15.1 Introduction to spherical geometry

Spherical Geometry Fundamentals

Spherical geometry studies what happens when you do geometry on the surface of a sphere instead of a flat plane. Many of the rules you've learned in Euclidean geometry break down on curved surfaces, and understanding how they break down is the core of this topic.

This matters beyond the classroom: navigation, aviation, and satellite communication all rely on spherical geometry because the Earth is (approximately) a sphere. The shortest flight path between two cities, for example, isn't a straight line on a flat map; it's an arc along a great circle.

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Properties of Spherical Geometry

On a flat plane, "lines" are straight and extend infinitely. On a sphere, the equivalent of a line is a great circle: the largest circle you can draw on a sphere's surface, dividing it into two equal hemispheres. The equator and all lines of longitude are great circles. A small circle, like a line of latitude other than the equator, does not count as a "line" in spherical geometry.

Key properties to know:

• Angles on a sphere are measured between the tangent lines to two great circles at their point of intersection.
• The sum of angles in a spherical triangle always exceeds 180°. This is a direct result of the surface curving beneath the triangle. The more area the triangle covers, the larger the angle sum.
• No parallel lines exist. Any two great circles will always intersect at exactly two antipodal points (think of how all lines of longitude meet at the North and South Poles). This means Euclid's Parallel Postulate simply doesn't hold on a sphere.

Spherical vs. Euclidean Geometry

The differences between these two geometries come down to curvature. Euclidean geometry assumes a flat surface with zero curvature; spherical geometry works on a surface with constant positive curvature.

A practical example: on a flat map, the shortest path from New York to Tokyo looks like it should go straight across the Pacific. But the actual shortest flight path arcs up near Alaska, because that route follows a great circle. The flat map distorts the geometry, making the true shortest path appear curved.

Spherical Geometry Applications

Great Circles and Spherical Triangles

A great circle passes through the center of the sphere, so its radius equals the sphere's radius. Any circle on the sphere that doesn't pass through the center is a small circle and is not the spherical equivalent of a line.

Two distinct points on a sphere determine a unique great circle unless those points are antipodal (directly opposite each other, like the North and South Poles). Antipodal points have infinitely many great circles passing through them.

A spherical triangle forms when three great circle arcs connect three non-collinear points on the sphere's surface. These triangles have some properties that feel strange at first:

• Each angle can range from just above 0° to just under 180°, so the angle sum falls strictly between 180° and 540°.
• The amount by which the angle sum exceeds 180° is called the spherical excess, and it's directly proportional to the triangle's area. A tiny spherical triangle has an angle sum barely above 180°; a triangle covering an entire hemisphere has an angle sum approaching 540°.

Airlines and shipping companies use great-circle routes to minimize fuel and travel time. If you've ever noticed that a flight path on a screen looks curved, that's the great-circle route displayed on a flat map projection.

Area and Circumference Formulas

Surface area of a sphere:

$A = 4\pi r^2$

where $r$ is the radius. This gives the total area of the entire sphere.

Circumference of a great circle:

$C = 2\pi r$

This is the same formula as the circumference of any circle whose radius equals the sphere's radius. For Earth ($r \approx 6{,}371$ km), the great-circle circumference is roughly 40,030 km.

Area of a spherical triangle:

$A = (\alpha + \beta + \gamma - \pi) \, r^2$

where $\alpha$, $\beta$, and $\gamma$ are the three angles measured in radians, and $r$ is the sphere's radius. The quantity $(\alpha + \beta + \gamma - \pi)$ is the spherical excess, often denoted $E$.

Here's how to use this formula step by step:

1. Convert all three angles to radians if they aren't already (multiply degrees by $\frac{\pi}{180}$).
2. Add the three angles: $\alpha + \beta + \gamma$.
3. Subtract $\pi$ (which is 180° in radians) to get the spherical excess.
4. Multiply the spherical excess by $r^2$.

Example: A spherical triangle on a sphere of radius 10 cm has angles of 90°, 90°, and 90°.

1. Each angle in radians: $\frac{\pi}{2}$

2. Sum: $\frac{\pi}{2} + \frac{\pi}{2} + \frac{\pi}{2} = \frac{3\pi}{2}$

3. Spherical excess: $\frac{3\pi}{2} - \pi = \frac{\pi}{2}$

4. Area: $\frac{\pi}{2} \times 10^2 = 50\pi \approx 157.1 \text{ cm}^2$

This triangle covers one-eighth of the sphere's total surface area, which you can verify: $\frac{50\pi}{4\pi(100)} = \frac{1}{8}$. You can visualize this triangle by thinking of three mutually perpendicular great circles carving the sphere into eight identical pieces.