# Calculation Method

## Overview - Designing Horizontal Alignment

### First Chainage

The first chainage is automatically put to 0.0 at the first element's fixed point. This can be changed with the function Start Chainage, which can be accessed from the menu Object.

### Design

The linear chainage direction is marked on the drawing with an arrow at every element transition. The direction of the alignment can be changed with the function Turn, which can be accessed from the menu Modify.

Figure 1: There is an arrow at every transition

### Designing several Alignments simultaneously

Several sessions of alignment design can be done simultaneously by opening several alignment design dialogs. This gives the option of editing individual alignments independently. This is useful when designing intersection areas.

Every dialog that is open refers to one single alignment and its illustration objects or external data. When working with multiple alignment design dialogs open, remember to update the content and data seen in the other dialogs with the Update button to get the latest design that is stored in Quadri.

### Offset - Horizontal Alignment Containing Clothoid

It is geometrically impossible to find an exact parallel copy of one clothoid.

The tolerance from a perfect parallel alignment becomes in the area of 0.01 - 0.5 meter

The accuracy is affected by the clothoid's length â€“ a longer clothoid results in less accuracy.

Several consecutive clothoids give the poorest accuracy.

Accuracy with several consecutive clothoids

The result will usually become poor in the clothoids, for example in the reverse transition (clothoid).

The accuracy is dependent on the new clothoids parameters.

The new parameters will have almost the same (mutual) relationship as between the parallel offset.

In some situations, it is possible to manually optimize the clothoids after the offset function.

Method for offsetting a transition (clothoid) element

Inserted transitions (clothoid) are treated as an unknown quantity until the adjacent elements are fixed.

It is not necessary to try and optimize the transitions (clothoids) after the offset.

Method for offsetting a number of transitions (clothoid), for example reverse transitions (clothoid)

A number of consecutive transitions (clothoid) are treated as an unknown quantity. The parameter relationship is declared as accurate as possible prior to the offset function.

Based on the fixed elements the new transition (clothoid) parameters are calculated with the aid of NADB.

The calculated transition (clothoid) parameters are displayed as new transition (clothoid) parameters in the alignment design.

The chainage division is adjusted according to the transition (clothoid) parameters.

Lastly, one last calculation is done to get those results, which are displayed, in the main dialog box Alignment Design Professional.

Adjacent adjustments can be done to the alignment, where there are consecutive transitions (clothoid). Try and change the relationship between the transition (clothoid) parameters, but it is recommended that the elements be locked before and after the transition (clothoid) combination.

Tips:

A reference point can be placed in the area where the distance needs to be checked while changing the transition (clothoid) parameters. Place the reference points on the source line.

Transitions (clothoids) are drawn in CAD as small circles. If the transition curves (clothoid) are not required and the parallel copy is to be as accurate as possible, use the CAD function Offset.

## Saving Alignment to Database

Alignments can be saved as described here:

Input data/Result - Input data (fixed points, reference points, alternatives) and Result is saved on the drawn horizontal alignment in CAD.

Result-data - The result can be saved into Quadri as well.

Export to File

Exclusively the user saves the alignment back into Quadri database (Digital Terrain Model), using the function Name, by giving a name to the alignment, group number, and feature code.

When this is done both the horizontal and vertical alignments are stored in a database and at the same time, a link between the construction line and the alignment in the database is established. When editing the alignment in the drawing the program will prompt the user if the equivalent alignment in the database is to be updated as well.

Note: To access the Input data, reference points and alternatives in history the starting point for information access would be the drawn horizontal alignment in the CAD drawing. In the database, only the result of the alignment is saved.

## Overview - Designing Vertical Alignment

Designing the Vertical Geometry can only be done on an existing Horizontal Alignment. Before creating the vertical alignment, the horizontal alignment is to be saved to the active database (Digital Terrain Model). Novapoint system cannot handle "free" profiles (vertical alignments); they always are linked to an alignment in the database.

Run the function Vertical Drawing from the menu Window in the floating window Alignment Design Professional. It's a prerequisite that the Overview - Horizontal Geometry is already read.

When the vertical alignment design function, Vertical Drawing, is activated, a new temporary drawing file is created automatically to house the vertical alignment and the terrain profile. The terrain profile is calculated along the horizontal alignment. This drawing is a temporary drawing, reserved only for the current vertical alignment.

The scale of the vertical drawing file is in relation to the scale of the horizontal drawing file, determined through the function Scale of the menu Novapoint.

