Modified Intersection Method for Solving Alignment Problems Containing Incomplete Transition Curves
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摘要:
为解决公路线形设计过程中交点法依赖的“非对称基本型”计算模型，在转弯曲线包含非完整缓和曲线时出现计算失效的问题，以“非对称基本型”计算模型为基础，通过分析计算模型在解算非完整缓和曲线场景下的失效原因，对计算模型结构与求解逻辑予以优化改进，进而提出“非对称通用型”计算模型. 此模型新定义了“缓和曲线方向”，通过判断缓和曲线曲率变化趋势与路线行进方向间的关系，将缓和曲线划分为正向和逆向两类，再根据缓和曲线在单曲线中的前后关系，建立特殊局部坐标系，通过几何推导，得出非完整缓和曲线切线增长值和曲线内移值，进而可以使用“非对称基本型”计算模型进行求解. 研究表明：“非对称通用型”计算模型突破了“非对称基本型”计算模型对线形组合类型的限制，允许缓和曲线起、终点曲率半径可为任意值；通过与传统线元法对同一复杂曲线段进行解算对比，控制桩号里程值及控制桩号坐标的计算差异均小于1 mm，满足工程精度要求.
Abstract:To address the issue that the “asymmetric basic type” calculation model, which the intersection method relies on in highway alignment design, fails to perform calculations when turning curves include incomplete transition curves, the “asymmetric basic type” calculation model was used as the foundation. By analyzing the causes of the model’s failure in solving incomplete transition curve scenarios, the structure and solution logic of the calculation model were optimized and improved, and an “asymmetric general type” calculation model was further proposed. This new model introduced a novel definition of “transition curve direction”. It classified transition curves into two categories, positive and negative, by judging the relationship between the curvature change trend of the transition curve and the route’s traveling direction. Then, based on the positional order of the transition curve within a single curve, a special local coordinate system was established. Through geometric derivation, the tangential growth value and curve offset value of the incomplete transition curve were obtained, enabling the subsequent use of the “asymmetric basic type” calculation model for further solution. The research has shown that the “asymmetric general type” calculation model eliminates the restrictions of the “asymmetric basic type” model on alignment combination types, allowing the curvature radii at the start and end points of transition curves to be arbitrary values. By comparing the calculation results of the same complex curve segment with those obtained using the traditional element method, the differences in the calculated mileage values and coordinates of the control stakes are both less than 1 mm, which meets the engineering accuracy requirements.
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图 2 完整缓和曲线与非完整缓和曲线图示
Figure 2. Illustration of complete transition curve and incomplete transition curve
图 4 正向缓和曲线内移值$ p $和切线增长值$ q $求解模型
Figure 4. Solving model of offset value $ p $ and tangential growth value $ q $ of positive transition curve
图 5 逆向缓和曲线内移值$ p $和切线增长值$ q $求解模型
Figure 5. Solving model of offset value $ p $ and tangential growth value $ q $ of negative transition curve
表 1 “非对称通用型”参数列表
Table 1. “Asymmetric universal type” parameters
曲线段落 参数名 参数说明 前缓和曲线 $ {L}_{{\mathrm{s}}1} $ 前缓和曲线长度 $ {R}_{1} $ 前缓和曲线在$ P_{\mathrm{ZH}} $点处曲率半径，当
为完整缓和曲线时，有$ {R}_{1}=\infty $$ {R}_{3} $ 前缓和曲线在$ P_{\mathrm{HY}} $点处曲率半径，等
于相接圆曲线半径$ {R}_{{\mathrm{c}}} $圆曲线 $ {R}_{{\mathrm{c}}} $ 相接圆曲线半径 后缓和曲线 $ {L}_{{\mathrm{s}}2} $ 后缓和曲线长度 $ {R}_{2} $ 后缓和曲线在$ P_{{\mathrm{HZ}}} $点处曲率半径，当
为完整缓和曲线时，有$ {R}_{2}=\infty $$ {R}_{4} $ 后缓和曲线在$ P_{{\mathrm{YH}}} $点处曲率半径，等
于相接圆曲线半径$ {R}_{{\mathrm{c}}} $交点坐标 （$ {x}_{0},{y}_{0} $） PJD0坐标 （$ {x}_{1},{y}_{1} $） PJD1坐标 （$ {x}_{2},{y}_{2} $） PJD2坐标 $ a $ 转向角 
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表 2 缓和曲线方向分类
Table 2. Classification of transition curve direction
曲线段 缓和曲线端曲率点半径关系 曲线方向 前缓和曲线 $ {R}_{1} $>$ {R}_{3} $ 正向 $ {R}_{1} $<$ {R}_{3} $ 逆向 后缓和曲线 $ {R}_{2} $>$ {R}_{4} $ 正向 $ {R}_{2} $<$ {R}_{4} $ 逆向
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表 3 正向缓和曲线各求解步骤间参数输入输出关系
Table 3. Parameter input and output relationship between each solving step of positive transition curve
