Topografía - Videoclase: ángulos y direcciones

Main ideas / lessons conveyed

• Purpose in topographic surveying: To perform a topographic survey, you take measurements that create geometric elements made from straight segments connected end-to-end.
• How angles and directions arise:
  • Connected segments share common vertices.
  • At each vertex, the connected straight lines form angles and define the directions of those lines.
• Terminology for straight lines (segments): These straight lines are called alignments (or topographic alignments).

Types of angles and directions

• Angles can be:
  • Internal
  • External
  • Deflection angles
• Directions can be expressed as:
  • Azimuth direction
  • Bearing direction
• Bearings/azimuths may be referenced to meridians as geographic, magnetic, or arbitrary (depending on the meridian chosen).

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Key definitions

1) Direction as reference to north (general concept)

• Direction is given relative to north.
• The north reference used may be:
  • Magnetic north
  • Arbitrary north
  • Geographic north (depending on which meridian is taken on the topographic map)

2) Bearing vs. Azimuth

Azimuth (Azimuth of an alignment)

• Definition: The direction/angle measured between:
  • the northern end of the reference meridian, and
  • the alignment line.
• Measurement rule:
  • Varies from 0 to 360°
  • Always measured clockwise
  • Always measured starting from the north axis
• Reference labels: Azimuths may be true/magnetic/arbitrary depending on the meridian reference.

Bearing (Bearing direction)

• Definition: Direction expressed using topographic quadrants relative to north–south axes.
• Quadrants:
  • Quadrant 1: NE (northeast)
  • Quadrant 2: SE (southeast)
  • Quadrant 3: SW (southwest)
  • Quadrant 4: NW (northwest)
• Angle magnitude rule: Bearing angles lie within 0 to 90° (each quadrant spans 90°).
• Measurement rule (important): Bearing is read from north or south depending on the quadrant:
  • Quadrant 1: measured from north
  • Quadrant 2: measured from south
  • Quadrant 3: measured from south
  • Quadrant 4: measured from north
• Final bearing format:
  • Angle value + quadrant name (e.g., “42° … NE”)

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Geometry/method for converting direction in opposite sense (“contra” / back azimuth)

Contra / back azimuth concept

• Contra-azimuth (or “back azimuth”):
  • The direction of the same straight line but in the opposite direction (opposite end).
• Why it changes: A line segment has an origin and endpoint; reversing it changes the direction reading.

Back azimuth computation (numeric rule given)

• If azimuth is in Quadrant 1 or 2: add 180°
• If azimuth is in Quadrant 3 or 4: subtract 180°
• (Equivalently: convert to the opposite end by shifting by 180°.)

Back bearing concept (directional reversal)

• Bearing magnitude stays the same, but the quadrant changes to the opposite one when viewing the line from the reverse endpoint.

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Traverse methodology: calculating azimuths using the contra method (detailed steps)

The video presents an open traverse (points like 1–2–3–4), using:

• An initial known azimuth (reference to north)
• Traverse angles at vertices (given as internal/external/deflection depending on traverse type; for open traverse they can be taken above/below/deflection as described)

Given/required inputs

• Starting alignment azimuth (example stated): 167°
• Vertex angles at each vertex (example stated):
  • angle at p2 = 71°
  • angle at p3 uses a complementary relation from a given angle of 129°
• Normalization requirement: azimuth must remain within 0–360° (subtract 360° if needed)

Step-by-step process (as demonstrated)

1. Identify the first alignment’s azimuth

   • Use the initial azimuth from field (example: 167°).

2. Compute the contra-azimuth of the current alignment

   • Determine quadrant/location logic (in the example, first azimuth is in quadrant 2).
   • Example: contra-azimuth = azimuth + 180°
   • Example result: 347°.

3. Compute the next alignment azimuth by adding the vertex angle

   • Take:
     • (contra-azimuth of previous alignment) + (angle at next vertex)
   • Example for p2–p3:
     • 347° + 71° = 418°
     • Normalize to 0–360:
       • 418° − 360° = 58°
     • Resulting azimuth: 58°.

4. Normalize and verify

   • Confirm “graphical correspondence” (calculated azimuth matches the sketch relative to north).

5. Repeat for subsequent sides

   • For each next alignment:
     • compute contra from current azimuth (add/subtract 180° as needed)
     • add the appropriate vertex angle
     • normalize to 0–360°
     • verify visually against the sketch

Special angle handling shown in the example

• At vertex p3, an “below” vs “above” angle requires converting using the 360° complement rule:
  • If one part is 129° below, then the “above” angle used is:
    • 360° − 129° = 230°
  • Then proceed with azimuth computations using the corrected “above” angle.

Outcome of the example

• Azimuths are computed for all alignments using the contra method, with normalization and verification.
• The final computed alignment azimuth stated is 109°, confirmed graphically.

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Analytic geometry relationships (coordinate-based computations)

The video transitions from graphical angles to Cartesian/topographic coordinate equations, using axes:

• East (E) coordinate
• North (N) coordinate

A point is written as (E, N) (or equivalent ordering depending on subtitle notation).

Main equations described

1. Distance between two points (Pythagorean theorem) If:

   • Point 1 = ((E_1, N_1))
   • Point 2 = ((E_2, N_2))

[ d=\sqrt{(E_2-E_1)^2+(N_2-N_1)^2} ]

1. Slope/angle relationship using tangent relative to the north axis [ \tan(\theta)=\frac{E_2-E_1}{N_2-N_1} ] Then: [ \theta = \arctan\left(\frac{E_2-E_1}{N_2-N_1}\right) ]

2. Projections using trigonometry

   • Projection on north axis: [ \Delta N = d \cos(\theta) ]

   • Projection on east axis: [ \Delta E = d \sin(\theta) ]

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Worked example from coordinates (as described)

Given coordinates

• Point 1:
  • (N_1 = 190.2,\quad E_1 = 137.419)
• Point 2:
  • (N_2 = 105.565,\quad E_2 = 50.327)

1) Compute distance (d)

• Substitute ((E_2-E_1)) and ((N_2-N_1)) into the distance formula.
• Result stated:
  • (d = 122{,}873) (Subtitles show likely scaling/precision noise, but the workflow is what matters.)

2) Compute bearing (quadrant direction via tangent)

• Use: [ \tan(\theta)=\frac{E_2-E_1}{N_2-N_1} ]

• Result stated:

  • (\theta \approx 45^\circ\ 8’\ 14’‘)
• Determine quadrant from sketch/graph:
  • Quadrant identified as Quadrant 3 (SW)
• Therefore, the bearing corresponds to SW with that angle.

3) Compute azimuth from bearing

• Rule used:
  • Azimuth = bearing reference angle + 180°
• Result stated:
  • Azimuth = 225° 8’ 14’‘.

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Speakers / sources featured

• Simon (referenced as the source/origin for direction definitions: “which Simon defines…”)
• No other named speakers are clearly identified in the subtitles.