GELE Photogrammetry & Cartography — Map ProjectionsRevision Notes
Revision notes for GELE Photogrammetry & Cartography — Map Projections. Short, focused, and designed for the week before exam day. Use these when you are already familiar with the chapter and need a quick refresh on the high-yield items Professional Regulation Commission (PRC) — Board of Geodetic Engineering tests.
Exam context
On the GELE 2026, the Photogrammetry & Cartography subtest carries a "Core" weight in Professional Regulation Commission (PRC) — Board of Geodetic Engineering's pattern. Map Projections lands at position 4th out of 6 in the standard review order. Target score is 70% weighted average, no sub-test below 50%, and roughly a meaningful share of items come from Photogrammetry & Cartography on a typical GELE paper.
Map Projections - Revision Notes
Map projections are mathematical transformations that convert the curved surface of the Earth (ellipsoid or sphere) onto a flat plane. Because the Earth is not developable (cannot be unrolled without tearing or stretching), every projection introduces distortion in at least one geometric property: area, shape (angles), distance, or direction. For the PRC Geodetic Engineer Licensure Examination, you must know the classification of projections, the property each preserves, the scale factor concept, and how the Philippine Plane Coordinate System (PPCS) and UTM apply Transverse Mercator to Philippine surveys. Key legal references include RA 4374 and RA 8560 (regulating geodetic engineering practice), PD 1529 (Property Registration Decree), and CA 141 (Public Land Act), all of which rely on accurate coordinate systems rooted in correct projection theory.
Sections
Exam Tips
- Board exams often ask: 'Which projection preserves [property]?' — memorize conformal → angles, equal-area → area, equidistant → distance.
- Remember: No projection preserves ALL properties simultaneously — this is a classic true/false trap.
- When a question says 'suitable for comparing province areas,' the answer is always equal-area (equivalent).
Key Points
- A map projection is a systematic mathematical transformation from the ellipsoidal (or spherical) surface to a flat plane.
- The Earth's surface is non-developable; flattening it always introduces distortion.
- Four geometric properties can be distorted: area, shape (angles), distance, and direction.
- No single projection can simultaneously preserve all four properties.
- The choice of projection depends on the purpose of the map: surveying, navigation, thematic analysis, etc.
- The Philippines (archipelago extending ~1,850 km N-S, ~1,100 km E-W) requires a projection system that minimizes distortion across many narrow longitude bands.
Definitions
Term
Map Projection
Definition
A systematic, mathematically defined method of representing the curved surface of the Earth on a flat plane, involving a geometric or analytical transformation of geographic coordinates (φ, λ) to plane coordinates (x, y) or (E, N).
Importance
Foundation concept for all plane coordinate systems used in Philippine surveying (PPCS, UTM).
Term
Developable Surface
Definition
A geometric surface that can be unrolled or 'developed' into a flat plane without tearing or stretching. The three classic developable surfaces are the cylinder, cone, and plane (azimuthal).
Importance
Determines the projection family (cylindrical, conic, azimuthal) and the pattern of distortion.
Term
Standard Line (Standard Parallel or Central Meridian)
Definition
The line along which the developable surface is tangent (or secant) to the ellipsoid; distortion is zero (k = 1) along this line.
Importance
Defines where the projection is most accurate; zones in UTM/PPCS are designed around central meridians.
Term
Distortion
Definition
The difference between a geometric property (area, angle, distance, direction) on the projected map and its true value on the ellipsoid.
Importance
Quantified by the scale factor k; must be corrected in high-precision surveys.
Section Title
1. Fundamentals of Map Projections
Common Mistakes
- Assuming that a 'good-looking' map has no distortion — visual appearance does not indicate accuracy.
- Confusing the developable surface (how the projection is constructed) with the property it preserves.
- Thinking that a globe (sphere model) also has projection distortion — a globe itself is a true, undistorted representation.
Exam Tips
- Transverse Mercator: cylinder axis is in the equatorial plane, tangent to a meridian. Board exams test this distinction from normal Mercator.
- Lambert Conformal Conic (LCC) is used in aeronautical charts in the Philippines — it preserves angles (conformal), not area.
- Albers Equal-Area Conic is used for area statistics maps — two standard parallels, preserves area.
- Gnomonic projection: great circles = straight lines → used for great-circle navigation route planning.