### Longitudinal Terrain Profile Calculation

When the function Vertical Drawing is activated, the longitudinal profile is calculated automatically. The longitudinal profile is updated continuously if the horizontal alignment is edited.

The longitudinal profile is represented with pre-defined line types.

Longitudinal profile in the vertical drawing can be calculated either from terrain model or road model.

Terrain from Terrain Model:

The longitudinal profile in the vertical drawing is derived from the terrain model and the calculation is based on the setting defined in the function Terrain.

The calculation and presentation will also follow the setting of the terrain model and will be based on the data available in the terrain model.

Wireframe models and the actual terrain surface beneath the wireframe models in the longitudinal profile presentation can be viewed with a drawing setup.

Alignments, Reference lines, and Alignments as Surface Edges will also be shown in the longitudinal profile with respective text labels (alignment name, elevations, and chainage).

The Terrain Model Properties (Groups, Feature Codes, and Profiles) can be used to administrate the data to be used/presented in the longitudinal profile.

Terrain from Road Model (when the current alignment is the active alignment of the road model):

The road model method is available only when the alignment is the active alignment of the road model.

The terrain is derived from the terrain model, but the calculation is limited to the option/settings described by the program Novapoint Road Design, for e.g., soil and mountain transitions are given chainage intervals.

When the current line is a "free" line any of the above methods can be adopted.

### Updating terrain profile

If the horizontal geometry is changed, modified, and/or edited, then the new terrain profile along the horizontal alignment is calculated automatically. This can take a few seconds, so it is recommended that only small changes be done to the horizontal geometry when the vertical drawing is open.

To toggle the longitudinal terrain profile calculation and presentation in vertical drawing; double-click the field TERRAIN in the status bar of the user interface. Alternatively, run the function Display. You can also use normal CAD layer functions.

When TERRAIN is indicated in black; the program calculates and displays the terrain profile.

When TERRAIN is displayed with grey; the terrain profile/section will be neither calculated nor displayed.

### Grid

Help lines in the grid are generated automatically. The X-axis is equivalent to the linear length in the grid. The Y-axis is equivalent to the absolute elevation in the grid.

The setting is done from the function Scale of the Novapoint menu (the scale in the vertical drawing is in relation to the scale in the horizontal drawing).

The program, depending on the zooming grade, updates the size on the grid text automatically.

### Signature in the grid

Chainage and elevation are displayed and updated by the program depending on the zooming grade.

### Points of tangency â€“ horizontal alignment

The horizontal alignments points of tangency are displayed in the grid of the vertical drawing. Whether element signature is to be displayed or not, can be controlled through CAD layer control. The element signature is drawn on the horizontal alignment layer.

### The construction lines for the vertical alignment

The initial principle for designing the vertical geometry is identical to the design procedure of the horizontal geometry, except for the following additional comments:

Clothoids cannot be used.

Fixed points are automatically converted to angular points.

As a starting point for the design, CAD lines, polylines, and arcs can be utilized to initiate vertical alignment, or lines and curves can be directly inserted through the table Input V in the main floating window Alignment Design Professional.

### Angular Points

The construction method of calculating angular points is often used in vertical geometry, even though the methodology for constructing the vertical geometry is almost identical to the construction methodology of horizontal geometry.

The calculate angular point methodology is simulated by giving one element the following section classification:

Fixed Line

Arc as an approximate element

The first element's second fixed point is placed on the third element's first fixed point. The common fixed point can be edited as an angular point.

There are different functions to create or remove angular points. These functions can be found in the Modify menu and through the shortcut menu in the drawing (right-click menu).

### Toggle between horizontal and vertical drawing

It is possible to work alternately between the horizontal and vertical alignment.

One can change from a vertical to a horizontal drawing with the aid of alignment design functions or activating the drawings by pointing at the desired drawing. Change projection can be done by activating the respective "Input" table in the main floating window, which changes active drawing automatically.

### Close the vertical construction drawing

The menu choice Close Vertical Drawing from the Window menu closes the vertical drawing window.

When Novapoint Alignment Design is closed, the vertical drawing is automatically closed as well and deleted from the hard drive. The drawing is only a temporary construction drawing.

When the program is terminated with the function OK - Draw, the vertical geometry is saved on the horizontal alignment in the database as well as on the horizontal alignment in CAD. The temporary vertical drawing is not saved but terminates and gets deleted from the hard drive.

Novapoint Drawing Generation functions are used to create profile projections, etc, drawings. With the alignment design, only the actual alignment is created, not drawings.

