According to the definition of the SG-Bézier curve, an SG-Bézier curve has one global and multiple local shape parameters. In view of this, we can modify the local or global shape under the precondition that the G1, G2 and C1, C2 smooth continuity of the composite curves remain unchanged. In what follows, we shall discuss the shape adjustability of the composite SG-Bézier curves with G1, G2 and C1, C2 continuity, which were constructed by two SG-Bézier curves. Moreover, a similar discussion can be carried out on the multiple curves case.
5.1. Shape Adjustment of the Composite SG-Bézier Curves with C1 or G1 Smooth Continuity
Proposition 1. For a composite SG-Bézier curve with C1smooth continuity, some significant conclusions can be obtained when keeping its control points and C1smooth continuity unchanged.
(1) We can adjust the local shape of the composite curve by altering(or),).
(2) We can modify the global shape of a composite curve by alteringand(or the parameters, and) to achieve the global shape adjustment of the splicing curve. Here, at least one of the shape parameters(respectively) should be changed.
Proof. As the C1 continuity conditions of Equation (18) just involve the shape parameters and , then we can modify the local shape by altering the shape parameters that are not contained in Equation (18). Thus, conclusion (1) is proved. On the other hand, if we change at least one of the shape parameters , then at least one of the shape parameters should be changed to ensure that the C1 smooth continuity is maintained unchanged for the splicing curve (to modify the shape parameters in terms of the second equation in Equation (18)). Therefore, the global shape adjustment of the splicing curve can be achieved by changing the values of the shape parameters:, and . This completes the proof of Proposition 1. □
Proposition 2. Keeping the G1smooth continuity between splicing curves unchanged, we have:
(1) We can modify the local shape of the composite curve by altering the shape parametersor.
(2) We can modify the global shape of the composite curve by altering the shape parametersandat the same time.
Proof. According to equations (15) and (3), G
1 continuity merely requires that the tangent directions at the common joint of the two splicing SG-Bézier curves are the same, that is:
This shows that altering any of the shape parameters of part of the curves simply exerts an influence on the size of the tangent vector without changing its direction. So, the global or local shape can be adjusted once we change the shape parameters of part of the curves while keeping the control points and G1 smooth continuity the same. Thus, Proposition 2 is proven. □
5.2. Modeling Examples of Shape Adjustment for the Composite SG-Bézier Curves with C1 or G1 Smooth Continuity
Figure 9 shows the
C1 smooth continuity of two SG-Bézier curves. In
Figure 9, the blue curve is the pre-given SG-Bézier curve, and the red SG-Bézier curve is now constructed by the continuity conditions in Equation (18). The shape parameters are
,
and
.
Figure 9b shows the local shape adjustment of the composite curve just by modifying one shape parameter
. The upper part of the splicing curve is now modified, where the shape parameters of solid lines, dashed lines, and dotted lines are
, respectively.
Figure 9c shows the local shape adjustment by altering the shape parameters
, and now, the lower part of the splicing curve is modified, with the shape parameters
(solid lines),
(dashed lines), and
(dotted lines). We see that the global shape adjustment can be achieved in
Figure 9d by changing the shape parameters
and
, with the shape parameters
(solid lines),
(dashed lines), and
(dotted lines). The rest of the shape parameters are equal to those in
Figure 9a, expect that they are modified in the other three figures.
Figure 10 shows the local and global shape adjustment of the composite SG-Bézier curves with G
1 smooth continuity. In
Figure 10a, the G
1 smooth continuity and the shape parameters are the same as that in
Figure 5b.
Figure 10b shows the global shape adjustment by altering the shape parameters
. Now, we only adjust the left part of the splicing curve with the shape parameters
(solid lines),
(dashed lines), and
(dotted lines).
Figure 10c shows the graph of the local adjustment for the splicing curve by modifying shape parameters
, where we only make a adjustment for the right part of the splicing curve with the shape parameters
(solid lines),
(dashed lines), and
(dotted lines).
Figure 10d shows the graph of the global adjustment by changing the shape parameters
, with the shape parameters
(solid lines),
(dashed lines), and
(dotted lines). The rest of the shape parameters are the same as those in
Figure 10a, expect for those that are modified in
Figure 10.
5.3. Shape Adjustment of the Composite SG-Bézier Curves with C2 or G2 Smooth Continuity
Proposition 3. Suppose that the control points and C2smooth continuity for the splicing curve remain unchanged, we have the following conclusions:
(1) We can alter the local shape of a composite curve by adjusting shape parametersor.
(2) We can alter the global shape of a composite curve by changing the parameters(or) and(or) simultaneously. In addition, in the sole situation where the C2smooth continuity is kept unchanged, the global shape of the composite curve can also be adjusted by modifying the following shape parameters and control points:
① By changing the two shape parametersand, as well as two control pointsand.
