The Central Idea
Navigation and surveying both solve a common class of problem:
\[ \boxed{\text{known position}+\text{distance}+\text{direction} \longrightarrow \text{new position}}. \]The process can also work in reverse. If two positions are known, geometry can determine the distance and direction between them.
These relationships connect maps, coordinates, triangles, vectors, trigonometry, and physical measurement.
Reference Directions
A direction must be measured relative to a reference.
Common reference directions include:
- north;
- south;
- east;
- west;
- a local horizontal axis;
- a grid direction;
- a previously established survey line.
Without a reference direction, an angle alone does not identify an absolute orientation.
The Cardinal Directions
The four primary compass directions are:
- north;
- east;
- south;
- west.
Intermediate directions such as northeast, southeast, southwest, and northwest divide the plane more finely.
In a standard coordinate model, east is often represented by the positive \(x\)-direction and north by the positive \(y\)-direction.
Azimuth
An azimuth is commonly measured clockwise from north.
Under this convention:
\[ \text{north}=0^\circ, \] \[ \text{east}=90^\circ, \] \[ \text{south}=180^\circ, \] \[ \text{west}=270^\circ. \]Angles continue around the full circle from \(0^\circ\) to \(360^\circ\).
Example: Interpreting an Azimuth
An azimuth of
\[ 60^\circ \]is measured \(60^\circ\) clockwise from north.
The direction therefore lies between north and east.
An azimuth of
\[ 225^\circ \]lies between south and west.
Quadrant Bearings
A quadrant bearing describes direction using north or south as the starting direction and then turning toward east or west.
Examples include:
\[ \text{N }35^\circ\text{ E}, \] \[ \text{S }20^\circ\text{ E}, \] \[ \text{S }48^\circ\text{ W}, \] \[ \text{N }12^\circ\text{ W}. \]The angle in a quadrant bearing is always between \(0^\circ\) and \(90^\circ\).
Reading a Quadrant Bearing
The bearing
\[ \text{N }40^\circ\text{ E} \]means:
- begin facing north;
- turn \(40^\circ\) toward east.
The bearing
\[ \text{S }25^\circ\text{ W} \]means:
- begin facing south;
- turn \(25^\circ\) toward west.
Converting Quadrant Bearings to Azimuths
Because both systems describe the same directions, they can be converted.
For
\[ \text{N }\theta\text{ E}, \]the azimuth is
\[ \boxed{\theta}. \]For
\[ \text{S }\theta\text{ E}, \]the azimuth is
\[ \boxed{180^\circ-\theta}. \]For
\[ \text{S }\theta\text{ W}, \]the azimuth is
\[ \boxed{180^\circ+\theta}. \]For
\[ \text{N }\theta\text{ W}, \]the azimuth is
\[ \boxed{360^\circ-\theta}. \]Example: Bearing to Azimuth
Convert
\[ \text{S }35^\circ\text{ E} \]to an azimuth.
Since the direction starts from south and turns toward east,
\[ 180^\circ-35^\circ=145^\circ. \]Therefore,
\[ \boxed{145^\circ}. \]Example: Azimuth to Bearing
Convert the azimuth
\[ 300^\circ \]to a quadrant bearing.
The direction lies between west and north.
Its angle west of north is
\[ 360^\circ-300^\circ=60^\circ. \]Therefore,
\[ \boxed{\text{N }60^\circ\text{ W}}. \]Direction as a Vector
A movement can be represented by a vector with horizontal and vertical components.
Suppose a displacement has magnitude \(d\) and azimuth \(\alpha\), measured clockwise from north.
Then the eastward component is
\[ \boxed{\Delta x=d\sin\alpha}, \]and the northward component is
\[ \boxed{\Delta y=d\cos\alpha}. \]These formulas differ from the standard unit-circle formulas because azimuth is measured from north rather than from the positive \(x\)-axis.
