Figure of the Earth and the Reference Ellipsoid - Memory Anchors
The human brain stores information as networks of associations — vivid images, stories, sounds, and emotions. By attaching abstract geodetic formulas to concrete mental hooks (stories, acronyms, analogies), you create multiple retrieval pathways. When exam pressure hits and your mind goes blank, these anchors act like mental search-and-rescue ropes, pulling the right formula or concept back to the surface. Use these anchors actively: close your eyes, replay the story, see the image, say the rhyme out loud. The more senses you engage, the deeper the memory trace. Every anchor here is engineered for one purpose — to make sure that on PRC exam day, you never forget the Figure of the Earth.
Memory anchors
The Earth is an oblate ellipsoid — flattened at the poles, bulging at the equator
Memory aid
Imagine a slightly squished balon (balloon) — round but pressed gently on the top and bottom. Now imagine a Pinoy mango that someone sat on lightly. The mango is wider at the middle (equator) and flatter at the stem ends (poles). That is exactly the Earth's shape — an oblate ellipsoid. 'Oblate' comes from ob- (against) + latus (side) meaning 'pressed against the side.' The Earth is that mango.
Why it works
Concrete, familiar Filipino object (mango) creates a vivid, tactile image that maps perfectly to the geometric concept of oblateness.
Example usage
Exam asks: 'What is the geometric shape of the Earth used in geodesy?' → See the mango → Answer: oblate ellipsoid of revolution, flattened at poles, bulging at equator.
Recall trigger
Think: 'Philippine mango pressed lightly at top and bottom'
Semi-major axis a (equatorial) and semi-minor axis b (polar) — a > b
Memory aid
Remember: 'A is for Around the equator, B is Below at the poles.' A (around) is the bigger radius; B (below) is the shorter one. Or use the alphabet: A comes before B, and A is always LARGER. Think 'A is Amazing and BIG; B is Beneath and smaller.'
Why it works
Alphabetic ordering + directional cues (Around/Below) create dual reinforcement for remembering which axis is larger.
Example usage
Given WGS84: a = 6,378,137 m and b = 6,356,752.31 m. Which is bigger? A is Amazing → a = 6,378,137 m → correct, it is the bigger one.
Recall trigger
A comes before B → A is larger → a is equatorial (bigger), b is polar (smaller)
Flattening formula: f = (a − b) / a
Memory aid
Recall the sentence: 'Flattening Finds how much the Big axis lost.' f = (a − b) / a. The numerator (a − b) is HOW MUCH was 'lost' from a to make b. The denominator a normalizes it. Think of f as the 'fraction of flatness' — how flat is it compared to the full equatorial radius? F = Flat fraction = (Axial difference) / (Anterior big axis).
Why it works
The word 'flattening' itself encodes the meaning of the formula — it IS the fraction of how much the ellipsoid has been flattened from a perfect sphere.
Example usage
Board problem: WGS84, a = 6,378,137 m, b = 6,356,752.31 m. f = (6,378,137 − 6,356,752.31) / 6,378,137 = 21,384.69 / 6,378,137 ≈ 1/298.257.
Recall trigger
Flattening = fraction of how much was flattened → (a − b) / a
WGS84 defining parameters: a = 6,378,137 m and 1/f = 298.257223563
Memory aid
Chunk the semi-major axis as: '6-37-8-137.' Read it as: 6 | 378 | 137. Story: 'Six jeepneys (6) carry 378 passengers to 137 barangays.' For 1/f = 298.257...: chunk as '298 point 257.' Say it as: 'Two-ninety-eight point two-fifty-seven.' Drill: '6,378,137 — sixty-three-seventy-eight — one-thirty-seven. And the flattening inverse is two-ninety-eight.'
Why it works
Chunking reduces working memory load from 10 individual digits to 3-4 meaningful chunks. The jeepney story adds an emotional Filipino anchor.
Example usage
Examiner asks: 'What is the semi-major axis of WGS84?' → Six jeepneys → 6,378,137 m. What is the inverse flattening? → Two-ninety-eight → 298.257223563.
