An Improved Model of Nonuniform Coleochaete Cell Division

4.1. Case 1

The cell in the first quadrant keeps its lower boundary (the upper boundary of the fourth quadrant) fixed because of the “death” of the cell in the fourth quadrant; at the same time, the former continues to grow and reaches to the area of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$2 \pi$$ \end{document} according to the theory in Section 2.

We can calculate and make comparisons of the length of the division arc (cell membrane, see Figure 7) specifically. When the area of the cell in the first quadrant reaches \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$2 \pi$$ \end{document} , the center of the circle is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$( 0 , b ) \dot = ( 0 , 1.04815 )$$ \end{document} , and its radius is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$r \dot = 2.25801$$ \end{document} ; the central angle with respect to this arc is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\delta \dot = 2.05351$$ \end{document} . We can find that the length of the arc l₁ in Figure 7 is the shortest as 2.20886. The detailed data about the radius, the center, and the length of each arc can be found in Table 1 in Appendix C.

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FIG. 7.

First discussion in the first quadrant.

4.2. Case 2

Now we continue to discuss the division of individual parts. In the first quadrant, the outward growth of the first division arc (BB₁ in Fig. 8) keeps the direction of that of the rays (tangent) of the arc BB₁. And the central point of the outer boundary for new cells located on the ray, and the boundary, namely, the outer membrane, passes through point A.

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FIG. 8.

Second division in the southeast corner for the first quadrant.

The boundary here being a union of three curves is different from the line boundary (y-axis), but the shape of the outer boundary is still an arc. This process can be deduced theoretically as in Appendix A. The detailed division data can be found in Table 2 in Appendix C.

4.3. Case 3

The third division of the cell in the first quadrant begins with its central point of the outer boundary at (3.144,1.86759), and its corresponding radius is equal to 2.19012. After calculation, we find that the shortest length of the division curve of the cell is 2.16159, specifically in Figure 9 and Table 3.

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FIG. 9.

Third division of the cell in the first quadrant.

4.4. Case 4

Next to case 3, the cell on the right of the division membrane w₁ continuously expands in a normal way as the growth of cells in the second quadrant, and the left cell is in the constrained division. When the area of the left cell reaches \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$2 \pi$$ \end{document} , the circle point P(3.97075, 0.160892) of the cell locates on the ray of the last cell division line, and the radius of the cell's outer membrane is 1.97731. Note that, at this moment, the outer boundary, namely the cell's outer membrane, only intersects with the boundary of the “dead” cell in the fourth quadrant at point A. In Figure 10, at this time there are two possible divisions, assumed l₄₁, l₄₂. By calculation, we can find that the length of division line l₄₁ is 2.19165, and there's no solution in l₄₂. See Table 4 for specific data.

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FIG. 10.

Fourth division of the cell in the first quadrant.

4.5. Case 5

After case 4, the division membrane l₄₁ coincides with the membrane of the mother cell. In this way the original cell (the right cell in Fig. 10) has already been an “inner cell,” and it will not grow and divide. Only the cell on the outside will continue to grow (see Fig. 11). The detailed division data can be found in Table 5 in Appendix C.

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FIG. 11.

Fifth division of the cell in the first quadrant.

4.6. Case 6

After case 3, the cell on the right of the division membrane w₁ in Figure 9 begins to grow; the left cell grows very slowly because it is pressed by its left “inner cell.” This division process is similar with case 1 (Fig. 7). The central point of the outer membrane for the lower right new cell locates at (3.144, 1.86759), and its radius is 3.06062 (Fig. 12). It can be found that the difference between this kind of division and case shows that this division membrane ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{63}}$$ \end{document} being the shortest in Fig. 12) bends down, rather than the reverse situation in case 1. See Table 6 in Appendix C for detailed division data.

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FIG. 12.

Sixth division of the cell in the first quadrant.