Example: Components from an Azimuth
A traveler moves \(10\) kilometers on azimuth
\[ 30^\circ. \]The eastward component is
\[ \Delta x=10\sin30^\circ=5. \]The northward component is
\[ \Delta y=10\cos30^\circ =5\sqrt{3} \approx8.66. \]Therefore, the displacement is approximately
\[ \boxed{5.00\text{ km east},\qquad8.66\text{ km north}}. \]Position from a Known Starting Point
If a starting position is
\[ (x_0,y_0) \]and a displacement has components
\[ (\Delta x,\Delta y), \]then the new position is
\[ \boxed{(x_1,y_1)=(x_0+\Delta x,\;y_0+\Delta y)}. \]This simple coordinate update is the basis of many navigation and surveying calculations.
Example: Updating Position
A survey point has local coordinates
\[ (120,80). \]A second point lies \(50\) units east and \(30\) units north.
Then
\[ x_1=120+50=170, \] \[ y_1=80+30=110. \]Therefore,
\[ \boxed{(170,110)}. \]Distance Between Survey Points
If two points have coordinates
\[ P(x_1,y_1) \]and
\[ Q(x_2,y_2), \]their planar distance is
\[ \boxed{ d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} }. \]Coordinate surveying therefore connects directly to the Pythagorean relationship.
Direction Between Coordinate Points
Suppose
\[ \Delta x=x_2-x_1, \qquad \Delta y=y_2-y_1. \]The ratio
\[ \frac{\Delta x}{\Delta y} \]can help determine the angle from north when the appropriate quadrant is known.
In practical computation, inverse trigonometric functions and quadrant-aware methods are used so that the resulting direction is placed correctly.
Back Bearings
A back bearing points in the direction exactly opposite a forward bearing.
In azimuth form, the back azimuth differs by
\[ 180^\circ. \]If the original azimuth is less than \(180^\circ\), add \(180^\circ\). If it is at least \(180^\circ\), subtract \(180^\circ\).
Example: Back Azimuth
A forward azimuth is
\[ 72^\circ. \]The back azimuth is
\[ 72^\circ+180^\circ=252^\circ. \]Therefore,
\[ \boxed{252^\circ}. \]Back Bearings in Quadrant Form
Opposite quadrant bearings exchange north with south and east with west while preserving the acute angle.
For example,
\[ \text{N }30^\circ\text{ E} \]has back bearing
\[ \boxed{\text{S }30^\circ\text{ W}}. \]Similarly,
\[ \text{S }15^\circ\text{ E} \]reverses to
\[ \boxed{\text{N }15^\circ\text{ W}}. \]Triangulation
Triangulation determines unknown positions or distances by forming triangles from measured baselines and angles.
The central idea is:
\[ \boxed{\text{known baseline}+\text{measured angles} \longrightarrow \text{unknown geometry}}. \]This is useful when the target point is inaccessible but visible from two known locations.
A Basic Triangulation Setup
Suppose survey stations \(A\) and \(B\) are separated by a measured baseline
\[ AB=c. \]An inaccessible point \(C\) is observed from both stations.
If the measured angles are
\[ \angle A=\alpha, \qquad \angle B=\beta, \]then the third angle is
\[ \boxed{\angle C=180^\circ-\alpha-\beta}. \]The Law of Sines can then determine the unknown distances \(AC\) and \(BC\).
Example: Triangulating an Inaccessible Point
Two survey stations are \(100\) meters apart.
The observed angles to an inaccessible point are
\[ \angle A=50^\circ, \qquad \angle B=60^\circ. \]Therefore,
\[ \angle C=180^\circ-50^\circ-60^\circ=70^\circ. \]Using the Law of Sines,
\[ \frac{AC}{\sin60^\circ} = \frac{100}{\sin70^\circ}. \]Hence,
\[ AC= \frac{100\sin60^\circ}{\sin70^\circ} \approx92.2\text{ m}. \]Thus,
\[ \boxed{AC\approx92.2\text{ m}}. \]Trilateration
Trilateration determines position primarily from measured distances rather than measured angles.
In a plane, a point can be constrained by its distances from known reference points.
