Scale, Relief Displacement and Parallax - Cheat Sheet
Your last-minute revision companion for photogrammetric height measurement. Master relief displacement, parallax calculations, and scale relationships in 30 minutes.
Photo Scale and Image Relationships
Formulas
Scale = f / H or 1 / (H/f)
Meaning
f = focal length (mm); H = flying height above datum (m or mm — keep units consistent); Scale is the ratio of photo distance to ground distance
Watch out
H MUST be above the datum (not above ground level in hilly terrain). Mixing units (mm and m without conversion) is the #1 mistake — convert to consistent units first
When to use
Any time you need to convert between photo measurements and real ground distances; fundamental to all photogrammetric work
Ground distance = Photo distance / Scale or Ground distance = Photo distance × (H/f)
Meaning
Photo distance = measured distance on the photo (mm); H/f is the scale denominator; gives actual ground separation
Watch out
Scale changes with terrain relief — use average flying height for hilly areas. This is NOT constant across a tilted photo
When to use
When you measure a distance on a photo and need its true ground equivalent (e.g., building width, road length)
Average scale = f / H_avg where H_avg = (H1 + H2) / 2
Meaning
H_avg accounts for terrain elevation variation; H1 = flying height at near end, H2 = at far end
Watch out
This is an approximation. Different points have slightly different scales — stated average scale is for reference only
When to use
When terrain has significant relief and you need a representative scale for the entire photo strip
Common values
Value
150 mm or 200 mm
Symbol
f
Quantity
Standard aerial camera focal length (common)
Value
1500–3000 m
Symbol
H
Quantity
Typical flying height for topographic surveys
Value
1:5,000 to 1:10,000
Symbol
—
Quantity
Standard photo scale for cadastral work (Philippines)
Important facts
- Vertical photos have uniform scale at the same elevation; oblique photos have varying scale across the frame
- Scale is inversely proportional to flying height — higher altitude = smaller scale (smaller photo features)
- For accurate ground distances, H must be measured vertically to the datum plane, NOT to visible terrain
- PRS92 datum (Philippine Geodetic Reference System 1992) is the standard for Philippine photogrammetric work
- Scale affects measurement precision — larger scale (lower H) gives more precise measurements but smaller coverage area
Key definitions
Flying height (H)
Example
If aircraft altitude is 2000 m MSL and datum is MSL, then H = 2000 m
Vertical distance from aircraft to the datum plane (usually mean sea level); the reference for all scale calculations
Photo scale
Example
f = 150 mm, H = 1500 m → Scale = 150 / 1500,000 = 1/10,000 = 1:10,000
Ratio of a distance on the photo to the corresponding distance on the ground; expressed as 1:n (e.g., 1:10,000 means 1 mm on photo = 10,000 mm on ground)
Principal point
Example
For a vertical photo, the principal point is typically at the geometric center (at the intersection of diagonal lines)
The point on the photo where the optical axis perpendicular to the photo plane intersects; the center of perspective projection
Radial distance (r)
Example
A building's top is 85 mm from the principal point → r = 85 mm
The distance measured from the principal point to any point on the photo, along a straight line; used as reference for relief displacement
Diagrams to know
- Vertical photo geometry showing aircraft, datum, flying height H, and principal point
- Radial distance r from principal point to image point
- Comparison of scale variation in vertical vs. tilted/oblique photos
Relief Displacement
Formulas
d = (r × h) / H
Meaning
d = relief displacement (mm); r = radial distance from principal point to image top (mm); h = object height (m); H = flying height above datum (m)
Watch out
H and h MUST be in the same vertical datum. Keep r, d in mm and H in m, OR convert all to mm. The displacement is RADIAL OUTWARD from the principal point — NOT vertical or horizontal
When to use
When you measure a tall object (tower, tree, building) on a photo and want to find the outward radial shift caused by height
h = (d × H) / r
Meaning
Rearranged form to find object height given measured displacement; solving for h
Watch out
The displacement d is typically small (1–5 mm). Measurement error is magnified — use precision instruments (parallelepiped, comparator)
When to use
When you measure displacement d at radial distance r on a photo from height H, and you want to find the object height
d_max occurs near the edge of photo; d = 0 at the principal point
Meaning
Relief displacement increases with distance from the principal point; this is a consequence of the conic projection geometry
Watch out