Key Points
- Projections are classified by (a) developable surface and (b) the geometric property preserved.
- CYLINDRICAL: Cylinder wrapped around the Earth; standard line is a great circle (equator for normal aspect, meridian for transverse). Examples: Mercator, Transverse Mercator, UTM.
- CONIC: Cone placed over the Earth; standard lines are one or two parallels. Examples: Albers Equal-Area Conic, Lambert Conformal Conic.
- AZIMUTHAL (PLANAR): Plane tangent to a pole or any point; standard line is a single point or small circle. Examples: Stereographic, Gnomonic, Orthographic.
- CONFORMAL (Orthomorphic): Preserves angles/shapes locally; graticule lines cross at right angles; scale factor same in all directions at a point. Examples: Mercator, Transverse Mercator, Lambert Conformal Conic.
- EQUAL-AREA (Equivalent): Preserves area; regions maintain correct size ratios. Examples: Albers Equal-Area Conic, Sinusoidal, Mollweide.
- EQUIDISTANT: Preserves distances along specific lines (e.g., from center, or along meridians). Examples: Azimuthal Equidistant, Plate Carrée.
- GNOMONIC: All great circles appear as straight lines — used for planning great-circle routes.
- A projection can be NEITHER conformal NOR equal-area (compromise projections) — e.g., Robinson, Winkel Tripel (used for world maps).
Definitions
Term
Conformal Projection
Definition
A projection that preserves local angles and shapes. At every point, the scale factor k is the same in all directions, though k varies from point to point across the map.
Importance
Essential for surveying and engineering: measured angles and bearings transfer to the grid correctly. PPCS and UTM are both conformal (Transverse Mercator).
Term
Equal-Area (Equivalent) Projection
Definition
A projection where the ratio of any mapped area to its true ellipsoidal area is constant across the entire map. Shape is sacrificed to preserve area.
Importance
Required for thematic maps comparing areas (e.g., forest cover by province, land-use statistics). Used by NAMRIA for area-based maps.
Term
Equidistant Projection
Definition
A projection preserving true distances along specified lines (e.g., all directions from the map center, or along meridians), but not all distances simultaneously.
Importance
Used for measuring distances from a fixed point (e.g., earthquake epicenter maps, aviation range circles).
Term
Transverse Mercator (TM)
Definition
A conformal cylindrical projection with the cylinder axis in the equatorial plane (cylinder tangent to a meridian rather than the equator). The standard line is the central meridian.
Importance
The mathematical basis for both PPCS and UTM — the two coordinate systems governing all Philippine cadastral and engineering surveys.
Section Title
2. Classification of Projections
Common Mistakes
- Confusing Mercator (normal cylinder, equatorial standard line) with Transverse Mercator (cylinder rotated 90°, meridional standard line).
- Assuming conformal = equal-area. Mercator famously distorts area at high latitudes (Greenland appears larger than Africa on Mercator, but Africa is ~14× larger in reality).
- Saying 'cylindrical projection preserves area' — it depends on the specific projection, NOT just the surface type.
- Classifying Lambert Conformal Conic as equal-area because it sounds like a conic — it is CONFORMAL, not equal-area.
Formulas
Example
A geodetic line is 5,000.00 m on the ellipsoid. At that location k = 0.99960. Grid distance = 0.99960 × 5,000.00 = 4,998.00 m. The grid is 2.00 m shorter than the true distance.
Formula
k = d_projected / d_ellipsoidal
Variables
k = scale factor (dimensionless); d_projected = distance on the map/grid (m); d_ellipsoidal = true geodetic distance on the ellipsoid (m)
Application
Convert between grid distances (plotted on maps or computed in coordinate systems) and true ellipsoidal distances. Required in every cadastral and engineering survey using PPCS or UTM.
Example
Ellipsoidal distance = 8,000.00 m, k = 1.00012. Grid distance = 1.00012 × 8,000.00 = 8,000.96 m.
Formula
d_grid = k × d_ellipsoidal
Variables
d_grid = projected (grid) distance (m); k = scale factor; d_ellipsoidal = ellipsoidal distance (m)
Application
Forward computation: from field-measured (or computed ellipsoidal) distances to grid distances for plotting and coordinate computation.