Geometrically, each known distance defines a circle:
\[ (x-a)^2+(y-b)^2=r^2. \]The unknown point lies where the relevant distance circles intersect.
Why Multiple References Are Needed
One distance from one known point places the unknown position somewhere on a circle.
A second distance usually reduces the possibilities to two intersection points.
Additional information or a third reference can identify the correct position.
The same geometric principle extends into three dimensions using spheres instead of circles.
Traverses
A traverse is a sequence of connected measured segments.
Each segment may be described by:
- length;
- bearing or azimuth;
- horizontal component;
- vertical or northing component.
Traverses are common in surveying because a long boundary or route can be reconstructed from a chain of measured directions and distances.
Open and Closed Traverses
An open traverse does not return to its starting point or another previously established point.
A closed traverse returns to the starting point or closes onto a known control point.
Closed traverses provide a built-in consistency check because the net displacement should ideally be zero.
Traverse Components
For segment \(i\) with length \(d_i\) and azimuth \(\alpha_i\),
\[ \Delta x_i=d_i\sin\alpha_i, \] \[ \Delta y_i=d_i\cos\alpha_i. \]After several segments, the total displacement is
\[ \boxed{ \Delta x_{\text{total}}=\sum_i \Delta x_i } \]and
\[ \boxed{ \Delta y_{\text{total}}=\sum_i \Delta y_i }. \]Example: Two-Leg Navigation
A traveler moves \(8\) kilometers east and then \(6\) kilometers north.
The net displacement has components
\[ \Delta x=8, \qquad \Delta y=6. \]Its magnitude is
\[ d=\sqrt{8^2+6^2}=10. \]Therefore, the traveler is
\[ \boxed{10\text{ km}} \]from the starting point in straight-line distance.
Finding the Direction of a Resultant Displacement
In the previous example, the displacement is \(8\) kilometers east and \(6\) kilometers north.
Let \(\theta\) be the angle east of north.
\[ \tan\theta=\frac{8}{6}. \]Therefore,
\[ \theta=\tan^{-1}\left(\frac{8}{6}\right) \approx53.1^\circ. \]The bearing is approximately
\[ \boxed{\text{N }53.1^\circ\text{ E}}, \]and the corresponding azimuth is approximately
\[ \boxed{53.1^\circ}. \]Closed-Traverse Check
For an ideal closed traverse,
\[ \sum_i \Delta x_i=0 \]and
\[ \sum_i \Delta y_i=0. \]Physical measurements rarely close perfectly because of observational and instrumental error.
The remaining discrepancy is called a closure error or misclosure.
Linear Misclosure
If a closed traverse produces residual components
\[ e_x \]and
\[ e_y, \]then the magnitude of the linear misclosure is
\[ \boxed{ e=\sqrt{e_x^2+e_y^2} }. \]A small misclosure suggests better agreement among the measured traverse components than a large one.
Survey Control
Control points are established reference positions used to organize later measurements.
A control point may have known:
- horizontal coordinates;
- elevation;
- both horizontal position and elevation.
New measurements can be tied to this known framework so that separate parts of a survey remain spatially consistent.
Horizontal and Vertical Position
Surveying often separates horizontal position from elevation.
A three-dimensional point can be represented by
\[ \boxed{(x,y,z)}, \]where \(x\) and \(y\) describe horizontal position and \(z\) describes elevation relative to a chosen datum.
Elevation Difference
If two points have elevations
\[ z_1 \]and
\[ z_2, \]then their elevation change is
\[ \boxed{\Delta z=z_2-z_1}. \]A positive result means the second point is higher relative to the chosen reference; a negative result means it is lower.
Slope Distance and Horizontal Distance
A distance measured along sloping terrain is generally longer than its horizontal projection.
If a slope distance \(s\) makes angle \(\theta\) above the horizontal, then the horizontal distance is
\[ \boxed{d_h=s\cos\theta}, \]and the elevation change is
\[ \boxed{\Delta z=s\sin\theta}. \]Example: Resolving a Slope Distance
A measured slope distance is
\[ 50\text{ m} \]at an angle of
\[ 12^\circ \]above the horizontal.