Recall trigger
Six jeepneys → 6,378,137 m. Two-ninety-eight → 1/f = 298.257
First eccentricity squared: e² = (a² − b²) / a² = 2f − f²
Memory aid
Remember the phrase: 'Eccentricity is Easy — it's the big minus small over big, all squared.' e² = (a² − b²) / a². Or the alternate form: '2f minus f-squared' — think '2 Flat minus Flat-Flat.' Imagine paying 2 times the flat rate but getting a discount of flat-times-flat: 2f − f². For WGS84: e² ≈ 0.00669438. Memory hook: '669 — six-six-nine after the decimal zeros' → e² ≈ 0.00669438.
Why it works
The phrase '2 Flat minus Flat-Flat' directly mirrors the algebra 2f − f² and makes the formula self-narrating.
Example usage
Given f = 1/298.257, compute e²: e² = 2(1/298.257) − (1/298.257)² = 0.006694380 − 0.0000112 ≈ 0.00669438.
Recall trigger
e² → '2 Flat minus Flat-Flat' → 2f − f². WGS84 value: 'six-six-nine' → 0.00669438
Second eccentricity: e'² = (a² − b²) / b²
Memory aid
First eccentricity uses 'a-squared' in the denominator — it sees the world from the equator's perspective (the big boss). Second eccentricity uses 'b-squared' in the denominator — it sees the world from the pole's perspective (the small one looking up). Think: 'First e uses the big house (a); Second e' uses the small house (b) as base.' They both have the same numerator (a² − b²) — same difference, different base.
Why it works
House size analogy (big vs. small house as denominator base) gives a concrete image for remembering which squared term goes in the denominator.
Example usage
e'² = (a² − b²)/b². For WGS84: e'² = (6,378,137² − 6,356,752.31²) / 6,356,752.31² ≈ 0.006739497.
Recall trigger
First e: big house (a²) below. Second e': small house (b²) below.
Prime vertical radius of curvature N — applies East–West direction
Memory aid
N is for 'North-East-West No — actually East-West!' But more powerfully: N is the radius that holds up the NORMAL to the ellipsoid — and the Normal in the prime vertical plane runs East-West. Memory hook: 'N is Ninety degrees from the meridian' — East and West are 90° from the North-South meridian. So N governs the perpendicular (East-West) direction. Also: N ≥ M always, so 'N is Never smaller than M' — N is the 'Nicer bigger' one.
Why it works
The letter N linked to 'Normal' (which is geometrically correct — N is the radius of curvature in the prime vertical normal section) plus the 'Never smaller' trick gives two independent memory hooks.
Example usage
Exam asks: 'Which radius of curvature is used for the East-West direction?' → N is Ninety from meridian → N (prime vertical radius).
Recall trigger
N = Normal direction (East-West), Never smaller than M
Meridian radius of curvature M — applies North–South direction
Memory aid
M is for Meridian and for Moving North-South. Think of M as 'Me going to Maynila (Manila) from Mindanao' — you travel North-South along a meridian. M is always the SMALLER radius (harder to bend N-S because the ellipsoid is flatter there). And M has a more complex formula with (1-e²sin²φ)^(3/2) — the 3/2 power makes it more 'bent' and thus smaller. 'M is the Modest, smaller radius, going between Mindanao and Maynila.'
Why it works
Filipino geographic reference (Mindanao to Maynila = North-South travel) makes the direction immediately intuitive. 'Modest/smaller' encodes the M < N relationship.
Example usage
Exam asks: 'Which curvature radius governs North-South geodetic computations?' → Me going to Maynila = Meridian = M.
Recall trigger
M = Meridian = Mindanao to Maynila (N-S). M is Modest (smaller than N).