Objects AT the principal point show zero displacement — this makes height measurement impossible there. Always measure objects well off-center
When to use
Explains why buildings and towers appear to lean outward, most noticeably at photo edges
Common values
Value
1.6 mm
Symbol
d
Quantity
Typical relief displacement for 30 m object at 1500 m altitude, r = 80 mm
Value
5.3 mm
Symbol
d
Quantity
Relief displacement for 100 m object at same altitude and radial distance
Important facts
- Relief displacement is RADIAL (along a line from principal point) and OUTWARD (away from principal point), NOT vertical or horizontal displacement
- The effect is strongest for tall objects far from the principal point; it vanishes at the principal point
- Relief displacement depends ONLY on object height and flying height, NOT on the object's ground position (x, y coordinates)
- For a given H and r, displacement is directly proportional to object height h
- RA 8560 (Geodetic Engineer Licensure Law) requires competence in photogrammetric height measurement; relief displacement is a core technique
Key definitions
Relief displacement (d)
Example
A 50 m tower on a 1500 m altitude photo appears displaced 2.67 mm from its base position, radially outward from the principal point
Radial outward shift of the image of a point at height h, due to the conic projection from altitude H; independent of horizontal ground position
Vertical photograph
Example
A photo from a vertical camera mount where the optical axis points straight down to the ground
An aerial photo taken with the optical axis perpendicular (vertical) to the ground; principal point coincides with nadir
Nadir
Example
For a vertical photo at 2000 m altitude, the nadir is the ground point 2000 m directly below the aircraft
The point on the ground directly below the aircraft (at the intersection of the vertical line through the camera with the ground plane)
Diagrams to know
- Cross-section diagram of a tower showing vertical height h, relief displacement d, and the conic projection from camera to image
- Plan view of photo showing radial displacement vectors pointing outward from principal point for objects of different heights
Stereoscopic Parallax
Formulas
P = f × B / H or P = b / m
Meaning
P = absolute parallax (mm); f = focal length (mm); B = air-base (ground distance between camera stations, m); H = flying height (m); b = parallax base on photo (mm); m = scale denominator
Watch out
Keep units CONSISTENT — if f and b are in mm, ensure B and H are related by the same scale (H/f = B/b). B is the GROUND distance, not the photo distance
When to use
To establish the absolute parallax of a point in a stereo model from flight geometry; used as reference for height calculations
h = (H × Δp) / (P + Δp) ≈ (H × Δp) / P (when Δp << P)
Meaning
h = object height (m); Δp = parallax difference (top minus base of object, mm); P = absolute parallax (mm); H = flying height (m)
Watch out
Δp is the DIFFERENCE in parallax between top and base, NOT the absolute parallax of either point. Sign matters — top parallax > base parallax. For small heights, the approximation (denominator = P) is valid; full formula is more accurate for tall objects
When to use
Most common exam formula — given parallax readings on a stereo pair, find the height of a tower, chimney, or building
Δp = p_top − p_base (always top minus base)
Meaning
Parallax difference is the signed difference; top of object is closer to camera, so has larger parallax value
Watch out
DIRECTION matters — top parallax is LARGER (object is higher, so closer to camera in stereo model). Reversing the sign gives negative height
When to use
When measuring parallax differences to determine object heights in stereo photogrammetry
Common values
Value
55–60%
Symbol
—
Quantity
Typical overlap percentage for stereo coverage
Value
300 m
Symbol
B
Quantity
Typical air-base for 1500 m altitude survey at 60 m/s, 5 s interval
Value
30 mm
Symbol
P
Quantity
Typical absolute parallax for 1500 m altitude, 150 mm lens, 300 m air-base
Value
1.0–1.5 mm
Symbol
Δp
Quantity
Parallax difference for 50 m object under same conditions
Important facts
- Parallax is measured PARALLEL TO THE FLIGHT LINE (x-direction), NOT perpendicular (y-direction or across-track)
- Parallax increases as an object gets closer to the camera; top of a tower has greater parallax than its base
- The relationship h = H×Δp/(P+Δp) is LINEAR in Δp for small Δp (typically valid for objects < 10% of flying height)
- Standard overlap between stereo photos is 50–60% for continuous strip coverage; sufficient overlap ensures parallax can be measured
- Modern digital photogrammetry (automated stereo matching) computes parallax at sub-pixel accuracy, enabling meter-level elevation accuracy
- PPCS (Philippine Plane Coordinate System) conversions assume heights are derived from stereo parallax or other surveyed elevations
