Scale, Relief Displacement and Parallax - Revision Notes
This chapter covers three exam-critical topics in aerial photogrammetry: photo scale, relief displacement, and stereoscopic parallax. On the PRC Geodetic Engineer board exam, these topics consistently appear as numerical problems requiring direct formula application. Relief displacement is the radial outward lean of tall objects on a vertical photo; parallax is the apparent shift of a point between two overlapping photos measured along the flight line. Both effects are exploited to compute object heights and terrain elevations photogrammetrically. Mastery of the governing formulas, their variables, and their sign conventions is essential for exam success.
1. Photo Scale
Formulas
S = f / H = d_photo / D_ground
Example
Camera f = 152 mm; H = 3040 m. Scale = 0.152 / 3040 = 1 / 20 000. A photo distance of 50 mm represents 50 × 20 000 = 1 000 000 mm = 1000 m on the ground.
Variables
S = photo scale (dimensionless ratio); f = principal focal length of aerial camera (m or mm); H = flying height above datum (m); d_photo = measured distance on photo; D_ground = corresponding horizontal ground distance
Application
Determine the ground distance from a measured photo distance, or compute the flying height required for a desired map scale.
Exam tips
- Memorise: S = f/H. If only ground and photo distances are given, use S = d_photo / D_ground.
- Convert focal length from mm to m before dividing by H in metres.
- Board exam frequently asks: 'What is the ground distance represented by X mm on the photo?' — simply multiply X by the scale denominator.
Key points
- Photo scale S = f / H, where f is the camera focal length and H is the flying height above the datum (or above mean terrain).
- Scale can also be expressed as S = photo distance / ground distance.
- A larger denominator means a smaller (less detailed) scale; a smaller denominator means a larger (more detailed) scale.
- For a tilted photo or when terrain elevation varies, the scale changes across the photo.
- Scale is dimensionless and must use consistent units (e.g., both in metres or both in mm).
Definitions
Principal Point
The geometric centre of the aerial photograph, where the optical axis of the camera intersects the photo plane. Marked by fiducial marks.
Importance
All radial distances for relief displacement are measured from the principal point. Misidentifying this point will cause systematic error.
Flying Height (H)
The altitude of the aircraft's camera station above the reference datum (usually mean sea level or a chosen datum).
Importance
H appears in both the scale and relief displacement formulas. Board exam problems specify whether H is above datum or above average ground.
Fiducial Marks
Reference marks (usually four or eight) imprinted on the edges or corners of each aerial photo by the camera frame, used to locate the principal point.
Importance
Required to establish the coordinate system of the photo before measuring r, d, or parallax.
Common mistakes
- Mixing units: f in mm while H is in m. Always convert to the same unit before computing scale.
- Confusing flying height above datum with flying height above ground (terrain clearance). Exam problems must specify which H is given.
- Inverting the scale ratio: S = 1/20000 means the photo is smaller, not larger, than ground reality.
2. Relief Displacement
Formulas
d = (r · h) / H
Example
H = 1500 m, r = 80 mm, h = 50 m → d = (80 × 50) / 1500 = 2.667 mm
Variables
d = relief displacement (mm); r = radial distance of the image top from the principal point (mm); h = height of object above datum (m); H = flying height above datum (m). Keep r and d in the same unit.
Application
Compute how much a tall object's image is displaced on the photo, or rearranged as h = dH/r to determine object height from a measured displacement.
h = (d · H) / r
Example
d = 2.667 mm, H = 1500 m, r = 80 mm → h = (2.667 × 1500) / 80 = 50.0 m
Variables
Same as above; this is the rearranged form used when d and r are measured on the photo and H is known from flight records.
Application
Most common board exam form — measure d and r on the photo, solve for unknown object height h.
Exam tips
- Memorise: d = rh/H and h = dH/r. These two forms cover essentially all relief displacement board exam items.
- When asked 'where is relief displacement greatest?' answer: at the edge of the photo (maximum r).
- When asked 'where is relief displacement zero?' answer: at the principal point.
- Check units: r and d should both be in mm; H and h in metres. The formula is dimensionally consistent because h/H is dimensionless.
Key points
- Relief displacement (d) is the radial outward shift of the image of an elevated point from where it would appear if it were at datum level.
- Formula: d = r·h / H, rearranged to h = d·H / r.
- r is the radial distance of the TOP of the object from the principal point (not the base).
- H is the flying height above the DATUM, not above the object.
- Displacement is ZERO at the principal point and MAXIMUM at the photo edges — heights are therefore best measured near the edges.
- Relief displacement is always directed RADIALLY OUTWARD from the principal point.