Formula for N: N = a / √(1 − e²sin²φ)
Memory aid
Walk through a house mentally. At the FRONT DOOR: you see 'a' (the big equatorial number, 6,378,137 — the address of the house). In the LIVING ROOM: you see a giant square root sign hanging from the ceiling like a chandelier. Under the chandelier: '1 minus e²sin²φ' written on the floor like a carpet. The fraction is the house itself: a (roof/numerator) over the square root (floor/denominator). Every time you think of N, walk through that house: door (a) → chandelier (√) → carpet (1 − e²sin²φ).
Why it works
Method of loci (memory palace) encodes the formula spatially. The house metaphor gives a physical structure to an abstract equation.
Example usage
φ = 45°, WGS84: N = 6,378,137 / √(1 − 0.00669438 × sin²45°) = 6,378,137 / √(0.99665281) ≈ 6,388,838.3 m.
Recall trigger
Walk into the house: front door = a, chandelier = √, carpet = 1 − e²sin²φ
Formula for M: M = a(1 − e²) / (1 − e²sin²φ)^(3/2)
Memory aid
Compare N and M side by side. N = a / W, where W = √(1−e²sin²φ). M = a(1−e²) / W³. The key differences: (1) M has an extra (1−e²) in the numerator — 'M has a Modifier up top'; (2) M uses W-cubed (power 3/2) instead of W-squared (power 1/2) — 'M is More powerful below (3/2 vs 1/2).' Rhyme: 'N is simple a-over-W; M modifies a and cubes the W.' The 3/2 power makes M smaller — more denominator, less result.
Why it works
Contrast learning (N vs M) is highly effective. The rhyme and structural comparison encode the differences between the two most-confused formulas in this chapter.
Example usage
At φ = 30°, WGS84: W = √(1 − 0.00669438 × sin²30°) = √(0.998326) = 0.999163. M = 6,378,137(1 − 0.00669438) / (0.999163)³ = 6,378,137 × 0.993306 / 0.997491 ≈ 6,351,376 m.
Recall trigger
M has Modifier (1−e²) on top AND cubed W below (3/2 power)
At the equator (φ = 0°): N = a and M = a(1−e²)
Memory aid
Story: Imagine you are standing exactly on the equator — the 'waistline' of the Earth-mango. At this special point, sin(0°) = 0, so the e²sin²φ term vanishes completely. The formula simplifies beautifully. N falls back to just 'a' — the equatorial radius itself — as if the Earth is briefly pretending to be a perfect sphere in the East-West direction. But M shrinks to a(1−e²) because the meridian curves MORE at the equator (sharper bend). Lesson: 'At the equator, N is at its minimum (= a) and M is at its minimum too, even smaller than N.'
Why it works
The story gives a physical intuition for why the formulas simplify at φ = 0°, preventing the common error of misapplying the general formula at special latitudes.
Example usage
Board shortcut: If φ = 0°, immediately write N = a = 6,378,137 m and M = a(1−e²) = 6,378,137 × 0.993306 ≈ 6,335,439 m without using the full formula.
Recall trigger
Equator: sin(0°) = 0 → N = a (minimum), M = a(1−e²) (also minimum, smaller than N)
At the poles (φ = 90°): M = N = a(1−e²)^(1/2) / (1−e²)^... = a(1−e²)/b² × b = b²/a
Memory aid
At the poles, M = N — the two radii of curvature become EQUAL. Visualize the two parallel rails of a railroad track (M and N) that are always separate, but at the very tip of the pole, the track converges to a single point. They merge. The common value at the poles = a²/b (a common result). Visual: two tracks → one point at the pole. Also memorize: at poles, curvature is MAXIMUM (smallest radius means tightest curve — the ellipsoid is most 'curved' at the pole tip).
Why it works
The railroad convergence metaphor is a powerful spatial image for why two normally different values become equal at the poles.
Example usage
At φ = 90°: M = N = a²/b = 6,378,137²/6,356,752.31 ≈ 6,399,594 m. Note this is LARGER than N at equator (6,378,137 m) — confirms N increases with latitude.