Key definitions
Absolute parallax (P)
Example
For a photo pair from 1500 m with 300 m air-base and 150 mm focal length, P = (150 × 300) / 1500 = 30 mm
The parallax of a point on the datum plane (ground level); defined by the flight geometry (f, B, H); constant reference for a stereo model
Parallax (p)
Example
A point appears 35 mm farther to the right in the right photo compared to the left photo → p = 35 mm (for this example)
The apparent shift of a point between left and right photos of a stereo pair, measured parallel to the flight line (x-direction)
Parallax difference (Δp)
Example
Building top has p_top = 32.5 mm, base has p_base = 30.0 mm → Δp = 2.5 mm
The difference in parallax between two points (top and base of an object); used to determine relative heights
Air-base (B)
Example
Aircraft flies at constant altitude and speed; if ground speed is 60 m/s and exposure interval is 5 s, air-base ≈ 300 m
The ground distance (3D vector) between the two camera exposure stations of a stereo pair; in practice, often the horizontal distance along the flight direction
Stereoscopic model
Example
Viewed through a stereo viewer, a tower appears to rise above the surrounding terrain, and its parallax shift is measurable
The 3D mental image created when viewing overlapping aerial photos in stereo (left and right eyes receive slightly different images); allows parallax measurement
Diagrams to know
- Stereo pair geometry showing left and right camera positions, object, and the parallax shift measured parallel to baseline
- Cross-section showing how parallax changes with altitude (higher objects = larger parallax shift)
- Parallelepiped or parallax bar measurement setup with markings for left and right image readings
Height Determination Methods Comparison
Formulas
Method 1 (Relief Displacement): h = (d × H) / r
Meaning
Single photo measurement; d from principal point, r to image top
Watch out
Less accurate than stereo parallax; object must be well off-center (r > 50 mm typical); measurement error in d is magnified
When to use
Quick method when you have only one photo; e.g., measuring a single tall building
Method 2 (Stereoscopic Parallax): h = (H × Δp) / (P + Δp)
Meaning
Stereo pair measurement; Δp from parallax readings at top and base
Watch out
Requires overlapping stereo pair and parallax measurement instrument; Δp must be measured carefully with parallelepiped or digital correlation
When to use
Preferred method; stereo coverage is standard in aerial surveys; more accurate and less sensitive to small measurement errors
Important facts
- Relief displacement works from a SINGLE photo; parallax requires a STEREO PAIR
- Parallax method is generally MORE ACCURATE (better precision) for moderate heights (10–200 m)
- Relief displacement is applicable even when stereo coverage is not available; useful for oblique or archived single photos
- Both methods give HEIGHT, not elevation — to get elevation, add height to the base/ground-point elevation
- Modern practice: digital photogrammetry automates parallax measurement across dense point clouds, giving sub-meter accuracy
Diagrams to know
- Comparison chart: Single-photo relief displacement setup vs. stereo pair parallax setup
- Diagram showing measurement points: principal point and r for displacement; left/right photos and parallax marks for stereo
Scale Relationships and Datum Corrections
Formulas
Scale at elevation E: Scale_E = f / (H − E)
Meaning
f = focal length; H = flying height above datum; E = elevation of terrain at point (positive upward from datum)
Watch out
If terrain is ABOVE datum, (H − E) is smaller, so scale is LARGER (finer detail). If you use absolute altitude without subtracting elevation, your scale is WRONG
When to use
When datum is at sea level and terrain is elevated; accounting for local scale variation
Scale correction: ΔScale = f × E / H(H − E) or relative: ΔScale/Scale = E / (H − E)
Meaning
The change in scale per unit elevation change; used to correct measured distances on sloping terrain
Watch out
For small E relative to H (e.g., E = 100 m, H = 1500 m), correction is small (~7%) but cumulative in large surveys. Often IGNORED in practice unless precision < 1% is required
When to use
When extrapolating a single average scale across hilly terrain; small but important correction for cadastral or precision mapping
Common values
Value
-0.5 to -1.5 m
Symbol
N
Quantity
Typical geoid undulation (N) in Philippines
Value
WGS84 (compatible)
Symbol
—
Quantity
PRS92 reference ellipsoid
Important facts
- PRS92 is the official Philippine reference system (RA 4374, RA 8560); all surveys should be referenced to PRS92
- UTM/PPCS coordinate projections assume known elevations to compute N/E coordinates correctly; scale varies with elevation