## Segment classification

### Segment Classification

A segment is a part of the alignment that lies between two consecutive fixed points. A segment may therefore span over one or more elements.

Fixed points and approximate points are located such that a mathematical solution exists.

An element contains several element factors. Within a segment, an element may have the following unknown factors:

Table 1: Element type and factor

The number of unknown element factors in every segment determines the mathematical solution.

In alignment design, the elements are described in consecutive fashion, from start to finish, with fixed, partly, and approximate point coordinates, which is the basis to determine segment classification. The element factors and the fixed points must be stated before the alignment can be solved mathematically. That is; all elements are tangent to each other and the placing is precisely determined. For a successful alignment, the alignment must be comprised of segments, which contain 0, 2, or 3 unknown element factors, the so-called 0, 2, and 3 segments.

Element factors, which are not openly given, but can be calculated based on other element factors, are not regarded as unknown elements. For example, the transition curve's length is regarded as a known factor when endpoints, parameters, and radii are known.

The composition of the different segments must be done according to set rules.

### Segment Type 0

One element, a circle with a known radius, is placed such that it passes through two fixed points. The location of the circle is now determined. The length of the arc between the fixed points is known when the radius is known. The segment has no unknown element factors and is, therefore, a 0-segment.

Generally, a 0-segment is defined by two fixed points and consists of only part of an element. The segment is calculated independently of the adjacent elements.

### Segment Type 2

Two elements, an arc, and a line are going to be calculated. There are two fixed points on the arc, which is a 0-segment and one fixed point on the line. The line placing in itself is not yet determined as it can rotate in one fixed point. In this element combination, the line is going to be a tangent to the arc (which is a 0-segment). Thus the line is determined and the point of tangency can be calculated.

The segment between the last two fixed points contains two unknown element factors:

The distance from the fixed point on the arc to the point of tangency.

The distance from the point of tangency to the fixed point on the line.

Thus the segment is a segment type 2. The line element cannot be calculated until the arc is calculated, which gives the following general rule:

Segment type 2 can only be calculated after the adjacent segment is a known factor.

In case, if the line is also having two fixed points, then both elements are fixed separately and it would only be an accidental circumstance if they touch. In other words, the line would be overruled. If the line had two fixed points, the only possible solution would be that the arc is having only one fixed point.

### Segment Type 3

Three elements are going to be calculated; one line with two fixed points, an arc with a known radius, and a line with two fixed points.

The two lines are comprised of two 0-segments, which means they are already stated (known). There is only one placement for an arc with a known radius, which is tangent to a line. The arcsâ€™ placing is then stated (known) and the point of tangency can be calculated.

The middle segment (the arc) includes three unknown element factors:

The length between the fixed point and the point of tangency.

The arcs length between the points of tangency.

The length between the point of tangency and the fixed point.

The segment is a type 3 segment.

An approximate point belongs to the circle. This is due to the fact that a 3-segment can have certain element combinations that result in two different solutions. The approximate point is placed so the solution, which is closer to the point, is chosen.

The arc cannot be calculated before the two line elements have been calculated and this gives the following rule:

First, a 3-Segment is calculated, when the adjacent segments are calculated. Thus on both sides of a 3-segment, there must be a 0-segment.

The arc would not have been tangent to the line elements if it had a fixed point. The line would have been overruled. There would have been a 2-segment instead of a 3-segment between the end segments, but this is not possible. The following rule can be applied:

There should be a 3-segment between two 0-segments.

### Location of Fixed Points

The fixed points should be placed such that the error in result gets reduced as much as possible.

Segment type 0 should be placed on long elements to make the distance between the fixed points as long as possible. The inaccuracy in the coordinates will be given a smaller rotation to the element if the points are closer together. The fixed points can be placed at the continuation of the element to increase the distance.

### Fixed point on the continuation of element - 0-Segment

If there is only one fixed point on the element, then this should be placed as far towards the start of the element as possible, when viewed in the direction of the calculation, i.e., as far as possible from the segment type 0 (where the calculation starts). The fixed point can be placed on the continuation of the element, forwards in the calculation direction but not in the opposite direction.

### Fixed point on the continuation of an element - 2-segment

### Segment Classification - Calculation Sequence

According to the segment type examples, the following was highlighted:

The 0-segment is determined by two given fixed points and consists of only part of an element. The calculation is independent of the other segments on the alignment.

The 2-segment contains two unknown element factors and is not possible to calculate until one of the adjacent segments is calculated.