Recall trigger
Railroad tracks converging at the pole tip → M = N at φ = 90°
Mean radius of curvature: R = √(MN) — the Gaussian mean
Memory aid
R = √(MN) is the geometric mean of M and N — the same formula you use for the geometric mean of two numbers in basic math. Think of it this way: M and N are two friends of different heights. R is the 'fair average height' that represents both — not the arithmetic average (M+N)/2, but the geometric mean √(MN). This R is used to approximate the Earth locally as a sphere for small-area computations. 'R is the Representative radius — the Reasonable compromise between M and N.'
Why it works
Connecting Gaussian mean to the familiar concept of geometric mean from basic mathematics creates a bridge to prior knowledge, making the formula intuitive rather than arbitrary.
Example usage
At φ = 14° (Manila area), compute M and N separately using WGS84 formulas, then R = √(MN) ≈ the local sphere radius used in plane surveying approximations.
Recall trigger
R = geometric mean of M and N = √(MN). Two friends, fair average height.
Flattening is tiny: f ≈ 1/298 ≈ 0.00335, NOT 0.298 or 1/3
Memory aid
Story: A board exam taker (let's call him Kuya Rodrigo) mistakenly wrote f = 0.298 — confusing 1/f = 298 with f = 298. His entire answer fell apart. The Earth looks almost perfectly round to the naked eye — it is only about 1 part in 298 flatter than a sphere. That is less than 0.4% flatter. If you scaled the Earth to a basketball, the flattening would be invisible to your eye. Kuya Rodrigo's lesson: '1/f = 298 means f is ONE over 298, not 298 itself. Big number below means tiny fraction above.'
Why it works
A cautionary story activates emotional memory and directly addresses the most common board exam mistake. The basketball analogy gives physical intuition for the smallness of f.
Example usage
If a problem gives 1/f = 298.257, immediately compute f = 1/298.257 = 0.003352811 — never use 298 as f itself.
Recall trigger
Kuya Rodrigo's mistake → 1/f = 298 → f = 1/298 ≈ 0.00335 (SMALL!)
N ≥ M everywhere on the ellipsoid (prime vertical radius always ≥ meridian radius)
Memory aid
Rhyme: 'N is Never beaten by M; M is always the Modest gem. East-West N stands big and tall; North-South M is not as tall. At the equator the gap is wide; at the poles they stand side by side.' This rhyme encodes: (1) N ≥ M always, (2) N is East-West, M is North-South, (3) maximum difference at equator, (4) they converge at poles.
Why it works
Rhymes engage phonological memory, making the entire relationship memorable as a unit. The rhyme encodes four facts simultaneously.
Example usage
Any latitude on WGS84: N = 6,388,838 m at 45° vs M = 6,367,489 m at 45°. N > M confirmed. At poles: N = M ≈ 6,399,594 m.
Recall trigger
Sing the rhyme: 'N is Never beaten by M...'
e² uses a² in the denominator (first eccentricity); e'² uses b² in the denominator (second eccentricity)
Memory aid
Use the phrase: 'FIRST is Famous (uses a — the famous equatorial radius). SECOND is Shy (uses b — the less famous polar radius).' Or: 'First e: A-grade denominator. Second e': B-grade denominator.' The prime symbol on e' (like a graduation mark) signals it uses b (also starts with a 'b' sound: b-prime). Together: e² = (a²−b²)/a², e'² = (a²−b²)/b².
Why it works
Grading analogy (A-grade vs B-grade) with the letter correspondence (a→A, b→B) creates a logical mnemonic that is self-checking.
Example usage
WGS84: e² = (a²−b²)/a² ≈ 0.00669438. e'² = (a²−b²)/b² ≈ 0.00673966. Note e'² > e² because b < a means smaller denominator means bigger result.
Recall trigger
First e: A-grade (a²) below. Second e': B-grade (b²) below.
