Geology Reference
In-Depth Information
4. Choose point O
A
anywhere on line AA' on the opposite side of Z from A. (If the length
O
A
-Z is very large, it is equivalent to drawing a straight line through A.)
5. On BB' locate point D such that BD = AO
A
.
6. Draw DO
A
connecting D and O
A
.
7. Erect the perpendicular bisector of DO
A
. This line intersects BB' at O
B
.
8. Draw O
A
E' through O
A
and O
B
, intersecting AB at E.
9. With center O
A
and radius AO
A
, draw an arc from A, intersecting O
A
E' at F.
10. With center O
B
and radius BO
B
, draw an arc from B intersecting O
A
E' at F. This com-
pletes the interpolation.
If a correct solution is not obtained, it may be because the sense of curvature changes
across an inflection point, causing the centers of curvature to be on opposite sides of
the key bed. Modify step 4 above by using the alternate position of O
A
(Fig. 6.29:
alternate O
A
), located between C and A.
6.4.2.3
Other Smooth Curves
Interactive computer drafting programs provide several different tools for drawing
smooth curves through or close to a specified set of points. Typically they are para-
metric cubic curves for which the first derivatives, that is the tangents, are continuous
where they join (Foley and Van Dam 1983). In this respect the curves are like the
method of circular arcs, for which the tangents are equal where the curve segments
join, but cubics are able to fit more complex curves than just segments of circular
arcs. Two different smooth curve types are widely available in interactive computer
drafting packages, Bézier and spline curves. The two curve types differ in how they
fit their control points and in how they are edited. Both types are useful in producing
smoothly curved lines and surfaces (Foley and Van Dam 1983; De Paor 1996).
A Bézier curve consists of segments that are defined by four control points,
two anchor points on the curve (P
1
and P
4
, Fig. 6.30a) and two direction points (P
2
and P
3
) that determine the shape of the curve. The curve always goes through the
anchor points. The shape is controlled in interactive computer graphics applications
by moving the direction points. In a computer program the direction points may
be connected to the anchors by lines to form handles (Fig. 6.30b) that are visible in
the edit mode. At the join between two Bézier segments, the handles of the shared
anchor point are colinear, ensuring that the slopes of the curve segments match at the
intersection.
A spline curve only approximates the positions of its control points (Fig. 6.31) but
is continuous in both the slope and the curvature at the segment boundaries, and so
the curve is even smoother than the Bézier curve (Foley and Van Dam 1983). The
shape is controlled in interactive computer graphics applications by moving the con-
trol points that are visible in the edit mode. This curve type should be drawn sepa-
rately from the actual data points because editing the curves changes the locations of
the points that define the curve. The control points can be manipulated until the match
between the curve and the data points is acceptable.