② By modifying the shape parameters,,(or,) and the control points,; or altering the shape parameters,,(or,) and the control points,; or changing the shape parameters,,(or,) and(or,), and the control points,.
③ By altering the shape parametersandand the control pointsand.
Here, we can calculate the control pointsandaccording to the second and third equation in Equation (27).
Proof. According to the C2 continuity conditions in Equation (27), if the control points and C2 smooth continuity remain unchanged, we can alter the local shape of the composite curve by changing these shape parameters, which are not included in the conditions in Equation (27). Thus, conclusion (1) is proved. □
As the conditions in Equation (27) involve the global shape parameters , and local shape parameters , , , , we merely change these shape parameters (apparently these parameters are not included in smooth continuity conditions) to adjust the shape of the composite curve while keeping the control points and C2 smooth continuity the same. Now, we only adjust the local shape of the composite curve. So, we cannot modify the global shape only by altering the shape parameters. As and are the global shape parameters, we might alter the two control points and simultaneously to ensure that C2 smooth continuity is unchanged once we adjust the global shape by changing and . Similarly, as the C2 smooth continuity conditions involve the parameters , , , , and , we can adjust the global shape of the two parts that go beyond the splicing curve by modifying the shape parameters , , and , , , as well as the two control points and to ensure that the C2 smooth continuity is unchanged. Obviously, the global shape adjustment can be achieved by modifying the shape parameters , , and the control points and . Thus, conclusion (2) is proved. This completes the proof of Proposition 3.
Proposition 4. For a composite SG-Bézier curve with G2smooth continuity, some significant conclusions can be obtained when keeping its control points and G2smooth continuity unchanged.
(1) We can modify the local shape of the composite curve by changing the shape parametersor.
(2) The global shape of a composite curve can be altered by changing the parameters(or) and(or) simultaneously. Moreover, if only the G2smooth continuity remains unchanged, we can modify the global shape of the composite curve by altering the following shape parameters and control points:
① By altering the shape parametersand, as well as the control pointsand.
② By modifying shape parameters,,(or,) and the control pointsand; or modifying shape parameters,,(or,) and the control pointsand; or modifying shape parameters,,(or,),(or,) and the control pointsand;
③ By modifying the shape parametersand, as well as two control pointsand.
Here, we can calculate the control pointsandaccording to the second and third equation in Equation (19).
Proof. Proposition 4 can follow by the similar argument in Proposition 3. Thus, we omit the proof. □
5.4. Examples of Shape Adjustment between SG-Bézier Curves with C2 and G2 Smooth Continuity
Figure 11 gives an example to show the C
2 smooth continuity between two SG-Bézier curves of degree 3. In this figure, the blue curve is the given SG-Bézier curve, and the red curve is constructed by the continuity conditions in Equation (27), which together with the blue curve reaches C
2 smooth continuity. The solid lines denote the curves before adjustment, while the dashed lines and dotted lines denote the curves after the adjustment of the shape parameters. The circular points denote the points before adjustment, and the asterisks indicate the modified control points. The shape parameters in
Figure 11a are
and
.
Figure 11b displays the global shape adjustment of the composite curve by altering the shape parameters
, where the shape parameters for the solid lines, dashed lines, and dotted lines are
,
, and
, respectively.
Figure 11c displays the local shape adjustment by using a single shape parameter
, where the shape parameters are
(solid lines),
(dashed lines), and
(dotted lines), respectively. Based on
Figure 11a,
Figure 11d illustrates the global shape adjustment by modifying the shape parameters
. From the figure, we realize the global shape adjustment for the dashed lines merely using the shape parameters
(with
), and the dotted lines by changing shape parameters
(with
) . The rest of the shape parameters are equal to those in
Figure 11a expect for the ones that are modified in the other three figures.
Figure 12 shows the shape adjustment of G
2 smooth continuity between the two SG-Bézier curves of degree 3. The shape parameters in
Figure 12a are the same as those in
Figure 7b. As argued in the paragraph above, the solid line means the curves before adjustment, while the dotted lines and dashed lines are the curves after the shape parameters are changed. The circular points in
Figure 12 denote the control points before adjustment, and the asterisks marked on the control polygon are the modified control points.
Figure 12b displays the local shape adjustment by changing the shape parameters
, with the shape parameters
(solid lines),
(dashed lines), and
(dotted lines).
Figure 12c illustrates the local shape adjustment by using a single shape parameter
, with the shape parameter
(solid lines),
(dashed lines) and
(dotted lines).
Figure 12d gives an example showing the global shape adjustment by modifying the shape parameters
. The dashed lines achieve the global shape adjustment merely by using the shape parameters
(with
). The dotted lines realize the global shape adjustment in
Figure 12d by altering the shape parameters
(with
). The rest of the shape parameters are all equal to those in
Figure 12a expect for the ones that are modified in the other three figures.