- Tilt displacement is a separate effect and follows a different geometry; do not confuse the two.
Definitions
Relief Displacement (d)
The distance by which the image of an elevated object is displaced radially outward from the principal point relative to the position the base of the object would occupy on the photo.
Importance
Understanding this concept is essential for interpreting aerial photos and for computing object heights on the board exam.
Radial Distance (r)
The straight-line distance from the principal point to the image of the TOP of the object on the photo.
Importance
Using r to the BASE instead of the TOP is a very common board exam trap — always use r to the top of the displaced image.
Datum
The reference elevation surface from which flying height H and object height h are measured. In the Philippines, this is typically mean sea level referenced to the Philippine Vertical Datum (PMRC 2000) or a local datum.
Importance
H must be consistent with h: if H is above MSL, then h is the object's elevation difference above MSL.
Common mistakes
- Using r from the base of the object instead of the top — the formula requires r measured to the top (displaced image).
- Using terrain clearance instead of H above datum in the formula.
- Forgetting that relief displacement is zero exactly at the principal point — exam may ask where it is maximum.
- Applying the relief displacement formula to tilted photos without correction.
- Not keeping d and r in the same unit (both must be mm, or both in m).
3. Stereoscopic Parallax and Height Determination
Formulas
h = (H · Δp) / (P + Δp)
Example
H = 1500 m, P = 90 mm, Δp = 1.5 mm → h = (1500 × 1.5)/(90 + 1.5) = 2250/91.5 = 24.59 m ≈ 24.6 m
Variables
h = height of object above datum (m); H = flying height above datum (m); Δp = parallax difference between top and base of object (mm); P = absolute parallax of the base of the object (mm). P and Δp must be in the same unit.
Application
Primary formula for computing object height or terrain elevation difference from stereo pair measurements. Used in analogue, analytical, and digital photogrammetry.
h ≈ (H · Δp) / P
Example
Using same values: h ≈ (1500 × 1.5)/90 = 25.0 m (compare exact 24.6 m — ~1.6% error)
Variables
Same as above. This approximation is valid when Δp is much smaller than P (typically Δp < 5% of P).
Application
Quick estimate in board exam problems when the approximate form is specifically requested or when Δp is very small.
P = B · f / H (for a level stereo pair)
Example
B = 900 m, f = 0.152 m, H = 1500 m → P = (900 × 0.152)/1500 = 0.0912 m = 91.2 mm
Variables
P = absolute parallax; B = air base (m); f = focal length (m); H = flying height above datum (m). This links absolute parallax to mission parameters.
Application
Compute absolute parallax from known mission parameters when it is not directly measured.
Exam tips
- Exact formula: h = HΔp/(P + Δp). Always use this unless told to approximate.
- Units check: H in m gives h in m if Δp and P are both in mm (they cancel out as a ratio).
- If the board exam gives B and f separately, first compute P = Bf/H, then apply the height formula.
- Parallax difference Δp is always measured along the x-axis (flight direction) — this is a frequently tested conceptual point.
- A chimney, building, or tower will have a larger absolute parallax at its top than at its base — hence Δp > 0.
Key points
- Parallax is the apparent displacement of a point when viewed from two different camera positions along the flight line.
- Absolute parallax (P) of a point = x_L - x_R, where x_L and x_R are the x-coordinates of the point on the left and right photos respectively (measured parallel to the flight line).
- Parallax difference (Δp) between the top and the base of an object is used to compute object height.
- Height formula: h = H·Δp / (P + Δp), exact form.
- Approximate form (when Δp << P): h ≈ H·Δp / P.
- Parallax is measured PARALLEL to the flight line (the x-direction in a stereo pair).
- B is the air base (distance between the two exposure stations); P relates directly to B and scale.
Definitions
Absolute Parallax (P)
The sum of the x-coordinates of a point measured on the left and right photos (using the standard sign convention: P = x_L + x_R when coordinates are set up so that the flight direction is positive). It represents the total lateral shift of the point image between the two photos.
Importance
P is the baseline parallax value for height computation. Error in measuring P directly propagates into h.
Parallax Difference (Δp)
The difference in absolute parallax between two points at different elevations: Δp = P_top − P_base. It is always positive when the top is higher than the base.
Importance
Δp is directly measured in the stereo model and is the key observable for height determination.
Air Base (B)
The horizontal distance between two successive camera exposure stations along the flight line.
Importance
B determines the stereo base and hence the depth perception in the stereo model. Longer B gives larger parallax differences and better height sensitivity.
Stereoscopic Vision
The perception of depth achieved by viewing two overlapping photos simultaneously, one with each eye (using a stereoscope), simulating the binocular vision of the human eye.