4.7. Case 7

This division is later than its left neighbor cell; its way of division is similar with case 6. There are three possible division forms when the cell grows to the threshold of the area \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$2 \pi$$ \end{document} . After calculation, we find that the new membrane \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{72}}$$ \end{document} being 2.61411 in vertical direction is the shortest in Figure 13, and the old boundary \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{71}}$$ \end{document} of the original cell is not the shortest. The detailed division data can be found in Table 7 in Appendix C.

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FIG. 13.

Seventh division of the cell in the first quadrant.

4.8. Case 8

The fast-developing group of outer cells formed in case 7 begins to divide; the central point of the outer arc still remains at (3.144, 1.86759), and its radius increases to 5.25889. It is easy to find that the division arc formed by the original outer boundary \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{81}}$$ \end{document} is the shortest one (Fig. 14 and Table 8).

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FIG. 14.

Eighth division of the cell in the first quadrant.

4.9. Case 9

When the new cell formed in Case 5 is separated by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $${l_{51}}$$ \end{document} into two daughter cells because of the cell's symmetric growth, the cells in the third quadrant will also grow accordingly, except that the outer membrane of cells will be pressed by the original “dead” cell membranes in the fourth quadrant. This leads to a common boundary in the fourth quadrant, the bisector of the fourth quadrant in Figure 15. In the expanding process, some of cells in the fourth quadrant have division arcs convex toward the center of the whole cell (refer to the right-hand pictures in Fig. 15).

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FIG. 15.

Ninth division of the cell in the first quadrant.

6.1. Pending endpoint on the x-axis

In Figure 3, we denote two curves (one is a line) by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} C: \left\{ { \begin{matrix}{x = \alpha ( t ) , } & \\ { y = \beta ( t ) , } & \\ \end{matrix} } \right.t \in [ {t_ * } , {t^ * } ]; \ \ {\rm and} \ \ L: \left\{ { \begin{matrix}{x = t - {t^ * } , } & \\ { y = 0 , }\hfill & \\ \end{matrix} } \right.t \in [ {t^ * } , {t_ * } ]. \tag{10} \end{align*} \end{document}

Using the Lagrangian multiplier method, we can obtain the conditional variational problem \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} J ( L ) = \int_ { { t_ * } } ^ { { t^ * } } \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } dt + \lambda \left( { \frac { 1 } { 2 } \oint_ { C \bigcup L } x { \rm { d } } y - y { \rm { d } } x - \Omega } \right). \tag { 11 } \end{align*} \end{document}

Suppose that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$( x , y ) = ( { \alpha _1} ( t ) , { \beta _1} ( t ) ) \ ( t \in [ {t_ * } , {t^ * } ] )$$ \end{document} is an arbitrary perturbation curve, then the perturbation problem is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} J ( L_ { \varepsilon } ) & = \int_ { t_* } ^ { { t^* } ( \varepsilon ) } \sqrt { ( \alpha^ { \prime } ( t ) + \varepsilon \alpha^ { \prime } _1 ( t ) ) ^2 + ( \beta^ { \prime } ( t ) ) ^2 + \varepsilon \beta^ { \prime } _1 ( t ) )^2 { \rm { d } } t } & ( 12 ) \\ & + \frac { \lambda } { 2 } \int_ { t_* } ^ { { t^* } ( \varepsilon ) } ( \alpha + \varepsilon { \alpha _1 } ) { \rm { d } } ( \beta + \varepsilon { \beta _1 } ) - ( \beta + \varepsilon { \beta _1 } ) { \rm { d } } ( \alpha + \varepsilon { \alpha _1 } ) - \lambda \Omega. & ( 13 ) \end{align*} \end{document}