Example
Grid distance = 4,998.00 m, k = 0.99960. Ellipsoidal distance = 4,998.00 / 0.99960 = 5,000.00 m.
Formula
d_ellipsoidal = d_grid / k
Variables
Same as above.
Application
Inverse computation: recover true ellipsoidal distance from a grid measurement.
Example
k = 0.99960, H = 200 m, R = 6,371,000 m. Elevation factor = 6,371,000 / (6,371,000 + 200) = 0.999969. CSF = 0.99960 × 0.999969 = 0.999569.
Formula
CSF = k × (R / (R + H))
Variables
CSF = combined scale factor; k = projection scale factor; R = mean Earth radius ≈ 6,371,000 m; H = orthometric height above MSL (m)
Application
Ground-truth distance reduction: converts ground (slope-corrected horizontal) distances to grid distances in one step.
Exam Tips
- Always state the formula first, substitute values, then compute — board exam solutions expect step-by-step presentation.
- k = 0.9996 for UTM → grid distance is SHORTER than ellipsoidal at the central meridian.
- k > 1 at zone edges → grid distance is LONGER than ellipsoidal.
- If the problem gives both k and elevation, compute CSF = k × (R/(R+H)) before multiplying by distance.
- Quick check: if k ≈ 1.0, grid ≈ ellipsoidal — useful for sanity-checking answers.
Key Points
- The scale factor k is the ratio of an infinitesimal distance on the projected plane to the corresponding true distance on the ellipsoid.
- k = 1 along the standard line(s) — no distortion there.
- For Transverse Mercator (UTM): k < 1 at the central meridian (intentionally set to k₀ = 0.9996), k = 1 at two lines parallel to the central meridian, and k > 1 beyond those lines toward the zone edges.
- Setting k₀ = 0.9996 (rather than 1.000) at the central meridian spreads distortion more evenly across the 6° UTM zone.
- Grid distance = k × ellipsoidal (geodetic) distance.
- Conversely: ellipsoidal distance = grid distance / k.
- For PPCS, k₀ varies by zone (most zones also use k₀ = 0.99995).
- The combined scale factor (CSF) in practice incorporates both projection scale factor and elevation factor: CSF = k × (R / (R + H)), where R is mean Earth radius and H is elevation.
Definitions
Term
Scale Factor (k)
Definition
The ratio of an infinitesimal projected distance to the corresponding ellipsoidal distance at a given point. k = 1 means no linear distortion at that point.
Importance
The single most tested numerical concept in map projections for the board exam.
Term
Central Scale Factor (k₀)
Definition
The scale factor assigned to the central meridian of a Transverse Mercator projection. For UTM: k₀ = 0.9996. For PPCS: k₀ = 0.99995 (most zones).
Importance
k₀ < 1 is set deliberately so that distortion is balanced across the zone width — a key board-exam explanation question.
Term
Elevation Factor
Definition
The ratio R/(R+H), which accounts for the reduction of a surface (ground) distance to its ellipsoidal equivalent. Always ≤ 1 (since H ≥ 0).
Importance
Must be combined with projection scale factor to get the total grid-to-ground correction in engineering surveys.
Term
Combined Scale Factor (CSF)
Definition
CSF = k × (R/(R+H)). Multiplying a ground distance by the CSF gives the equivalent grid distance directly.
Importance
Tested in advanced survey computations; bridges field measurements to grid coordinate systems.
Section Title
3. Scale Factor (k) — The Core Formula
Common Mistakes
- Forgetting that k < 1 at the UTM central meridian (k₀ = 0.9996), which means grid distance < ellipsoidal distance there — students sometimes flip the formula.
- Applying k > 1 universally — k only exceeds 1.000 near the zone edges, not at the center.
- Ignoring the elevation factor when computing ground-to-grid reduction — in mountainous areas of the Philippines (e.g., Cordillera), H can be 1,500–2,900 m, making the elevation factor significant.
- Using the wrong k₀: UTM uses 0.9996; PPCS uses 0.99995 — these are NOT interchangeable.
Exam Tips
- Memorize: PPCS zones I–V, central meridians 117°, 119°, 121°, 123°, 125°E, zone width 2°, k₀ = 0.99995.
- UTM: zone width 6°, k₀ = 0.9996, Philippines in Zones 51N and 52N.