The 3-segment contains three unknown element factors and is not possible to calculate until both the adjacent segments are calculated.

Between two 0-segment, there has to be a 3-segment.

The segments that are dependent on each other result in only certain segment combinations that can create a solution. A simple segment classification (division) is the segment sequence 0 - 2 - 2 - 2 etc. The 0-segment is known (value/factor). This indicates that the consecutive 2-segment can be calculated since the adjacent segment is known. Because the 2-segment is now a known factor the second 2-segment can be calculated.

In those cases where the alignment only consists of 0 and 2-segments, only one 0-segment can occur, but it can occur anywhere in the segment combination.

A different segment classification (division) is 0 - 3 - 0 - 3 - 0 etc. In this case, rules apply; one segment on either side of a 3-segment must be known and a 3-segment must be between 0-segments.

The segment classification can also be given from a combination of two methods, for example 0 - 2 - 2 - 3 - 2 - 0. The 2-segments are calculated from adjacent 0-segments. Known 2-segments will surround a 3-segment when the 2-segments have been calculated, thus enabling the 3-segment to be calculated.

Segments that are dependent on each other result in a particular calculation sequence as we can see.

In a segment classification (division) with one 0-segment and the rest 2-segments, the 0-segments are calculated first as previously mentioned, and then the 2-segments. In the situation where the 0-segment is placed in the middle of the line, the 2-segments placed in front of the classification (division), from the 0-segment and to the start point, are calculated first and subsequently from the 0-segment and to the end point.

### Example

Segment classification (division) 2 : 2 : 2 : 0 : 2 : 2

Calculation sequence 4 : 3 : 2 : 1 : 5 : 6

When using 0- and 3-segments 1:a 0-segment are calculated first, then 2:a 0-segment and the 3-segments in-between. After that; the 3:a 0-segment, 2:a 3-segment etc.

### Example

Segment classification (division) 0 : 3 : 0 : 3 : 0

Calculation sequence 1 : 3 : 2 : 5 : 4

With combinations of 0-, 2- and 3-segments 1:a 0-segment is calculated first, then the closest 2-segment, after that it jumps to the nearest 0-segment. The rear 2-segment is calculated and last the 3-segment is calculated. The next interval between 0-segment is calculated accordingly (the same way as explained previously).

### Example

Segment classification (division) 0 : 2 : 3 : 2 : 0 : 2 : 3 : 0

Calculation sequence 1 : 2 : 5 : 4 : 3 : 6 : 8 : 7

### Calculation Method

This is a general section, which deals with both the horizontal and vertical alignment. Novapoint Alignment Design employs Reference Point Method to calculate the alignment geometry. One of the most important conditions for a successful result is the continuity requirement, i.e., the alignment is continuous without brakes.

### Successful Calculation

The status bar in the main dialog box Alignment Design Professional indicates if the calculation has resulted in a successful calculation (Calc OK) and the drawing/graphic is updated with the new alignment.

### Failed Calculation

The status bar in the main dialog box Alignment Design Professional indicates if the calculation fails on the status bar (Calc not OK). The drawing/graphic is then based on the latest successful alignment. When the alignment is edited without success, alignment in CAD is indicated with colors, which derives from the standard color setting (yellow, cyan, and magenta).

### Segment Classification - With Transition Curve Elements

Previously segment classification (division) using lines and arcs has been explained. The same principles are applied when using transition curves. Transition curves should not be given fixed points when this complicates the calculation. In most cases this can be avoided and thus transition curves with fixed points is not be explained here.

Based on previously mentioned rules for segment classification (division) the following rules can be applied when using transition curves.

If the transition curve parameter is given then the transition curve is stated and the segment classification can be done as if the transition curve was non-existent.

Insert a new transition curve if its parameters are not present making this transition curve an unknown factor. This demands another lock on the line, which is done by adding one more fixed point on the adjacent element.

### Transition Curve with one known parameter

Transition curve when its parameter is a known factor. This example is equivalent to the examples given for segment types 0, 2, and 3 where transition curves were not focused. The transition curve length can be calculated if the endpoints, parameters, and radian are known. Thus the transition curve does not add any unknown factors, and the two unknowns in the 2-segment becomes the length from the fixed points to the point of tangency with the transition curve (represented as L1 and L2 in the figure shown below).

The geometrical situation is illustrated in the figure shown below. The transition curve prohibits the circle extension to be tangent to the line element but is displaced and tangent to the line elements parallel in the distance D. When the transition curves parameters and the end radius are known, the distance D is also set, and on the principal of the geometric situation, it is equivalent to the case without transition curve.