By calculating the derivative with respect to \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\varepsilon$$ \end{document} , at the point \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\varepsilon = 0$$ \end{document} , the equation above becomes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { { \rm { d } } ( J ( { L_ \varepsilon } ) ) } { { \rm { d } } \varepsilon } } { \Big \vert_ { { \varepsilon } = 0 } } & = \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } { \frac { \alpha^ { \prime } ( t ) { \alpha _1^ { \prime } } ( t ) + \beta^ { \prime } ( t ) { \beta _1^ { \prime } } ( t ) } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } { \rm { d } } t + \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \bigg \vert_ { { t^* } ( 0 ) } } { \frac { { \rm { d } } { t^* } ( 0 ) } { { \rm { d } } \varepsilon } } \\ & + \frac { \lambda } { 2 } \int_ { { t_* } } ^ { { t^* } ( 0 ) } ( { \alpha_1 } ( t ) \beta^ { \prime } ( t ) + \alpha ( t ) { \beta _1^ { \prime } } ( t ) - { \beta _1 } ( t ) \alpha^ { \prime } ( t ) - \beta ( t ) { \alpha_1^ { \prime } } ( t ) ) \ { \rm { d } } t \\ & + \frac { \lambda } { 2 } ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } ( t { ) ) \Big \vert_ { { t^* } ( 0 ) } } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } \\ & = \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } \bigg \vert_ { { t_* } } ^ { { t^* } ( 0 ) } \\ & - \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} { \frac { \rm { d } } { { \rm { d } } t } } { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } \\ & + \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \Big \vert_ { { t^ * } ( 0 ) } } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } + \frac { \lambda } { 2 } ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } ( t { ) ) \Big \vert_ { { t^* } ( 0 ) } } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } \\ & + \frac { \lambda } { 2 } \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } \{ { \alpha_1 } ( t ) , { \beta _1 } ( t ) \} \ { \rm { d } } \{ \beta ( t ) , - \alpha ( t ) \} + \frac { \lambda } { 2 } \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} \{ - \beta ( t ) , \alpha ( t ) \} \Big \vert_ { { t_ * } } ^ { { t^ * } ( 0 ) } \\ & - \frac { \lambda } { 2 } \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } \{ { \alpha_1 } ( t ) , { \beta _1 } ( t ) \} \ d \{ - \beta ( t ) , \alpha ( t ) \} . \end{align*} \end{document}

After simplification, one has \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { { \rm { d } } ( J ( { L_ \varepsilon } ) ) } { { \rm { d } } \varepsilon } } { \Big \vert_ { \varepsilon = 0 } } & = - \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} { \frac { \rm { d } } { { \rm { d } } t } } \left( { { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } - \lambda \{ \beta ( t ) , - \alpha ( t ) \} } \right) \\ & + \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } \bigg \vert_ { { t_ * } } ^ { { t^ * } ( 0 ) } + \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \Big \vert_ { { t^ * } ( 0 ) } } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } \\ & + \frac { \lambda } { 2 } \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} \{ - \beta ( t ) , \alpha ( t ) \} \Big \vert_ { { t_ * } } ^ { { t^ * } ( 0 ) } + \frac { \lambda } { 2 } ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } ( t ) ) \Big \vert_ { { t^ * } ( 0 ) } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } \tag { 14 } \end{align*} \end{document}

in which \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\displaystyle { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } = { \frac { { \rm { d } } { t^ * } ( \varepsilon ) } { { \rm { d } } \varepsilon } } { \Big \vert_ { \varepsilon = 0 } } $$ \end{document} . From the arbitrary of the vector \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\{ { \alpha _1} , { \beta _1} \} $$ \end{document} , it follows that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { \rm { d } } { { \rm { d } } t } } \left( { { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } - \lambda \{ \beta ( t ) , - \alpha ( t ) \} } \right) = 0. \tag { 15 } \end{align*} \end{document}

One can find that the general solution of Equation (14) is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \{ \alpha ( t ) , \beta ( t ) \} = \frac { 1 } { \lambda } \{ \sin ct , \cos ct \} + \{ { c_1 } , { c_2 } \} . \tag { 16 } \end{align*} \end{document}