The horizontal component is
\[ d_h=50\cos12^\circ\approx48.9\text{ m}. \]The elevation change is
\[ \Delta z=50\sin12^\circ\approx10.4\text{ m}. \]Therefore,
\[ \boxed{d_h\approx48.9\text{ m},\qquad \Delta z\approx10.4\text{ m}}. \]Angles of Elevation and Depression
An angle of elevation is measured upward from a horizontal reference line.
An angle of depression is measured downward from a horizontal reference line.
These angles help connect horizontal distance and elevation difference through right-triangle trigonometry.
Example: Elevation from Horizontal Distance
A surveyor stands \(80\) meters horizontally from a point on a hillside. The angle of elevation is
\[ 18^\circ. \]The elevation difference is
\[ \Delta z=80\tan18^\circ. \]Numerically,
\[ \Delta z\approx26.0\text{ m}. \]Therefore,
\[ \boxed{\Delta z\approx26.0\text{ m}}. \]Leveling
Leveling is the process of determining elevation differences between points.
In its simplest geometric form, the method establishes a horizontal line of sight and compares vertical readings at different positions.
Repeated elevation differences can be accumulated to transfer a known elevation to new points.
Accumulating Elevation Changes
If a sequence of measured elevation changes is
\[ \Delta z_1,\Delta z_2,\ldots,\Delta z_n, \]then the total elevation change is
\[ \boxed{ \Delta z_{\text{total}} = \sum_{i=1}^{n}\Delta z_i }. \]If the starting elevation is \(z_0\), then the final elevation is
\[ \boxed{ z_n=z_0+\Delta z_{\text{total}} }. \]Navigation by Dead Reckoning
Dead reckoning estimates current position by starting from a known position and accumulating measured or estimated motions.
In a simplified planar model,
\[ x_n=x_0+\sum_i \Delta x_i, \] \[ y_n=y_0+\sum_i \Delta y_i. \]The method is geometrically straightforward, but errors can accumulate as more segments are added.
Error Accumulation
A small error in distance or direction on one segment changes the computed endpoint.
Later segments begin from that already shifted position, so the discrepancy can grow.
This is why practical navigation and surveying use:
- repeated observations;
- control points;
- back checks;
- closed traverses;
- independent position references.
Redundancy Improves Reliability
A position determined from only the minimum amount of information may be difficult to check.
Additional measurements provide redundancy.
For example, a survey point might be checked using:
- two independent angle observations;
- several measured distances;
- a traverse closure;
- a known control point;
- coordinate comparison.
Agreement among independent constraints increases confidence in the result.
Field Notes and Data Organization
Navigation and surveying calculations depend on clear records.
Useful field information may include:
- station names;
- measured distances;
- bearings or azimuths;
- angles;
- elevations;
- instrument heights;
- units;
- observation order.
A mathematically correct method can still fail if measurements are recorded under the wrong station or interpreted with the wrong reference direction.
True, Magnetic, and Grid North
The word north can refer to different reference systems.
- true north points toward the geographic North Pole;
- magnetic north follows the local direction indicated by a magnetic compass;
- grid north follows the north direction of a mapping or coordinate grid.
These directions may not coincide exactly.
A bearing is meaningful only when its reference north is understood.
Magnetic Declination
Magnetic declination is the angular difference between magnetic north and true north at a location and time.
Therefore, a magnetic compass bearing is not automatically the same as a true bearing.
Practical navigation may require converting between these reference systems using an appropriate local declination.
Maps, Bearings, and Coordinates Work Together
A map supplies a spatial representation, a scale connects map lengths to actual lengths, and bearings describe direction.
Coordinates provide another way to represent the same position.
Together,
\[ \boxed{ \text{scale}+\text{direction}+\text{distance} \longleftrightarrow \text{coordinates} }. \]Switching among these forms is a major practical skill in spatial problem-solving.