- RA 8560 (Geodetic Engineering Act) → requires use of NAMRIA-prescribed coordinate systems. RA 4374 is its predecessor.
- PD 1529 → land titles require survey plans with prescribed coordinates.
- CA 141 → public land surveys must follow NAMRIA standards.
- NAMRIA = prescribing authority for geodetic reference systems in the Philippines.
Key Points
- Both PPCS (Philippine Plane Coordinate System) and UTM (Universal Transverse Mercator) use the Transverse Mercator projection — conformal, suitable for surveying.
- PRS92 (Philippine Reference System of 1992) is the geodetic datum for both PPCS and Philippine UTM; it is aligned with WGS84 to within ~1 m.
- UTM divides the world into 60 zones of 6° longitude each. The Philippines (117°E–127°E) falls in UTM Zones 51N and 52N.
- PPCS divides the Philippines into 5 zones (I–V) with 2° width each, with central meridians at 117°E, 119°E, 121°E, 123°E, 125°E.
- UTM k₀ = 0.9996 (central meridian scale factor); PPCS k₀ = 0.99995 — PPCS has narrower zones so less distortion is needed at the center.
- UTM false easting = 500,000 m; false northing = 0 m (northern hemisphere). PPCS false easting = 500,000 m; false northing = 0 m.
- RA 4374 (1965) and RA 8560 (1998) mandate that geodetic engineers use NAMRIA-prescribed coordinate systems (PPCS/UTM on PRS92) for all survey work submitted to government.
- PD 1529 requires all land titles to be supported by survey plans using the prescribed coordinate system.
- CA 141 governs disposition of public lands — surveys must conform to NAMRIA standards (PPCS/UTM/PRS92).
- NAMRIA (National Mapping and Resource Information Authority) is the government agency responsible for prescribing and maintaining the national coordinate reference systems.
Definitions
Term
PPCS (Philippine Plane Coordinate System)
Definition
A Transverse Mercator grid system covering the Philippines in five 2°-wide zones (I to V), using PRS92 as datum, k₀ = 0.99995, false easting = 500,000 m. Prescribed by NAMRIA for cadastral and engineering surveys.
Importance
The primary coordinate system for Philippine land surveys; directly tested in board exams.
Term
UTM (Universal Transverse Mercator)
Definition
A global Transverse Mercator grid system with 60 zones of 6° each, k₀ = 0.9996, false easting = 500,000 m. Philippines is in Zones 51N and 52N on PRS92/WGS84.
Importance
Used for topographic mapping, GPS positioning, and international data exchange; frequently tested alongside PPCS.
Term
PRS92 (Philippine Reference System of 1992)
Definition
The current national geodetic datum of the Philippines, defined by a network of GPS-observed control points aligned with WGS84. Replaced the old Luzon Datum (Clarke 1866 spheroid).
Importance
All modern Philippine surveys and coordinates are referenced to PRS92. Board exams test the distinction from the old Luzon Datum.
Term
False Easting / False Northing
Definition
Constants added to grid coordinates to keep all values positive within a zone. For UTM and PPCS: false easting = 500,000 m (so the central meridian has E = 500,000 m).
Importance
Prevents negative coordinate values; required knowledge for coordinate computation problems.
Section Title
4. Philippine Coordinate Systems: PPCS and UTM
Common Mistakes
- Mixing up UTM k₀ (0.9996) and PPCS k₀ (0.99995) — always identify the system first.
- Saying the Philippines is in UTM Zone 51 only — it spans both Zone 51N (117°E–123°E) and Zone 52N (123°E–129°E).
- Confusing PRS92 with WGS84 — they are practically identical for most survey purposes, but PRS92 is the official Philippine datum (not WGS84 itself).
- Forgetting that PPCS has 5 zones (I–V), not 3 or 6.
- Citing the old Luzon Datum (Clarke 1866) for current surveys — it was superseded by PRS92.
Exam Tips
- Board exam question pattern: 'A cartographer needs to show areas correctly → equal-area.' 'A navigator needs constant bearing → conformal (Mercator).' 'Great-circle route planning → Gnomonic.'
- Mercator: straight lines = rhumb lines (constant bearing), NOT great circles.
- Gnomonic: straight lines = great circles.
- Lambert Conformal Conic: used in Philippine aeronautical charts — conformal, suitable for mid-latitude countries.