### Transition Curve with two known parameters

When transition curve parameters are not published, an unknown element factor will appear in segment type 2 in addition to the two distances. This way the segment becomes a 3-segment. With this, a 0-segment is required on both sides of the transition curve, which can be done by stating two fixed points on the two adjacent elements.

The geometrical situation is illustrated in the figure shown below. The distance D is set when both the line and the circle are locked with two fixed points and the transition curve parameters can be calculated.

### Reverse Transition Curve

A reverse transition curve consists of two transition curves that are tangent to each other and with opposite curvature. A reverse transition curve can be handled the same way as one single transition curve by stating both parameters of both the transition curves. Both the segment types 0 and 2 gained with displayed fixed points.

If one of the parameters is published and the other one is unknown, segment type 3 is to be created and then declare two fixed points on both circles.

If none of the parameters are published, we shall have 4 unknown factors on the segment, which cannot be calculated. To solve this situation the ratio between the parameters can be stated. The number of unknown factors is reduced to resemble the situation where one parameter is unknown and the other parameter is known.

### Examples of segment classification with transition curves

The placing of fixed points and segment classification can be performed in a number of different ways. The aim is to get the closest solution/result, thus making the calculation easy to accomplish.

This figure displays an example of a horizontal geometry drawn on a map. Below are 4 alternatives for segment classification.

Example 1

Two fixed points are assigned to the line element and one fixed point to every arc. All the transition curve parameters are stated. The segment classification becomes as shown in the figure below. This is a simple method, which often is the best. The 0-segment is placed anywhere on the longest element.

Example 2

This solution is applicable in the transition between long and short arcs and lines. Two fixed points lock long elements, thus avoiding short elements with fixed points. There are two possible solutions for placing the first arc, thus declare an approximate point to allow for the program to choose the "right" solution.

Generally, when all the unknown element factors in segment type 3 are lengths, an approximate point near the middle element is stated. The program will require an approximate point in these kinds of situations even though geometrically there is only one solution, thus an approximate point is stated near an arc with an unknown radius in a 3-segment.

Example 3

Here two fixed points are given to all lines and arcs. Transition curve parameters are not stated, but the reverse transition curves parameter ratio is stated. If possible, one of the reverse transition curves parameters is declared.

This method allows for good control over the line and can be declared, where the alignment needs an accurate placement. This method however can create problems for the calculation. With short elements, an overlap between the elements can occur if the transition curve parameter control is too poor. Especially in the transition between line and arc, small discrepancies in the fixed points placing can result in large outcomes on the transition curve parameters. This method requires an accurate placement of fixed points.

Example 4

Two fixed points are stated on the line and the last arc, one fixed point is stated on the middle arc. The transition curve between the line and the first arc is given and the reverse transition curves parameter.

This segment division allows for good control of the alignment at the same time as the calculation problems have decreased since the last example because the transition curve parameters between the line and the arc are given.

### Stipulation of Unknown Element Factors

The elements that have fixed points will be part of two or three segments. In some cases, it can be hard to decide which unknown element factor belongs to which segment. The following rules apply to the stipulation of the unknown element factors placing:

Element with an unknown length:

If the element's total length is unknown, two unknown element factors are calculated; the length from the fixed point to both of the points of tangency. This is the most common incident.

Element with an identified (known) length:

The total length of the element is known, but the allocation of the element's lengths is unknown. When the element has a fixed point an unknown element factor is added to the segment, which is calculated first, but no unknown element factor is added to the second segment.

Element with an unknown radius:

When the element contains an unknown radius, the radian is computed as an unknown factor in the segment, which is calculated first. There are some limitations when it comes to the use of an unknown radius: the length should be stated and/or the arc must be given a fixed point. Both a fixed point and a known length are not possible and two fixed points cannot be placed on the arc.

Under special circumstances (unknown radius) an arc with an unknown radius is going to go through one fixed point and be tangent to two identified (known) elements. There are two unknown lengths on both the two segments between the segment type 0, thus an unknown radius that normally can be linked to the first calculated segment. In this situation, we cannot determine which segment is going to be calculated first, but a segment combination can be achieved as long as the radian is only placed to one of the segments and this will result in two alternative solutions in the figure. The result is very clear if we regard the mathematical correct solution since only one arc can go through a given point and be tangent to two line elements.

The following line can be calculated even though the calculation sequel and the segment separation are not precisely set.

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