Common Error: Measuring Azimuth Counterclockwise
In the common navigation convention used here, azimuth is measured clockwise from north.
This differs from the standard mathematical angle convention, which usually measures counterclockwise from the positive \(x\)-axis.
Confusing the two conventions can place a direction in the wrong quadrant.
Common Error: Reversing East and West in Bearings
The bearing
\[ \text{N }25^\circ\text{ E} \]points northeast.
The bearing
\[ \text{N }25^\circ\text{ W} \]points northwest.
The angle is the same, but the final letter changes the side of north on which the direction lies.
Common Error: Using the Wrong Trigonometric Components
If azimuth \(\alpha\) is measured from north, then
\[ \Delta x=d\sin\alpha \]and
\[ \Delta y=d\cos\alpha. \]Using cosine automatically for the horizontal component assumes the angle was measured from the horizontal axis instead.
Always identify where the angle begins.
Common Error: Ignoring the Quadrant
An inverse tangent can return an acute reference angle without fully identifying direction.
The signs of
\[ \Delta x \]and
\[ \Delta y \]determine whether the displacement points northeast, southeast, southwest, or northwest.
Common Error: Treating Slope Distance as Horizontal Distance
A measured line on sloping terrain is generally longer than its horizontal projection.
If the slope angle is not negligible, using slope distance directly in a horizontal plan can introduce error.
Resolve the line into horizontal and vertical components when the application requires them.
Common Error: Forgetting Back Bearings
Traveling from \(A\) to \(B\) and returning from \(B\) to \(A\) uses opposite directions.
The back azimuth differs by
\[ 180^\circ. \]Reusing the forward bearing for the return direction would continue the motion rather than reverse it.
A Reliable Procedure for Bearing Problems
- Identify the direction convention being used.
- Draw north, south, east, and west reference directions.
- Place the bearing or azimuth in the correct quadrant.
- Identify the known distance or displacement.
- Resolve the motion into components if coordinates are needed.
- Apply signs according to east-west and north-south direction.
- Update the coordinates.
- Check that the final position lies in the expected direction.
A Reliable Procedure for Basic Survey Triangulation
- Establish a measured baseline.
- Identify the target point.
- Measure the relevant angles from the baseline endpoints.
- Draw the resulting triangle.
- Find the third angle if necessary.
- Use the Law of Sines or another valid triangle relationship.
- Calculate the unknown distances.
- Check that the triangle geometry is consistent.
Example: Full Navigation Workflow
A vehicle starts at local coordinate
\[ (200,300) \]and travels \(40\) kilometers on azimuth
\[ 120^\circ. \]Resolve the displacement:
\[ \Delta x=40\sin120^\circ =20\sqrt{3} \approx34.64, \] \[ \Delta y=40\cos120^\circ =-20. \]Update the position:
\[ x_1=200+34.64=234.64, \] \[ y_1=300-20=280. \]Therefore, the new position is approximately
\[ \boxed{(234.64,\;280)}. \]The signs are reasonable because azimuth \(120^\circ\) points southeast: eastward displacement is positive and northward displacement is negative.
Practice 1: Azimuth Interpretation
Which cardinal direction corresponds to each azimuth?
- \(0^\circ\)
- \(90^\circ\)
- \(180^\circ\)
- \(270^\circ\)
Practice 2: Bearing to Azimuth
Convert
\[ \text{N }42^\circ\text{ W} \]to an azimuth.
Practice 3: Azimuth to Bearing
Convert
\[ 210^\circ \]to a quadrant bearing.
Practice 4: Back Azimuth
Find the back azimuth of
\[ 138^\circ. \]Practice 5: Components
A displacement of \(20\) kilometers has azimuth
\[ 60^\circ. \]Find its eastward and northward components.
Practice 6: Position Update
A point begins at
\[ (50,75). \]A displacement moves it \(12\) units east and \(9\) units south. Find the new coordinates.