The relationship b = a(1 − f)
Memory aid
Think of 'a' as your original salary. Flattening 'f' is the percentage cut your boss takes from you. Your take-home pay is a(1 − f) = b. The Earth's polar radius b is the 'take-home pay' after flattening takes its cut from the full equatorial radius a. If f is small (like 1/298), then b is almost as large as a — just slightly less, like a tiny salary deduction.
Why it works
Financial analogy is universally relatable. Salary and deductions are concrete real-world concepts that map naturally to a, f, and b.
Example usage
WGS84: b = 6,378,137 × (1 − 1/298.257) = 6,378,137 × 0.996647 = 6,356,752.31 m.
Recall trigger
b = salary after deduction = a(1 − f)
Geodesy definition: science of Earth's size, shape, and precise point positions
Memory aid
Use the acronym SSP: 'Geodesy = Size, Shape, Positions (precise).' Or expand: 'Geo-DESY studies: Does Earth's Size yield Positions?' Alternatively: think of a GPS device — it gives you a POSITION. How does GPS work? It uses a model of Earth's SIZE and SHAPE. So: GPS → Size, Shape, Position → Geodesy. The word 'Geodesy' itself: Geo (Earth) + desia (division/measurement) → measuring/dividing the Earth.
Why it works
The GPS connection is immediately relevant to practicing geodetic engineers and makes the definition memorable through a tool they use daily.
Example usage
Exam: 'Define Geodesy.' → GPS mnemonic → 'The science that determines the Size and Shape of the Earth and the precise Positions of points on its surface.'
Recall trigger
GPS = needs Size + Shape + Position → Geodesy
Clarke 1866 ellipsoid — used by old Philippine surveys before PRS92/WGS84
Memory aid
Story: In the old days (1866), a British geodesist named Alexander Ross Clarke measured the Earth very carefully. His ellipsoid (a = 6,378,206.4 m, b = 6,356,583.8 m) was used by Philippine survey maps for over a century — it powered the old Luzon Datum of 1911. Think of Clarke 1866 as the 'old NAMRIA maps lolo/lola used' — the pre-digital Philippine cadastral surveys. When the Philippines shifted to PRS92 (based on GRS80/WGS84), Clarke was retired. 'Clarke is Classic; WGS84 is Current.'
Why it works
Philippine historical context (Luzon Datum, NAMRIA, PRS92 transition) makes this directly relevant and emotionally resonant for Filipino geodetic engineers.
Example usage
Clarke 1866: a = 6,378,206.4 m, b = 6,356,583.8 m. f = (a−b)/a = (21,622.6)/6,378,206.4 ≈ 1/295.0. Used in pre-PRS92 Philippine cadastral work.
Recall trigger
Clarke = Classic old Philippine surveys (Luzon Datum 1911). WGS84 = Current PRS92.
The denominator term W = √(1 − e²sin²φ) appears in both N and M formulas
Memory aid
Visualize W as a 'weight machine' at the gym. When latitude φ = 0° (equator), sin(0°) = 0, so W = √1 = 1 — the machine shows 'no weight added.' When φ = 90° (pole), sin(90°) = 1, W = √(1−e²) — maximum weight is added. The W value lies between √(1−e²) and 1. In the N formula, W is in the denominator once (W¹). In M, W is cubed (W³). Higher power of W below = smaller result = M < N. 'W is the Weight that reduces both N and M, and it Weights M more (W³) than N (W¹).'
Why it works
The gym weight machine metaphor makes the variable W concrete and shows intuitively why it appears in both formulas and why its higher power in M produces a smaller value.
Example usage
At φ = 14°: W = √(1 − 0.00669438 × sin²14°) = √(1 − 0.00669438 × 0.05843) = √(0.999609) = 0.999804. Use W to compute both N and M efficiently.
Recall trigger
W = weight machine at gym. W¹ for N, W³ for M → M gets heavier burden → M < N
Revision game
I am the equatorial waistline of the Earth. On WGS84, I am exactly 6,378,137 meters. What letter names me?
