{BI}* 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 * n^{X}* without recalculation of the matrix.

The **A**F normal vector (* N^{X}*), the potential (

The defined field is a complete **A**F because it provides obtaining all the **A**F indices at a point; vectors * n^{X}* and

This implementation of **A**F 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; * λ^{PR}* is a distance function of the kind introduced by (E5.1.1).

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