Importance
The geometric basis for all parallax measurement and stereo height determination.
Overlap (End Lap)
The percentage of the photo area that is duplicated on the successive photo along a flight strip, typically 60% for stereo coverage.
Importance
Ensures every ground point is visible on at least two photos, enabling stereo measurement and parallax computation.
Common mistakes
- Using the approximate formula h = HΔp/P when Δp is NOT negligible compared to P — always check the ratio first.
- Measuring parallax perpendicular to the flight line instead of parallel — parallax is always the x-direction shift.
- Confusing absolute parallax P with parallax difference Δp — P is for one point; Δp is the difference between two points.
- Forgetting that h computed by the parallax formula is the HEIGHT of the top above the BASE elevation, not above datum, if P is measured at the base.
- Using focal length in mm while air base is in metres in the formula P = Bf/H — convert to consistent units.
4. Comparison and Integration of Concepts
Exam tips
- If the problem gives ONE photo and mentions a displaced image: use d = rh/H.
- If the problem gives a STEREO PAIR and mentions parallax values: use h = HΔp/(P + Δp).
- Remember: board exam problems almost always involve vertical photos unless explicitly stated otherwise.
Key points
- Both relief displacement and parallax allow height determination from aerial photos — they are complementary methods.
- Relief displacement uses a SINGLE photo: d = rh/H. Parallax uses a STEREO PAIR: h = HΔp/(P + Δp).
- Relief displacement is largest at photo edges (large r); parallax measurement quality improves with longer air base B.
- In modern digital photogrammetry (used in Philippine mapping projects by NAMRIA), both effects are handled automatically by photogrammetric software, but the fundamental geometry is the same.
- Under RA 8560 (Philippine Geodetic Engineering Law), photogrammetric surveys are within the professional scope of registered Geodetic Engineers.
- NAMRIA uses aerial photogrammetry extensively for topographic mapping of the Philippines at 1:10 000 and 1:50 000 scales.
- Photo scale S = f/H governs the ground sample distance (GSD) and map scale achievable from a flight mission.
Definitions
NAMRIA
National Mapping and Resource Information Authority — the primary government agency responsible for geodetic surveys, topographic mapping, and photogrammetric mapping in the Philippines.
Importance
NAMRIA produces the official topographic maps of the Philippines using aerial photogrammetry, directly relevant to the application of these formulas.
Common mistakes
- Confusing single-photo height determination (relief displacement) with stereo-pair height determination (parallax) — different formulas, different data requirements.
- Applying the parallax formula to a single photo — parallax requires at least two photos with stereo overlap.
Connections
- Photo Scale (S = f/H) is the foundation — it determines how ground distances translate to photo distances, and the same H appears in both the scale and relief displacement formulas.
- Relief Displacement and Parallax are both height-measurement tools: relief displacement works from a single photo (d = rh/H), while parallax requires a stereo pair (h = HΔp/(P + Δp)). They give consistent results when applied correctly.
- Absolute Parallax P is related to photo scale and air base via P = Bf/H — connecting the stereo geometry to the mission parameters (B and H) and camera specification (f).
- The principal point is central to both scale and relief displacement: scale uses it as the reference for a truly vertical photo, and r in the relief displacement formula is always measured from this point.
- In practical Philippine mapping under NAMRIA, both effects are handled by aerotriangulation software, but understanding their geometry underpins the quality checks and interpretation of photogrammetric products.
- Under PD 1529 (Property Registration Decree) and CA 141 (Public Land Act), accurate topographic data derived from photogrammetric surveys support land classification and titling — connecting this technical chapter to Philippine land law.
- The concept of overlap (end lap ~60%, side lap ~30%) directly determines the stereo base B and hence the parallax sensitivity — linking flight planning to height accuracy.
Quick review questions
A vertical aerial photo is taken from H = 1800 m above datum. A building's rooftop image is at r = 95 mm from the principal point, and the building is 60 m tall. What is the relief displacement?
Answer
d = (r · h) / H = (95 × 60) / 1800 = 5700 / 1800 = 3.167 mm
Direct application of d = rh/H. Keep r in mm and h, H in metres — the h/H ratio is dimensionless, so d comes out in mm. This is a standard board exam numerical problem.
On the same photo (H = 1800 m), a pole shows a relief displacement d = 3.0 mm at radial distance r = 100 mm from the principal point. What is the pole height?
Answer
h = (d · H) / r = (3.0 × 1800) / 100 = 5400 / 100 = 54.0 m
Rearranged formula h = dH/r. The relief displacement formula and its inverse are the two most frequently tested computations in this topic.