Answer
Semi-major axis a
Memory link
A2 — A is Amazing and BIG (equatorial radius)
I am a tiny fraction — about 1 part in 298. I measure how squashed the Earth is. Kuya Rodrigo once wrote me as 0.298 — and failed. What am I?
Answer
Flattening f (f ≈ 0.00335, not 0.298)
Memory link
A14 — Kuya Rodrigo's cautionary tale: 1/f = 298, so f is ONE over 298
I govern East-West curvature. My formula has 'a' on top divided by a square root. I am never smaller than my meridian cousin. Who am I?
Answer
Prime vertical radius of curvature N
Memory link
A7 — N is Never smaller, N is Ninety degrees from meridian (East-West)
I am the geometric mean of M and N. I approximate the ellipsoid as a local sphere. My formula is simple — just take the square root of two friends multiplied. What am I?
Answer
Mean radius of curvature R = √(MN)
Memory link
A13 — R is the Representative radius, geometric mean, two friends of different heights
At the equator, I equal exactly 'a' — nothing more. At the poles, I grow to my maximum. I am the BIGGER radius of curvature at every latitude. What am I?
Answer
Prime vertical radius N
Memory link
A11 — At equator: N = a (minimum), grows toward poles
I use b-squared in my denominator — the 'B-grade' base. My first cousin uses a-squared. We share the same numerator: (a²−b²). Who am I?
Answer
Second eccentricity e'²
Memory link
A16 — Second e': B-grade denominator (b²). First e²: A-grade denominator (a²)
I was the reference ellipsoid for Philippine surveys before PRS92 arrived. My semi-major axis is 6,378,206.4 m — a bit larger than WGS84's. Lola's cadastral map used me. Who am I?
Answer
Clarke 1866 ellipsoid (Luzon Datum 1911)
Memory link
A19 — Clarke is Classic (old lola's maps), WGS84 is Current
I appear inside every radius of curvature formula. I equal √(1 − e²sin²φ). At the equator I equal 1; at the poles I equal √(1−e²). I am the gym machine that adds weight to the denominator. What letter am I called by?
Answer
W — the denominator factor W = √(1 − e²sin²φ)
Memory link
A20 — W is the Weight machine at the gym
Formula mnemonics
f = (a − b) / a
Mnemonic
Flattening is the Fraction of the gap (a minus b) relative to the Full equatorial radius (a). 'Flat Fraction = Gap over Grand radius.'
When to use
When you need to find flattening from the two axes, or when deriving b from a and f (rearranged as b = a(1−f)).
What each part means
f = flattening (dimensionless, ≈ 1/298 for WGS84); a = semi-major axis (equatorial radius); b = semi-minor axis (polar radius); (a − b) = the actual polar shortfall in meters
e² = 2f − f² = (a² − b²) / a²
Mnemonic
'2 Flat minus Flat-Flat' for the f-form. 'Big-squared minus Small-squared over Big-squared' for the axes form. WGS84 value: remember '669' → e² ≈ 0.00669438.
When to use
Use e² = 2f − f² when only f is given. Use e² = (a²−b²)/a² when both axes are given. e² appears inside every radius of curvature formula.
What each part means
e² = first eccentricity squared (dimensionless); the two forms are algebraically equivalent; e measures how much the ellipse deviates from a circle (e = 0 is a perfect sphere)
e'² = (a² − b²) / b²
Mnemonic
Second eccentricity = same numerator as e², but B-grade denominator (b²) instead of A-grade (a²). 'e-prime uses the Prime (polar) base.' e'² > e² always because b < a.
When to use
When problems specifically ask for second eccentricity, or in advanced geodetic computations involving series expansions for arc length.
What each part means
e'² = second eccentricity squared; b² in denominator makes this larger than e²; used in some geodetic series expansions and in certain conformal projection formulas
N = a / √(1 − e²sin²φ)
Mnemonic
'a over root of (1 minus eccentricity-latitude blend).' The HOUSE formula: a (roof) over √(1−e²sin²φ) (foundation). N governs East-West. At φ=0°: N=a (simplest value).