- Large-scale cadastral plans (RA 4374 lot surveys) must reference both horizontal (PPCS/UTM) and vertical (MSL/ellipsoid) datums
- GPS-derived ellipsoidal heights (WGS84) must be converted to orthometric heights (MSL-referenced) using geoid models (PHGeoid or similar)
- Geoid undulation in Philippines ranges ~1 to ~2 m; not negligible for high-precision surveying
Key definitions
Datum plane
Example
Philippine surveys use PRS92 (Philippine Geodetic Reference System 1992) with ellipsoidal heights; MSL is approximated by local sea-level datum
Reference surface (usually mean sea level, MSL) to which all vertical heights and elevations are referred; WGS84 ellipsoid or PRS92 ellipsoid for Philippine surveys
Elevation (E)
Example
A hilltop at 250 m above MSL has elevation E = 250 m; a 50 m tall tower on that hilltop has height h = 50 m
Vertical distance of a ground point above the datum plane; different from HEIGHT (which is object height or relative difference)
Diagrams to know
- 3D diagram: datum plane, flying height H, terrain elevation E, and scale variation with height
Exam Formulas Summary & Quick Reference
Formulas
Scale = f / H
Meaning
f (focal length, mm); H (flying height above datum, m); result is 1/n where n is scale denominator
Watch out
Unit mismatch is the #1 error — f in mm, H in m: convert H to mm first or f to m. Example: f=150 mm, H=1500 m → f/H = 0.15/(1500) = 1/10,000
When to use
First calculation in any photogrammetric problem; determines ground distance equivalent of photo measurements
d = rh / H
Meaning
Relief displacement from single photo
Watch out
r from principal point; h and H in same units (meters); d and r in same units (mm)
When to use
When h, H, r are known; find displacement d. Or h = dH/r when d, r, H are known
h = H × Δp / (P + Δp)
Meaning
Height from stereo parallax; full formula
Watch out
When Δp ≪ P (typical for h < 0.1H), approximation h ≈ H×Δp/P is valid and simpler
When to use
Preferred exam formula; accounts for non-linearity for tall objects
Important facts
- Three core formulas: Scale = f/H, Relief d = rh/H, Parallax h = HΔp/(P+Δp) — memorize these exactly
- Units: f in mm, H in m (convert as needed); r, d, Δp, P in mm; h in m
- Principal point is the reference for relief displacement; radial distance r is measured FROM it
- Parallax is measured parallel to flight line; Δp = p_top − p_base (always top minus base)
- Absolute parallax P depends on flight geometry; parallax difference Δp depends only on object height
Must remember
- Relief displacement d = rh/H is RADIAL OUTWARD from the principal point; object height h = dH/r. Displacement is zero at principal point.
- Stereoscopic parallax h = H×Δp/(P+Δp) where Δp = p_top − p_base (parallax DIFFERENCE between top and base). Approximation h ≈ H×Δp/P valid when Δp ≪ P.
- Photo scale = f/H (focal length / flying height above datum). Keep units consistent (f in mm, H in m: convert H to mm or f to m). Scale ≠ constant on sloping terrain.
- Flying height H is ABOVE DATUM, NOT above terrain. Neglecting elevation E causes systematic error in computed heights and scales.
- Parallax is measured PARALLEL TO FLIGHT LINE (x-direction in stereo model), NOT across-track. Left and right photo shifts along the baseline give the parallax reading.
- Absolute parallax P = f×B/H where B is air-base (ground distance between camera stations). This is the reference parallax for datum elevation; all object parallax is relative to P.
- Relief displacement method works from SINGLE PHOTO; parallax method requires STEREO PAIR. Relief displacement is less accurate but applicable when stereo is unavailable.
- PRS92/WGS84 ellipsoidal heights must be converted to MSL-referenced orthometric heights using geoid model (PHGeoid, etc.); geoid undulation in Philippines ≈ −0.5 to −1.5 m.
- Exam errors: (1) mixing units (mm and m), (2) using absolute altitude instead of H above datum, (3) reversing top−base order in Δp (causes sign error), (4) measuring r to object base instead of top.
- Standard stereo overlap 55–60%; exposure interval and ground speed determine air-base B. Longer B (farther apart photos) = larger parallax shifts = larger Δp for same height.
Last minute tips
- UNITS CHECK FIRST: In d=rh/H, write r(mm), h(m), H(m), d(mm). Before solving, convert to consistent units — most exam mistakes stem from mixed units (mm vs m, km, etc.).
- RELIEF DISPLACEMENT DIRECTION: Always say 'radial outward from principal point' — not 'vertical' or 'horizontal.' Draw a vector from the principal point through the image toward the edge; displacement is along this vector, away from the principal point.
- PARALLAX DIFFERENCE SIGN: When reading parallax on a stereo bar or parallelepiped, p_top (closer object) > p_base (farther object). Δp = p_top − p_base is ALWAYS positive for a tall object above the base point. Reversed signs mean your reading is backward.
