Curve Mileage Inversion Using Normal Line Intersection
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Solution Overview
Problem
Existing methods for calculating the inverse line mileage and offset based on known coordinate points often result in non-convergent and complex calculations, particularly for curve elements with large rotation angles, leading to infinite loops and difficulties in determining the feasibility of design lines in engineering projects.
Innovation Solution
A method that divides curve elements into segments based on a preset rotation angle, calculates the straight-line rotation angle and length, and iteratively corrects and replaces curve sub-elements until a cyclic condition is met, ensuring convergent and effective calculation of mileage and offset for any rotation angle.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the back calculation method using tangent perpendicular to the starting point of the curve element is used to accumulate tangent vertical distance, then the mileage can be calculated for known points, but the calculation may form an infinite loop when the vertical foot of P point relative to the new tangent is in the direction of small mileage again
Solution Approach 1:
The patent applies inversion by switching from forward calculation (starting from curve beginning and accumulating distances) to backward calculation (starting from known point P and working backwards to find mileage). This is achieved by calculating the intersection of the normal line at point P with the curve, then backtracking along the curve to determine the mileage and offset. This inverted approach avoids the infinite loop problem inherent in forward accumulation methods.
Solution Approach 2:
The patent introduces the normal line at point P as an intermediary element to find the intersection point with the curve. This intermediary construction allows the calculation to start from the known point P rather than from the curve beginning, enabling backward calculation and avoiding the convergence issues of forward accumulation methods.
2Ease of operation
If the vertical line of the tangent of some known points for calculating mileage is used, then the calculation can proceed, but the vertical line may exceed the curve element, resulting in failure of the inverse calculation
Solution Approach 1:
The patent applies preliminary action by first constructing the normal line at point P and finding its intersection with the curve before attempting any mileage calculation. This preliminary geometric construction ensures that the calculation starts from a valid point on the curve and proceeds backwards, avoiding the problem of vertical lines exceeding curve elements and ensuring calculation success.
3Measurement precision
If analytical method is used to solve the equations, then the mileage and offset can be calculated, but higher-order equations need to be solved making the overall calculation very difficult
Solution Approach 1:
The patent replaces the complex analytical mechanical calculation system (solving higher-order equations) with a geometric construction system. By using normal lines, intersection points, and curve tracing, the method achieves the same mileage and offset calculation without requiring complex algebraic equations, thereby reducing calculation complexity while maintaining accuracy.
Data Source
AI summary
A method for inversion of a path mileage and offset by using a known coordinate point, comprising: based on coordinates of a known point P and curve elements of a start location and an end location of a curve element, first segmenting or not segmenting the curve element according to a corner of the curve element to obtain a plurality of curve sub-elements (AiBi), and calculating curve elements of the curve sub-elements (AiBi); taking any curve sub-element (AiBi) to calculate a half chord length S and a straight corner θ, and performing precision determination and convergence correction according to S and θ, the point P being always effective with respect to the curve sub-element (AiBi) in the correction process; and performing cyclic convergence calculation to finally obtain a distance from the point P to any point on a chord line or an arc line as an offset of the curve sub-element (AiBi) corresponding to the point P and to obtain a mileage of the curve sub-element (AiBi) corresponding to the point P. By calculating different curve sub-elements (AiBi), an effective offset and an effective mileage of the curve sub-element (AiBi) corresponding to the point P can be finally obtained. The method is suitable for mileage inversion of curve elements which comprise a large-corner curve element, the calculation process is always converged, the situation that calculation cannot be performed is avoided, and the calculation process is relatively simple.