步骤 输入 输出 1 $ {L}_{{\mathrm{S}}} $、$ {R}_{{\mathrm{max}}} $、$ {R}_{{\mathrm{min}}} $ $ {L}_{{\mathrm{t}}} $、$ A $、$ {L}_{{\mathrm{m}}} $ 2 步骤1输出 $ x $、$ {y} $、$ {\beta }_{{\mathrm{n}}} $、$ {p}_{0} $、$ {q}_{0} $ 3 步骤1输出 $ \beta $ 4 步骤3输出 $ {k}_{AC} $、$ {k}_{OD} $ 5 步骤2、4输出 $ {x}_{O} $、$ {y}_{O} $、$ {x}_{D} $、$ {y}_{D} $ 6、7 步骤2、3、5输出 $ p $、$ q $、$ {\beta }_{{\mathrm{d}}} $
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表 4 截取段落路线设计数据
Table 4. Design data of route section
点位 前缓和曲线 圆曲线 后缓和曲线 转角 设计坐标值 切线方位角 $ {L}_{S1} $/m $ {R}_{1} $/m $ R $/m $ {L}_{S2} $/m $ {R}_{2} $/m $ \alpha $/（°） $ x $/m $ y $/m $ \varphi $（°） PJD1 52.747 1000 85 1e-7 1e7 48.4839 482328.7674 3450066.5663 346.4092 PJD2 46.886 85 52 1e-7 1e7 104.1498 482395.2261 3450161.8571 34.8931 PJD3 1e-7 1e7 52 58.516 1e7 94.5106 482478.8952 3450065.4611 139.0429
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表 5 待校核点设计数据
Table 5. Design data of points to be checked
m 点位 坐标设计值 桩号里程 $ x $ $ y $ PYH, 1 482343.0828 3450007.3520 478.846 PHY, 2 482336.9930 3450059.4841 531.593 PYH, 3 482351.8541 3450099.6689 574.905 PHY, 4 482389.0167 3450126.5220 621.791 PYY, 5 482440.2371 3450109.9994 678.359 PYH, 6 482449.3910 3450056.9642 734.927 PHZ, 7 482410.2139 3450014.7386 793.342
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表 6 各交点前、后缓和曲线参数计算结果
Table 6. Calculation results of initial and terminal transition curve parameters at each intersection
交点 待求参数 前缓和曲线 后缓和曲线 $ {L}_{{\mathrm{t}}} $/m 57.64699454 1e-7 $ A_0 $/m 69.99996097 3e-3 $ {L}_{{\mathrm{m}}} $/m 4.899994536 0 $ {\beta }_{{\mathrm{n}}} $（$ {\text{°}} $） 0.339099968 0 $ {p}_{0} $/m 1.62233421 0 $ {q}_{0} $/m 28.71337018 0 PJD1 $ \beta $（$ {\text{°}} $） 0.002449997 0 切点坐标/m （ 4.899991594 ,0.004001656 ）（0,0） $ {k}_{AC} $ 0.002450002 0 $ {k}_{OD} $ − 408.1629038 0 圆心$ O $坐标/m （ 28.71337018 ,86.62233421 ）（0,85） $ q $/m 24.02552159 0 $ p $/m 1.559729938 0 $ {\beta }_{{\mathrm{n}}} $/rad 0.336649971 0 $ q $/m 8.609745433 0 PJD2 $ p $/m 1.872194183 0 $ {\beta }_{{\mathrm{n}}} $/rad 0.729598901 0 $ q $/m 0 28.90335233 JD3 $ p $/m 0 2.703741562 $ {\beta }_{{\mathrm{n}}} $/rad 0 0.561695228
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表 7 各交点切线值计算结果
Table 7. Calculation results of tangential values at each intersection
切线长/m JD1 JD2 JD3 $ {T}_{1} $ 60.92011083 75.81834455 58.97586509 $ {T}_{2} $ 40.35839291 68.66738024 85.38036815
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表 8 各交点控制点坐标计算结果
Table 8. Calculation results of coordinates of control points at each intersection
m 点位 坐标项 JD1 JD2 JD3 直缓点 $ x $ 482343.0826 482351.8544 482440.2369 $ y $ 3450007.3517 3450099.6693 3450109.9997 缓直点 $ x $ 482351.8539 482440.2371 482410.2142 $ y $ 3450099.6688 3450109.9995 3450014.7390
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表 9 各控制桩坐标计算结果
Table 9. Calculation results of coordinates of each control stake
m 验证点位 坐标项 设计值 计算值 误差绝对值 PYH, 1 $ x $ 482343.0828 482343.0826 2 e-4 $ y $ 3450007.3520 3450007.3517 3 e-4 PHY, 2 $ x $ 482336.9930 482336.9929 1 e-4 $ y $ 3450059.4841 3450059.484 2 e-4 PYH, 3 $ x $ 482351.8541 482351.8539 2 e-4 $ y $ 3450099.6689 3450099.6688 1 e-4 PHY, 4 $ x $ 482389.0167 482389.0165 2 e-4 $ y $ 3450126.5220 3450126.5223 3 e-4 PYY, 5 $ x $ 482440.2370 482440.2369 1 e-4 $ y $ 3450109.9994 3450109.9997 3 e-4 PYH, 6 $ x $ 482449.3910 482449.3908 2 e-4 $ y $ 3450056.9642 3450056.964 3 e-4 PHZ, 7 $ x $ 482410.2139 482410.2142 3 e-4 $ y $ 3450014.7386 3450014.7390 4 e-4
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表 10 各控制桩坐里程桩号计算结果
Table 10. Calculation results of mileage station of each control stake
m 验证点位 桩号校准值 计算值 误差绝对值 PYH, 1 478.846 478.846 <1e-3 PHY, 2 531.593 531.593 <1e-3 PYH, 3 574.905 574.905 <1e-3 PHY, 4 621.791 621.791 <1e-3 PYY, 5 678.359 678.359 <1e-3 PYH, 6 734.342 734.927 <1e-3 PHZ, 7 739.342 793.342 <1e-3
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