- ABSOLUTE PARALLAX REFERENCE: P is fixed by flight geometry (f, B, H) and depends on the datum level chosen. All height calculations assume a reference parallax P at the lowest point. If P is wrong, all heights are wrong by a constant.
- QUICK APPROXIMATION: When Δp is small compared to P (typically Δp < P/10), the approximation h ≈ H×Δp/P is accurate to >2%. Use this for speed; use full formula h = H×Δp/(P+Δp) if Δp is large (tall object, small flying height).
Comparison tables
Relief Displacement vs. Stereoscopic Parallax
| Characteristic | Relief Displacement (d = rh/H) | Stereoscopic Parallax (h = HΔp/(P+Δp)) |
|---|---|---|
| Data required | Single vertical photo; r (radial distance to top), h (height), H (flying height) | Stereo pair (left & right photos); Δp (parallax difference), P (absolute parallax), H (flying height) |
| Measurement tool | Ruler or stereoscope; measure r and estimate/measure d | Parallelepiped (parallax bar) or digital stereo correlation; measure p_left and p_right |
| Accuracy | Lower; ~2–5% typical; sensitive to r measurement and principal point identification | Higher; ~1–2% typical; less sensitive to small relative errors in Δp |
| Applicable height range | Any height, but best for h > 20 m (small d easier to measure); poor at principal point (d=0) | Any height < 0.2H; best for h = 10–100 m (Δp = 1–10 mm typical) |
| Availability of data | Works with single archived photos; no stereo requirement | Requires stereo coverage; standard in modern aerial surveys |
| Key equation | h = dH / r | h = HΔp / (P + Δp) ≈ HΔp / P |
| Common exam context | Single-photo height problems; building/tower on tilted photo | Stereo-pair problems; topographic survey height extraction; DEM generation |
Photo Scale Variation with Terrain
| Condition | Formula | Example (f=150 mm, H=1500 m, E_max=500 m) | Impact on measurement |
|---|---|---|---|
| Flat terrain (E≈0, datum-referenced) | Scale = f / H | Scale = 150 / 1500 m = 1:10,000 (uniform across photo) | Simple; all photo distances are multiplied by 10,000 to get ground distance |
| Sloping terrain (E varies) | Scale at E = f / (H − E) | At E=0: 1:10,000; at E=500m: 150/(1000) = 1:6,667 (coarser detail at higher elevation) | Must account for local elevation; higher ground = finer scale = measured distances appear shorter |
| Average scale (hilly area) | Scale_avg = f / (H − E_avg) | With E_avg = 250 m: 150 / 1250 = 1:8,333 (compromise) | Use for rough measurements; individual point scale varies from this by up to ±17% in 500 m relief |
Common Parallax & Relief Displacement Exam Scenarios
| Scenario | Given | Find | Primary Formula | Expected answer range |
|---|---|---|---|---|
| Tower height from relief displacement | d=2.5 mm, r=80 mm, H=1500 m | Tower height h | h = dH/r = 2.5×1500/80 | h = 46.9 m ≈ 47 m |
| Displacement of known tower | h=50 m, r=75 mm, H=1500 m | Relief displacement d | d = rh/H = 75×50/1500 | d = 2.5 mm |
| Building height from stereo parallax | Δp=1.8 mm, P=90 mm, H=1500 m | Building height h | h = HΔp/(P+Δp) = 1500×1.8/91.8 | h ≈ 29.4 m ≈ 30 m |
| Ground distance from photo | Photo distance = 85 mm, f=150 mm, H=1500 m | Ground distance | Ground = 85 × (H/f) = 85 × 10,000 | Ground distance = 850,000 mm = 850 m |
| Absolute parallax from flight geometry | f=150 mm, B=300 m, H=1500 m | Absolute parallax P | P = f×B/H = 150×300/1500 | P = 30 mm |
Diagrams
Photogrammetric Height Measurement Methods
Mind map showing the two primary methods for photogrammetric height determination (relief displacement and stereoscopic parallax), their key formulas, measurement approaches, accuracy ranges, and the foundational photo scale and datum concepts that underpin both.
Photogrammetric Height Problem-Solving Workflow
Step-by-step flowchart guiding a solver through either the relief displacement or stereoscopic parallax method, emphasizing the critical unit consistency check and verification steps common to exam problems.
Photogrammetric Data States and Transformations
State diagram showing the progression from raw photogrammetric measurements (r, d, or parallax) through height computation to final elevation data and coordinate transformation for map production or survey records.