- When in doubt for Philippine surveys: the answer is Transverse Mercator (PPCS or UTM).
Key Points
- The selection of a projection depends primarily on: (1) the purpose of the map, (2) the geographic extent/location, and (3) the property that must be preserved.
- SURVEYING and ENGINEERING: Use conformal (Transverse Mercator/UTM/PPCS) — angles and bearings transfer correctly to the grid.
- LAND-AREA STATISTICS / THEMATIC MAPS: Use equal-area (Albers, Sinusoidal) — province/region areas can be compared.
- NAVIGATION (sea/air): Use conformal (Mercator for sea; Lambert Conformal Conic for aviation) — constant bearing = straight line on Mercator.
- GREAT-CIRCLE ROUTES: Use Gnomonic — great circles appear as straight lines.
- DISTANCE FROM A POINT: Use Azimuthal Equidistant — distances from the center are true.
- WORLD/HEMISPHERE OVERVIEW: Use compromise projections (Robinson, Winkel Tripel) for visual balance.
- Small areas (e.g., a city, a barangay): Almost any projection works — distortion is negligible over small extents.
- Large areas (national, continental): Choice of projection critically affects area, shape, and distance accuracy.
Definitions
Term
Rhumb Line (Loxodrome)
Definition
A line that crosses all meridians at the same angle. On a Mercator projection, a rhumb line appears as a straight line — the basis for traditional sea navigation with Mercator charts.
Importance
Explains why Mercator is used for marine navigation despite its severe area distortion at high latitudes.
Term
Great Circle
Definition
The shortest path between two points on a sphere, lying on a plane that passes through the Earth's center. On a Gnomonic projection, great circles appear as straight lines.
Importance
Aviation and long-distance navigation use great-circle routes for fuel efficiency; the Gnomonic projection facilitates planning.
Term
Compromise Projection
Definition
A projection that is neither perfectly conformal nor perfectly equal-area, but attempts to minimize all distortions for aesthetic and general-purpose world maps.
Importance
Examples: Robinson (used by National Geographic for decades), Winkel Tripel. Not suitable for precise measurement.
Section Title
5. Choosing the Right Projection
Common Mistakes
- Selecting Mercator for a Philippine land-area statistics map — Mercator is conformal (angle-preserving), NOT equal-area.
- Using UTM/PPCS (conformal) to compute areas directly from grid coordinates without applying correction factors — conformal projections distort area, so k² corrections are needed for precise area computations.
- Claiming Mercator shows shortest routes — it shows constant-bearing (rhumb) routes, not great-circle (shortest) routes.
Connections
- Chapter 5 (PPCS/UTM Coordinate System): Direct extension of map projection theory — the specific Transverse Mercator parameters for Philippines grid computations (false easting, false northing, k₀, zone limits).
- Geodesy (Ellipsoidal Computations): Ellipsoidal (geodetic) distances are the input to the scale factor formula; understanding the ellipsoid is prerequisite to understanding projection distortion.
- Cadastral Surveying (PD 1529, RA 4374/RA 8560): All survey plans for land titles must use PPCS/UTM on PRS92 — projection knowledge is legally mandated in practice.
- Traverse Computation: Arc-to-chord correction (t – T correction) in TM projections corrects for the difference between the geodetic azimuth and the grid bearing — a direct application of TM projection theory.
- Photogrammetry (Aerial Triangulation): Exterior orientation and georeferencing of aerial photos require transformation to the national grid (PPCS/UTM), connecting projection theory to remote sensing.
- GIS and Digital Mapping: All GIS software requires specification of the coordinate reference system (CRS); incorrect projection selection causes spatial errors in analysis — core competency for geodetic engineers.
- RA 8560 and Professional Practice: The law mandating that Geodetic Engineers use NAMRIA-prescribed projection systems (PPCS/UTM/PRS92) in all official surveys.