6.2. Pending endpoint on the y-axis

Suppose that the angle between \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$x$$ \end{document} -axis and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$y$$ \end{document} -axis is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta$$ \end{document} for the curve (II) in Figure 3 ( \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\theta$$ \end{document} may be acute angle, right angle, and obtuse angle), then one has that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} C: \left\{ { \begin{matrix}{x = \alpha ( t ) , } & \\{ y = \beta ( t ) , } & \\\end{matrix} } \right.t \in [ {t_ * } , {t^ * } ]; \tag{17} \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {L_1}: \left\{ { \begin{matrix}{x = ( {t_1} - t ) \cos \theta , } & { } \\ { y = ( {t_1} - t ) \sin \theta , } & { } \\\end{matrix} } \right.t \in [ {t^ * } , {t_1} ]; \tag{18} \end{align*} \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} {L_2}: \left\{ { \begin{matrix}{x = t - {t_1} , } & \\ { y = 0 , }\hfill & \\\end{matrix} } \right.t \in [ {t_1} , {t_2} ]. \tag{19} \end{align*} \end{document}

Let us see the variational problem: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} J ( L ) = \int_ { { t_ * } } ^ { { t^ * } } \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \rm { d } } t + \lambda \left( { \frac { 1 } { 2 } \oint_ { C \bigcup { L_1 } \bigcup { L_2 } } x \ { \rm { d } } y - y \ { \rm { d } } x - \Omega } \right). \tag { 20 } \end{align*} \end{document}

Suppose that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$( x , y ) = ( { \alpha _1} ( t ) , { \beta _1} ( t ) ) \ ( t \in [ {t_ * } , {t^ * } ] )$$ \end{document} is an arbitrary perturbation curve, the problem (20) becomes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} J ( { L_ \varepsilon } ) & = \int_ { { t_* } } ^ { { t^* } ( \varepsilon ) } \sqrt { ( \alpha^ { \prime } ( t ) + \varepsilon \alpha^ { \prime } _1 ( t ) ) ^2 + ( \beta^ { \prime } ( t ) ) ^2 + \varepsilon \beta^ { \prime } _1 ( t ) ) ^2 { \rm { d } } t } \\ & + \frac { \lambda } { 2 } \int_ { { t_ * } } ^ { { t^ * } ( \varepsilon ) } ( \alpha ( t ) + \varepsilon { \alpha _1 } ( t ) ) { \rm { d } } ( \beta ( t ) + \varepsilon { \beta _1 } ( t ) ) - ( \beta ( t ) + \varepsilon { \beta _1 } ( t ) ) { \rm { d } } ( \alpha ( t ) + \varepsilon { \alpha _1 } ( t ) ) \\ & + \frac { \lambda } { 2 } \int_ { { t^ * } } ^ { { t_1 } } ( { t_1 } - t ) \cos \theta \ { \rm { d } } ( { t_1 } - t ) \sin \theta - ( { t_1 } - t ) \sin \theta \ { \rm { d } } ( { t_1 } - t ) \cos \theta - \lambda \Omega. \end{align*} \end{document}

At the critical point \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\varepsilon = 0$$ \end{document} , it has \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { { \rm { d } } ( J ( { L_ \varepsilon } ) ) } { { \rm { d } } \varepsilon } } { \Big \vert_ { \varepsilon = 0 } } & = \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } { \frac { \alpha^ { \prime } ( t ) \alpha _ { 1 } ^ { \prime } ( t ) + \beta^ { \prime } ( t ) \beta _ { 1 } ^ { \prime } ( t ) } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } { \rm { d } } t + \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \bigg \vert_ { { t^ * } ( 0 ) } } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } \\ & + \frac { \lambda } { 2 } \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } ( { \alpha _1 } ( t ) \beta^ { \prime } ( t ) + \alpha ( t ) \beta _ { 1 } ^ { \prime } ( t ) - { \beta _1 } ( t ) \alpha^ { \prime } ( t ) - \beta ( t ) { \alpha_ { 1 } ^ { \prime } } ( t ) ) { \rm { d } } t \\ & + \frac { \lambda } { 2 } ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } ( t { ) ) \Big \vert_ { { t^ * } ( 0 ) } } { \frac { { \rm { d } } { t^ * } ( 0 ) } { d \varepsilon } } . \end{align*} \end{document}

