Auxiliary meshing techniques - Finite Element Mesh Generation - page 439

8.1.5.2 Touch
Edge Q 1 Q 2 is said to touch a triangular facet P 1 P 2 P 3 if it comes to within a very close dis-
tance with the triangle. Detailed formulas for the determination of the distance between a
line segment and a triangular facet are given in Section 2.4.8, and the following are the steps
in the determination of touches:

1. Calculate the barycentre co-ordinates of Q 1 and Q 2 w.r.t. triangle P 1 P 2 P 3 , (e 1 , f 1 , g 1 ) and
(e 2 , f 2 , g 2 )

e =⋅ e P Qf , =⋅ f PQ , g =⋅ g P Q ,

1 2 1 1 3 1 1 1 1

e =⋅ e P Qf , =⋅ f P QQg , =⋅ g PQ ,

2 2 2 2 3 2 2 1 2

where e , f and g are unit vectors along co-ordinate axes.

The minimum distance is given by the distance from an end point of the line segment to
the triangular facet, as shown in Figure 8.13a.

e ≥ 000 , f ≥ , g ≥ and hh ( − h ) ≥ 0 , thend = h

1 1 1 1 2 1 min 1

e ≥ 0 ,f ≥ 0 , g ≥ 0 andhhh ( −≥ ) 0 , then dh =

2 2 2 2 1 2 min 2

The minimum distance is given by the distance from an interior point of the line segment
to the triangular facet, as shown in Figure 8.13b.

QQ is on thesideofPP fe ≤ 0 or e ≤ , 0 d = ( dQQPP , )

12 23 12 2 3 1 2 m in

QQ is on thesideofPP if f ≤ 0 rf ≤ 0 , d = ( QQ PP , )

12 3 1 1 2 min 12 31

QQ is on the side of PP if g ≤ 0 org ≤ 0 , d = ( QQ PP , )

12 1 2 1 2 min 1 212

Segment Q 1 Q 2 is said to be in touch with triangle P 1 P 2 P 3 if

<× (

)

d ε min P PPPPPQQ , , , with et arbi ε  trarily to 1%

min 12 23 31 12

The distance between two line segments d(Q 1 Q 2 , P 1 P 2 ), etc., can be found in Section 2.4.3.

Q 2

P 1 P 3

Q 1

d min

(a)

Q 1

P 2

f

P 3

P 2

P 1

d min

(b)

Q 2

Figure 8.13 Determination of touches: (a) nearest point found at the interior; (b) nearest point found on
an edge.