When to use
For any computation involving East-West distances, parallel lengths, geographic coordinate conversions, and UTM/PPCS projection scale factors.
What each part means
N = prime vertical radius of curvature (m); a = semi-major axis; e² = first eccentricity squared; φ = geodetic latitude; the denominator W = √(1−e²sin²φ) varies from 1 (equator) to √(1−e²) (pole)
M = a(1 − e²) / (1 − e²sin²φ)^(3/2)
Mnemonic
'M has a Modifier (1−e²) on top, and cubed W below (power 3/2).' Compare to N: M adds (1−e²) numerator and raises denominator to 3/2 instead of 1/2. M governs North-South. M < N always.
When to use
For North-South distance computations, arc-to-chord corrections in N-S direction, meridian arc length calculations, and geographic coordinate to metric conversions.
What each part means
M = meridian radius of curvature (m); a(1−e²) = semi-latus rectum of the meridian ellipse; denominator (1−e²sin²φ)^(3/2) = W³ where W = √(1−e²sin²φ); M is smallest at equator and largest at poles
R = √(MN)
Mnemonic
'R is the Root of M times N' — geometric mean. 'R is the Representative sphere radius.' Used locally to approximate the ellipsoid as a sphere for small-area plane surveying.
When to use
For converting between geodetic and plane coordinates in small areas, for computing corrections in local plane surveys, and as the 'Earth radius' in first-order approximations.
What each part means
R = mean radius of curvature (m), also called Gaussian mean radius; M = meridian radius; N = prime vertical radius; R is the radius of the sphere that best fits the ellipsoid at a given latitude
b = a(1 − f)
Mnemonic
'b is the pay after deduction': start with full salary a, deduct fraction f, get take-home b. Rearranging: f = (a−b)/a and a = b/(1−f).
When to use
When you have a and f (like in WGS84 definition) and need b. Or to check consistency between given a, b, and f values.
What each part means
b = semi-minor axis (polar radius, m); a = semi-major axis (equatorial radius, m); f = flattening (dimensionless fraction); (1−f) is the fraction of a that b represents
Quick recall chains
WGS84 Key Numbers Chain
Recall test
Without looking: What is the WGS84 semi-major axis? Inverse flattening? Semi-minor axis? e²? Recite the jeepney story from start to finish.
Memory chain
Story chain: 'Six jeepneys (6) with 378 seats (378) carry 137 barangay captains (137) — that is a = 6,378,137 m. Their meeting is at Room 298 (1/f = 298.257). The meeting discusses a 0.335% budget cut (f ≈ 0.00335). After the cut, they have 6,356 pesos and 752 centavos (b ≈ 6,356,752 m). The eccentricity of their argument: 669 decibels (e² ≈ 0.00669).'
Items to remember
- a = 6,378,137 m
- 1/f = 298.257223563
- f ≈ 0.00335281
- b ≈ 6,356,752.31 m
- e² ≈ 0.00669438
Ellipsoid Parameter Derivation Sequence
Recall test
Starting only from a and f, can you write all 8 derived quantities in order without notes? Time yourself — should take under 3 minutes in an exam setting.
Memory chain
The ROAD to curvature: 'A Fine Base (a, f) → Build the B-road (b) → E-squared on the road (e²) → E-Prime diverges (e'²) → W is the Weight on the road (W) → N is the North-East overpass (N) → M is the Meridian underpass (M) → R is the Roundabout (R = √MN).'
Items to remember
- Start with a and f (defining parameters)
- Compute b = a(1 − f)
- Compute e² = 2f − f²
- Compute e'² = e²/(1 − e²)
- Compute W = √(1 − e²sin²φ)
- Compute N = a/W
- Compute M = a(1 − e²)/W³
- Compute R = √(MN)
Radii of Curvature at Special Latitudes
Recall test
Fill in the blanks: At φ = 0°, N = ___ and M = ___. At φ = 90°, N = M = ___. Which is always larger, N or M?