Using integrating by part, we can find that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}{ \frac { { \rm { d } } ( J ( { L_ \varepsilon } ) ) } { { \rm { d } } \varepsilon } } { \Big\vert _ { \varepsilon = 0 } } & = - \int_ { { t_ * } } ^ { { t^ * } ( 0 ) } \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} { \frac { \rm { d } } { { \rm { d } } t } } \left( { { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } - \lambda \{ \beta , - \alpha \} } \right) \\ & + \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } \Bigg \vert_ { { t_ * } } ^ { { t^ * } ( 0 ) } \\ & + \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \Big \vert_ { { t^ * } ( 0 ) } } { \frac { d { t^ * } ( 0 ) } { d \varepsilon } } + \frac { \lambda } { 2 } ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } (t) ) \Big \vert_ { { t^ * } ( 0 ) } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } \\ & + \frac { \lambda } { 2 } \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} \{ - \beta ( t ) , \alpha ( t ) \} \Big \vert_ { { t_ * } } ^ { { t^ * } ( 0 ) } . \tag { 21 } \end{align*} \end{document}

Because of the arbitrary of \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\{ { \alpha _1} , { \beta _1} \} $$ \end{document} , we have an equation system as (15), \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { \rm { d } } { { \rm { d } } t } } \left( { { \frac { \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } - \lambda \{ \beta ( t ) , - \alpha ( t ) \} } \right) = 0. \tag { 22 } \end{align*} \end{document}

6.3. Boundary condition on pending endpoint

By combining (14) and (21), we find that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} & { \frac { \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} \cdot \{ \alpha^ { \prime } ( t ) , \beta^ { \prime } ( t ) \} } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } \Big \vert_ { { t_ * } } ^ { { t^ * } ( 0 ) } + \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \bigg \vert_ { { t^ * } ( 0 ) } } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } \\ & + \frac { \lambda } { 2 } \{ { \alpha _1 } ( t ) , { \beta _1 } ( t ) \} \{ - \beta ( t ) , \alpha ( t ) \} \Big \vert_ { { t_ * } } ^ { { t^ * } ( 0 ) } + \frac { \lambda } { 2 } ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } ( t) ) \Big \vert_ { { t^ * } ( 0 ) } { \frac { { \rm { d } } { t^ * } ( 0 ) } { { \rm { d } } \varepsilon } } = 0. \tag { 23 } \end{align*} \end{document}

At the pending point t*, it has \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha ( {t^ * } ) + \varepsilon { \alpha _1} ( {t^ * } ) = ( {t_1} - {t^ * } ) \cos \theta; \quad \beta ( {t^ * } ) + \varepsilon { \beta _1} ( {t^ * } ) = ( {t_1} - {t^ * } ) \sin \theta. \tag{24} \end{align*} \end{document}

We calculate the derivative with respect to ɛ, and note that the derivative is zero at \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\varepsilon = 0$$ \end{document} , so we have that, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} { \frac { { \rm { d } } ( { t^ * } ( 0 ) ) } { { \rm { d } } \varepsilon } } = - { \frac { { \alpha _1 } ( t ) } { \alpha^ { \prime } ( t ) + \cos \theta } } = - { \frac { { \beta _1 } ( t ) } { \beta^ { \prime } ( t ) + \sin \theta } } . \end{align*} \end{document}

Substitute \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\displaystyle { \frac { { \rm { d } } ( { t^* } ( 0 ) ) } { { \rm { d } } \varepsilon } } $$ \end{document} into the boundary condition (24), one gets that at t = t*, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} & 0 = { \frac { \alpha^ { \prime } ( t ) } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } - \frac { \lambda } { 2 } \beta ( t ); & ( 25 ) \\ & 0 = { \frac { \beta^ { \prime } ( t ) } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } + \frac { \lambda } { 2 } \alpha ( t ) - { \frac { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } { \beta^ { \prime } ( t ) + \sin \theta } } \\ & + \frac { \lambda } { 2 } ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } ( t ) ) { \frac { - 1 } { \beta^ { \prime } ( t ) + \sin \theta } } . & ( 26 ) \end{align*} \end{document}