Exam Strategy
For Map Projections questions in the PRC Geodetic Engineer board exam, use this approach: (1) IDENTIFY the property asked (area → equal-area, angle/shape/surveying → conformal, distance → equidistant). (2) IDENTIFY the system (UTM: k₀=0.9996, zones 51N/52N; PPCS: k₀=0.99995, zones I–V). (3) APPLY the formula: d_grid = k × d_ellipsoidal — never invert without reason. (4) For CSF problems: compute elevation factor = R/(R+H) first, then CSF = k × elevation factor, then d_grid = CSF × d_ground. (5) CHECK your answer: if k ≈ 1, grid ≈ ellipsoidal; if H is large, elevation factor < 1, so CSF < k. Common point allocation: 2–4 problems on scale factor/grid distance, 1–2 multiple choice on projection types, 1 question on Philippine legal references. Always write the formula, substitute values with units, box the final answer. Memorize: UTM k₀=0.9996, PPCS k₀=0.99995, PPCS zones I(117°E) to V(125°E), UTM zones 51N–52N, false easting=500,000 m for both systems.
Quick Review Questions
A line measured on the ellipsoid is 8,000.00 m. The projection scale factor at that location is k = 1.00012. What is the grid distance?
Apply d_grid = k × d_ellipsoidal. Since k > 1.000, the grid distance is slightly longer than the ellipsoidal distance — this occurs near the edge of a UTM/PPCS zone where the secant cylinder intersects the ellipsoid. The excess is 0.96 m over 8 km.
Why does UTM set the central meridian scale factor at k₀ = 0.9996 instead of 1.0000?
A secant Transverse Mercator cylinder intersects the ellipsoid at two lines inside the zone, rather than touching at the central meridian. This reduces the maximum distortion by roughly half compared to a tangent cylinder.
A national thematic map is needed to compare rice-production areas among all Philippine provinces. Which projection property is required, and name one suitable projection?
Area comparisons are only valid if the projection preserves area relationships. A conformal projection (e.g., Mercator, TM) distorts area systematically, making province areas appear incorrectly sized relative to each other.
What are the five PPCS zones covering the Philippines, and what are their central meridians?
PPCS uses a Transverse Mercator projection with k₀ = 0.99995 and five narrow (2°) zones to minimize distortion. Zone III (121°E) is used for surveys in Metro Manila and central Luzon. Zone selection is based on the longitude of the survey area.
What Philippine law requires geodetic engineers to use NAMRIA-prescribed coordinate systems for surveys submitted to the government?
RA 8560 defines the practice of geodetic engineering and mandates compliance with NAMRIA standards. PD 1529 ensures all Torrens title surveys use official coordinate systems. CA 141 governs classification and disposition of public lands, requiring NAMRIA-compliant surveys.
A Transverse Mercator conformal projection is used for Philippine surveys. If a traverse angle is measured in the field as 95°30'20'', will this angle be the same on the grid? Why?
The defining property of a conformal projection is preservation of local angles (shape). This is why PPCS and UTM are chosen for surveying: bearings, angles, and shapes of small figures are accurately represented on the grid. For long lines, an arc-to-chord correction (t – T correction) is applied.
Compute the combined scale factor (CSF) for a survey in Baguio City where k = 0.99997 and the mean elevation is H = 1,500 m. Use R = 6,371,000 m.
At 1,500 m elevation, the elevation factor reduces distances by about 1:4,247. Combined with k ≈ 1, the CSF ≈ 0.999734, meaning grid distances are about 26.6 cm shorter per 1,000 m of ground distance. In precision surveys, this must be applied.
On which projection does a constant compass bearing (rhumb line) appear as a straight line, and which property does this projection preserve?
Mercator's conformal property means angles between the line and meridians are correctly represented everywhere on the map. A rhumb line maintains a constant angle with meridians, so it projects as a straight line. Great circles, however, appear as curved lines on Mercator (except the equator and meridians).
Which UTM zones cover the Philippines, and what is the approximate longitude range of each?
UTM zones are 6° wide. Zone 51 spans 114°–120°E, Zone 52 spans 120°–126°E. Since the Philippines extends from about 116°E (Palawan) to 127°E (eastern Mindanao), it straddles both zones. Zone 51N covers Palawan and most of Mindanao's western portion; Zone 52N covers most of Luzon, Visayas, and eastern Mindanao.
What is the key difference between the normal Mercator projection and the Transverse Mercator projection?
This rotation makes Transverse Mercator ideal for regions with greater N-S extent than E-W extent, like the Philippines. Normal Mercator is best for equatorial regions and nautical charts; Transverse Mercator is the basis for UTM and PPCS.
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Philippine Plane Coordinate System and UTM
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