📚 Further Maths 5D3: The Earth | 进阶数学 5D3：地球
This module explores how advanced mathematics models our planet, focusing on spherical geometry, great-circle distances, time zones, and navigation bearings. It combines trigonometry, calculus, and coordinate geometry to solve real-world problems related to the Earth’s surface.
本模块探讨高等数学如何为地球建模，重点关注球面几何、大圆距离、时区以及导航方位角。它结合三角学、微积分和坐标几何来解决与地球表面相关的实际问题。
1. Modelling the Earth | 地球的数学建模
In A-level Further Maths, the Earth is modelled as a perfect sphere with a mean radius R ≈ 6371 km.
在A-level进阶数学中，地球被模拟为一个半径为 R ≈ 6371 公里的完美球体。
This simplification allows us to apply spherical trigonometry to find distances, angles, and speeds.
这种简化使我们能够应用球面三角学来计算距离、角度和速度。
Although the Earth is an oblate spheroid, the spherical model gives sufficient accuracy for typical navigation and time-zone problems.
尽管地球是一个扁球体，但球面模型对于典型的导航和时区问题可以给出足够的精度。
2. Latitude and Longitude | 纬度与经度
Latitude φ is the angle measured from the equator to a point, ranging from 0° at the equator to 90°N (North) or 90°S (South).
纬度 φ 是从赤道到某一点的角距离，范围从赤道的 0° 到北纬 90°N 或南纬 90°S。
Longitude λ is the angle measured from the Prime Meridian (0°) eastwards or westwards, ranging from 0° to 180°E or 180°W.
经度 λ 是从本初子午线（0°）向东或向西量测的角度，范围从 0° 到 180°E 或 180°W。
Coordinates are usually written as (λ, φ); for example, London is approximately (0°W, 51.5°N).
坐标通常写作 (λ, φ)；例如伦敦大约在 (0°W, 51.5°N)。
3. Great Circles and Small Circles | 大圆与小圆
A great circle is the intersection of the sphere with a plane that passes through the centre of the sphere, giving the largest possible circumference.
大圆是球体与一个通过球心的平面相交所产生的圆，它给出了最大的可能圆周。
Small circles are intersections with planes that do not pass through the centre, such as lines of latitude other than the equator.
小圆是与不通过球心的平面相交所产生的圆，例如除赤道以外的纬线。
The shortest path between two points on a sphere lies along the arc of a great circle. This is why great-circle routes are used in aviation and shipping.
球面上两点间的最短路径位于大圆弧上，这就是为什么航空和航海使用大圆航线的原因。
4. Angular Distance and the Spherical Cosine Rule | 角距离与球面余弦定理
The central angle d (in radians) subtended by two points A(λ₁, φ₁) and B(λ₂, φ₂) is found using the spherical cosine rule:
由两点 A(λ₁, φ₁) 和 B(λ₂, φ₂) 所对应的圆心角 d（以弧度计）可用球面余弦定理求得：
cos d = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos(Δλ)
where Δλ = λ₂ − λ₁ (taking east as positive). Both latitudes and longitudes must be in degrees, but the cosine function handles them correctly if converted to radians in calculations.
其中 Δλ = λ₂ − λ₁（东经取正）。纬度和经度均以度为单位，但在计算中若转换为弧度，余弦函数可正确处理。
If one point is in the southern hemisphere, its latitude is negative (e.g., 34°S → −34°). Similarly, west longitudes can be treated as negative.
若一点位于南半球，则其纬度为负（例如 34°S → −34°）。类似地，西经可视为负值。
This rule derives from the spherical law of cosines for a spherical triangle with vertices at the two points and the North Pole.
这条规则源于以两点和北极点为顶点的球面三角形的球面余弦定律。
5. Calculating Great-Circle Distance | 大圆距离的计算
The great-circle distance s between two points on the Earth’s surface is s = R × d, where d is in radians.
地球表面两点间的大圆距离 s 为 s = R × d，其中 d 以弧度为单位。
To convert d from degrees to radians, multiply by π/180. Thus, the distance in kilometres is:
要将 d 从度转换为弧度，乘以 π/180。因此，以公里为单位的距离为：
s = (πR/180) × d°
or more commonly, calculate d in radians first using the arccosine function, then multiply by R.
或更常用的是先用反余弦函数计算出弧度 d，然后乘以 R。
Example: Find the distance from London (51.5°N, 0°W) to New York (40.7°N, 74.0°W).
示例：求伦敦 (51.5°N, 0°W) 到纽约 (40.7°N, 74.0°W) 的距离。
Here φ₁=51.5°, φ₂=40.7°, Δλ = −74°. Compute cos d = sin51.5° sin40.7° + cos51.5° cos40.7° cos74°.