And substitute \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\displaystyle { \frac { \alpha^ { \prime } ( t ) } { \beta ( t ) \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } = \frac { \lambda } { 2 } $$ \end{document} in (25) into (26), we obtain that \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}& { \frac { \beta^ { \prime } ( t ) } { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } + { \frac { \alpha ( t ) \alpha^ { \prime } ( t ) } { \beta ( t ) \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } - { \frac { \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } { \beta^ { \prime } ( t ) + \sin \theta } } \\ & - { \frac { ( \alpha ( t ) \beta^ { \prime } ( t ) - \beta ( t ) \alpha^ { \prime } ( t ) ) \alpha^ { \prime } ( t ) } { ( \beta^ { \prime } ( t ) + \sin \theta )\ \beta ( t ) \sqrt { { { ( \alpha^ { \prime } ( t ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } } } = 0. \tag { 27 } \end{align*} \end{document}

That is, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \beta^ { \prime } ( t ) + { \frac { \alpha ( t ) \alpha^ { \prime } ( t ) } { \beta ( t ) } } - { \frac { \alpha^ { \prime } ( t { { ) ) } ^2 } + { { ( \beta^ { \prime } ( t ) ) } ^2 } } { \beta^ { \prime } ( t ) + \sin \theta } } - { \frac { \alpha ( t ) \alpha^ { \prime } ( t ) \beta^ { \prime } ( t ) - \beta ( t ) ( \alpha^ { \prime } ( t { { ) ) } ^2 } } { \beta ( t ) ( \beta^ { \prime } ( t ) + \sin \theta ) } } = 0. \tag { 28 } \end{align*} \end{document}

By multiplying with \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$\beta ( t ) ( \beta^{ \prime} ( t ) + \sin \theta )$$ \end{document} in both sides, it becomes \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*}\begin{split} & \beta ( t ) \beta^{ \prime} ( t ) ( \beta^{ \prime} ( t ) + \sin \alpha ) + \alpha ( t ) \alpha^{ \prime} ( t ) ( \beta^{ \prime} ( t ) + \sin \theta ) \\ & - \beta ( t ) ( \alpha^{ \prime} ( t)) {^2} - \beta (t) ( \beta^{ \prime} (t)) {^2} - \alpha ( t ) \alpha^{ \prime} ( t ) \beta^{ \prime} ( t ) + \beta ( t ) ( \alpha^{ \prime} (t)) ^{2} = 0. \end{split} \tag{29} \end{align*} \end{document}

And then, \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \beta ( t ) \beta^{ \prime} ( t ) \sin \theta + \alpha ( t ) \alpha^{ \prime} ( t ) \sin \theta = 0 \tag{30} \end{align*} \end{document}

Because at the point \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$t = {t^* }$$ \end{document} \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} \alpha ( {t^ * } ) = ( {t_1} - {t^ * } ) \cos \theta \ \ \beta ( {t^ * } ) = ( {t_1} - {t^ * } ) \sin \theta , \tag{31} \end{align*} \end{document}

we find \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} \begin{align*} & ( { t_1 } - { t^ * } ) \sin \theta \beta^ { \prime } ( t ) \sin \theta + ( { t_1 } - { t^ * } ) \cos \theta \ \alpha^ { \prime } ( t ) \sin \theta = 0 \\ & \qquad \quad \,\; i.e. \qquad \qquad \qquad \qquad \quad { \frac { \beta^ { \prime } ( t ) } { \alpha^ { \prime } ( t ) } } \tan \theta = - 1. \tag { 32 } \end{align*} \end{document}

Therefore, the arc is vertical to the boundary at \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$t = {t^ * }$$ \end{document} .

Let us summarize the two cases mentioned above: the shortest curve we are looking for should be an arc and is vertical to the boundary at \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland, xspace}\usepackage{amsmath, amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document} $$t = {t^*}$$ \end{document} .