Memory chain
'Zero at equator: N is just a, M is a with modifier. Forty-five is middle ground: N = 6.388M and M = 6.367M (remember 8 and 7 at the end — N ends in 8, M ends in 7). Ninety at poles: the rails merge, N = M = a²/b.'
Items to remember
- At φ = 0° (equator): N = a, M = a(1−e²), M < N
- At φ = 45°: N ≈ 6,388,838 m, M ≈ 6,367,489 m for WGS84
- At φ = 90° (poles): N = M = a(1−e²)^(1/2) / (1−e²)^(1/2) = a²/b
Common Ellipsoids in Philippine Geodesy
Recall test
Which ellipsoid is the basis of PRS92? Which is used by GPS/GNSS? What is common between GRS80 and WGS84? What was the Luzon Datum 1911 ellipsoid?
Memory chain
'Philippine ellipsoid history: CLASSIC Clarke (old lola's maps, 1866) → GRS80 (basis of PRS92, same a as WGS84 but slightly different 1/f) → WGS84 (GPS era, current). Same a = 6,378,137 for both GRS80 and WGS84 — the equator never changed, only the fine details of flattening differ by the 7th decimal place.'
Items to remember
- Clarke 1866: a = 6,378,206.4 m — used in Luzon Datum 1911
- GRS80: a = 6,378,137 m, 1/f = 298.257222101 — basis of PRS92
- WGS84: a = 6,378,137 m, 1/f = 298.257223563 — basis of GNSS
- Clarke 1880: used in some old Mindanao surveys
Eccentricity Forms and Relationships
Recall test
Write the formula for e² in two ways. Write the formula for e'². What is the relationship between e'² and e²? Which is larger?
Memory chain
'Eccentricity family: FIRST born uses the A (a²) surname. SECOND born uses the B (b²) surname. They have the same numerator (a²−b²) — same parents, different last names. The relationship: e-prime-squared = e-squared divided by (1 minus e-squared) — second born is always slightly bigger because the family name B is smaller, making the fraction bigger.'
Items to remember
- e² = (a² − b²) / a² [first, A-grade denominator]
- e'² = (a² − b²) / b² [second, B-grade denominator]
- e² = 2f − f² [from flattening]
- e'² = e² / (1 − e²) [relationship between first and second]
- e'² > e² always
Diagrams
Figure of the Earth: Complete Concept Map
A comprehensive mind map showing all major concepts of the chapter: Earth's shape, ellipsoid defining parameters, eccentricity types, radii of curvature (N, M, R) with formulas and special values, and Philippine geodetic reference context. Use this map to orient yourself before diving into individual topics.
Step-by-Step: Computing Radii of Curvature N and M
A step-by-step flowchart for computing N (prime vertical radius) and M (meridian radius) from the defining ellipsoid parameters a and f. Special branches handle the common special cases of equator (φ=0°) and poles (φ=90°). Follow this flow on every board exam problem involving radii of curvature to avoid missing steps.
Decision Tree: Which Radius of Curvature to Use?
A decision flowchart for selecting the correct radius of curvature in different geodetic problems. The key decision is direction: East-West problems use N (prime vertical), North-South problems use M (meridian), and omnidirectional or local sphere approximations use R = √(MN). This prevents the most common board exam error of swapping N and M.
Timeline of Reference Ellipsoids Used in Philippine Geodesy
A timeline showing the historical progression of reference ellipsoids used in Philippine geodesy, from the old Clarke 1866 (Luzon Datum 1911) through to the modern PRS92/WGS84 era. Understanding this history is important for RA 8560 (Geodetic Engineering Act) context and for interpreting old vs new survey documents.
Size Comparison: Key WGS84 Ellipsoid Parameters (Scaled Proportions)
Approximate distribution of board exam questions on this chapter topic. Radii of curvature (N and M) dominate at 35% of questions, followed by basic ellipsoid parameters (25%) and eccentricity (20%). Prioritize mastering N, M, and the underlying parameters for maximum board exam score.