此处 φ₁=51.5°, φ₂=40.7°, Δλ = −74°。计算 cos d = sin51.5° sin40.7° + cos51.5° cos40.7° cos74°。
After calculation, d ≈ 0.935 rad or 53.6°, giving s ≈ 6371 × 0.935 ≈ 5957 km.
计算后得 d ≈ 0.935 rad 或 53.6°，因此 s ≈ 6371 × 0.935 ≈ 5957 km。
6. Time Zones and Longitude | 时区与经度
The Earth rotates 360° in approximately 24 hours, so 15° of longitude corresponds to 1 hour of time difference.
地球约 24 小时自转 360°，因此经度 15° 对应 1 小时的时差。
Local time advances by 1 hour for every 15° eastward, and falls back by 1 hour for every 15° westward.
每向东 15°，当地时间提前 1 小时；每向西 15°，当地时间推后 1 小时。
Problems often ask to find the local time at a given longitude when another location’s time is known.
问题常常要求已知某地的时间，求另一经度上的当地时间。
For example, if it is 12:00 noon at Greenwich (0°), a location at 45°E would have local time 12:00 + 3 hours = 15:00.
例如，如果格林尼治（0°）是中午 12:00，那么位于 45°E 的地方当地时间为 12:00 + 3 小时 = 15:00。
7. Speed from Earth’s Rotation | 地球自转产生的线速度
Each point on the Earth, except the poles, travels in a small circle of latitude with radius r = R cos φ.
除极点外，地球上的每一点都在一个半径为 r = R cos φ 的纬度小圆上运动。
The angular speed ω for Earth’s rotation is 2π/T, where T = 24 hours (or 23h 56min for sidereal day, but standard problems use 24 h).
地球自转角速度 ω = 2π/T，其中 T = 24 小时（恒星日为 23 时 56 分，但标准题目使用 24 小时）。
The linear speed v of a point due to Earth’s rotation is therefore v = ω r = ω R cos φ.
因此，某一点因地球自转而产生的线速度 v 为 v = ω r = ω R cos φ。
Substituting ω = 2π/(24×3600) rad/s and R = 6371 km, we can express v in km/h:
代入 ω = 2π/(24×3600) rad/s 和 R = 6371 km，可将 v 表示为 km/h：
v = 1670 cos φ km/h
At the equator (φ=0°), v ≈ 1670 km/h; at London (φ≈51.5°), v ≈ 1670 cos51.5° ≈ 1040 km/h.
在赤道（φ=0°），v ≈ 1670 km/h；在伦敦（φ≈51.5°），v ≈ 1670 cos51.5° ≈ 1040 km/h。
8. Bearings on a Great Circle | 大圆上的方位角
The initial bearing (or azimuth) α from point A to B along a great circle can be found from spherical trigonometry.
从点 A 到点 B 沿大圆的起始方位角 α 可通过球面三角学求出。
One common formula is:
一个常用公式是：
cos α = (sin φ₂ − sin φ₁ cos d) / (cos φ₁ sin d)
or using the sine rule: sin α = cos φ₂ sin(Δλ) / sin d
或者使用正弦定理：sin α = cos φ₂ sin(Δλ) / sin d
These bearings are measured clockwise from true north, ranging from 0° to 360°.
这些方位角从正北方向顺时针测量，范围 0° 至 360°。
It is essential to use the atan2 function (or quadrant checks) to determine the correct bearing from the sine and cosine values.
必须使用 atan2 函数（或象限检查）根据正弦和余弦值确定正确的方位角。
9. Practical Applications | 实际应用
Great-circle calculations are crucial for air traffic and ship navigation to minimise fuel consumption and time.
大圆计算对于空中交通和船舶导航至关重要，可以最大限度地减少燃料消耗和时间。
Satellite ground-track projections and GPS systems rely on these spherical models, often refined by ellipsoidal corrections.
卫星地面轨迹投影和 GPS 系统依赖这些球面模型，并常通过椭球校正加以精化。
In examination, questions may combine time-zone differences with positions on the Earth to test integrated problem-solving skills.
在考试中，题目可能会将时区差异与地球上的位置相结合，以考察综合解决问题的能力。
10. Summary of Key Formulae | 关键公式总结
Here is a quick reference for the essential relationships used in this module:
以下是本模块中用到的基本关系速查表：
| Quantity | Formula |
| Angular distance d | cos d = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos(Δλ) |
| Great-circle distance s | s = R × d (d in rad) |
| Longitude–time relation | 15° = 1 hour |
| Rotational speed v | v = 1670 cos φ km/h |
| Initial bearing α | 更多咨询请联系16621398022（同微信）
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