A stereo pair taken from H = 1600 m has an absolute parallax P = 88 mm for the base of a tower. The parallax difference between the tower top and base is Δp = 2.2 mm. Compute the tower height.
Answer
h = (H · Δp) / (P + Δp) = (1600 × 2.2) / (88 + 2.2) = 3520 / 90.2 = 39.02 m ≈ 39.0 m
Use the exact parallax-height formula. Note P + Δp = 90.2 mm. The approximate form would give h ≈ 1600 × 2.2 / 88 = 40.0 m — a ~2.5% overestimate, illustrating why the exact form is preferred.
At which location on a vertical aerial photo is relief displacement equal to zero, and why?
Answer
At the principal point (r = 0), because d = rh/H = 0·h/H = 0.
Since relief displacement is directly proportional to r, it vanishes at the photo centre (principal point) and increases radially outward. This is a key conceptual question on the board exam.
An aerial camera has a focal length of 152 mm. The desired photo scale is 1:20 000. What flying height above datum is required?
Answer
S = f/H → H = f/S = 0.152 / (1/20000) = 0.152 × 20000 = 3040 m
Rearrange S = f/H to H = f/S. Convert focal length to metres first: 152 mm = 0.152 m. The scale denominator 20 000 is the multiplier.
In a stereo pair, the air base B = 900 m, focal length f = 152 mm, and flying height H = 1500 m. What is the absolute parallax of a ground point?
Answer
P = B·f / H = (900 × 0.152) / 1500 = 136.8 / 1500 = 0.09120 m = 91.20 mm
Use consistent units: B and H in metres, f in metres. Convert f = 152 mm = 0.152 m. The result P = 91.20 mm is the absolute parallax, directly usable in the height formula.
Why are object heights measured near the EDGES of an aerial photo rather than near the centre when using relief displacement?
Answer
Because relief displacement d = rh/H is proportional to r. Near the edges, r is large, so d is large enough to measure accurately. Near the centre (small r), d is very small and difficult to measure precisely.
This is a conceptual board exam question. The relief displacement is zero at the principal point and grows toward the photo edges, making measurement more reliable (larger signal) at greater radial distances from the centre.
A photo taken from H = 2000 m shows a tower with relief displacement d = 3.0 mm at r = 100 mm. What government agency in the Philippines is primarily responsible for such photogrammetric mapping operations?
Answer
Tower height h = (3.0 × 2000) / 100 = 60.0 m. The agency is NAMRIA (National Mapping and Resource Information Authority).
This integrates computation with professional knowledge. NAMRIA is the primary mapping authority under the Department of Environment and Natural Resources (DENR). Photogrammetric surveys are within the professional scope of Geodetic Engineers under RA 8560.
Diagrams
Scale, Relief Displacement and Parallax – Chapter Overview
A mind map showing the hierarchical relationships among the three main topics — photo scale, relief displacement, and stereoscopic parallax — along with key parameters and Philippine professional context.
Decision Flowchart: Choosing the Correct Height Formula
A decision flowchart guiding the student to select the correct height determination formula based on whether a single photo or stereo pair is used, and whether to use the exact or approximate parallax formula.
Step-by-Step: Solving a Relief Displacement Problem
A step-by-step procedural flowchart showing how to systematically solve any relief displacement board exam problem, including the unit-checking step that prevents common errors.
Geometry of Relief Displacement on a Vertical Photo
A conceptual diagram illustrating the geometry of relief displacement on a vertical aerial photo, showing the relationship between the principal point, radial distance r, displacement d, object height h, and flying height H.
Development of Photogrammetric Height Measurement Methods
A timeline showing the historical development of photogrammetric height measurement methods, from early balloon photography to modern UAV photogrammetry, contextualising the board exam formulas within professional practice.
Exam strategy
For the PRC board exam on this chapter: (1) Identify the problem type immediately — single photo with displaced image = relief displacement formula; stereo pair with parallax values = parallax-height formula. (2) Write the formula first, then substitute — this earns partial credit and prevents arithmetic errors. (3) Always check unit consistency: r and d in mm; H, h, B in metres; f may be in mm and must be converted to metres. (4) For relief displacement: r goes to the TOP of the object. (5) For parallax: always use the exact form h = HΔp/(P + Δp) unless the approximate form is explicitly requested. (6) Conceptual questions on where displacement is maximum (photo edge) or zero (principal point) are frequent and require no computation. (7) Know NAMRIA and RA 8560 for the professional/legal context questions. (8) Practice the three standard worked examples in the reference document until you can solve them in under 2 minutes each.