Page 211 - Basic Structured Grid Generation

200 Basic Structured Grid Generation

D
F
R ABD

A M B

E
R ABC

C

Fig. 8.13 Non-accepted and accepted triangles.

Writing lengths AM = p and MF = q, it is straightforward to show that insertion
of a new point at the triangle circumcentre F (by choosing X to coincide with F) would
2
2
give a triangle AFB with circumradius given by (p +q )/2q. Since the smallest radius
of any circle passing through the points A and B is clearly p (the circle with centre
M), we must have
2 2
(p + q )/2q p
(which is also true for simple algebraic reasons). If f M p, the best choice for X
is such that the circumradius R ABX of the triangle ABX is equal to p. This occurs
when MX = p and the angle AXB is a right-angle. If f M >p, however, we locate X
so that
2 2
R ABX = min(f M ,(p + q )/2q)
which will give a position for X between the previously stated position and F. In this
case we have

2
2
MX = R ABX + (R ABX ) − p . (8.6)
In either case we can write as in Liseikin (1999)
2
2
R ABX = min{max(f M , p), (p + q )/2q}, (8.7)
with the position of X still given by (8.6).
Suppose that f M <p < 1.5f M and q is large compared with p. Then the point X is
chosen such that the angle AXB is a right-angle; moreover, the triangle AXB satisfies
the criterion to be accepted. The edges XA and XB will now be candidates for the
next active edge, assuming that the triangle DAX is non-accepted. Here we consider