Practice 7: Slope Distance
A \(30\)-meter line is measured at \(20^\circ\) above the horizontal. Find its horizontal component and elevation change.
Practice 8: Triangulation
A baseline is \(80\) meters long. The angles to an inaccessible point are
\[ 45^\circ \]and
\[ 65^\circ. \]Find the third angle.
Practice 9: Closed Traverse
A closed traverse has residual components
\[ e_x=0.30\text{ m}, \qquad e_y=-0.40\text{ m}. \]Find the magnitude of the linear misclosure.
Practice 10: Identify the Error
A student says that an azimuth of \(45^\circ\) is measured \(45^\circ\) counterclockwise from east.
Explain the convention error.
Practice Answers
Practice 1
- \(0^\circ\): north.
- \(90^\circ\): east.
- \(180^\circ\): south.
- \(270^\circ\): west.
Practice 2
For
\[ \text{N }42^\circ\text{ W}, \]the azimuth is
\[ 360^\circ-42^\circ=318^\circ. \]Therefore,
\[ \boxed{318^\circ}. \]Practice 3
The azimuth \(210^\circ\) lies between south and west.
Its angle west of south is
\[ 210^\circ-180^\circ=30^\circ. \]Therefore,
\[ \boxed{\text{S }30^\circ\text{ W}}. \]Practice 4
\[ 138^\circ+180^\circ=318^\circ. \]Therefore,
\[ \boxed{318^\circ}. \]Practice 5
\[ \Delta x=20\sin60^\circ =10\sqrt{3} \approx17.32, \] \[ \Delta y=20\cos60^\circ=10. \]Therefore,
\[ \boxed{17.32\text{ km east},\qquad10\text{ km north}}. \]Practice 6
East is positive \(x\), and south is negative \(y\):
\[ x=50+12=62, \] \[ y=75-9=66. \]Therefore,
\[ \boxed{(62,66)}. \]Practice 7
\[ d_h=30\cos20^\circ\approx28.19\text{ m}, \] \[ \Delta z=30\sin20^\circ\approx10.26\text{ m}. \]Therefore,
\[ \boxed{d_h\approx28.19\text{ m},\qquad\Delta z\approx10.26\text{ m}}. \]Practice 8
\[ 180^\circ-45^\circ-65^\circ=70^\circ. \]Therefore,
\[ \boxed{70^\circ}. \]Practice 9
\[ e=\sqrt{(0.30)^2+(-0.40)^2}. \] \[ e=\sqrt{0.09+0.16} =\sqrt{0.25}. \]Therefore,
\[ \boxed{e=0.50\text{ m}}. \]Practice 10
Under the navigation convention used in this lesson, azimuth is measured clockwise from north.
Therefore, an azimuth of
\[ 45^\circ \]points northeast, halfway between north and east. It is not defined here as an angle measured counterclockwise from east.
What to Remember
- A bearing describes direction relative to a stated reference.
- Azimuth is commonly measured clockwise from north from \(0^\circ\) to \(360^\circ\).
- Quadrant bearings use north or south as a starting direction and turn toward east or west.
- Distance and azimuth can be resolved into east-west and north-south components.
- Coordinates can be updated by adding displacement components.
- Back bearings point exactly opposite forward bearings.
- Triangulation uses known baselines and measured angles to determine inaccessible positions or distances.
- Trilateration uses distances from known reference points.
- Traverses reconstruct routes or boundaries from sequences of measured segments.
- Closed traverses provide a geometric consistency check through closure.
- Slope distance can be resolved into horizontal distance and elevation change.
- True north, magnetic north, and grid north are different reference directions and should not be confused.
The central structure is:
\[ \boxed{ \text{reference}+\text{direction}+\text{distance} \longleftrightarrow \text{position} }. \]Where We Go Next
Navigation and surveying determine where objects and boundaries are located. The next class of problems uses those measurements to create, lay out, and verify physical structures.
The next lesson, Architecture and Construction, applies spatial geometry to plans, dimensions, slopes, layouts, structural form, quantities, and construction measurement.