MASTERY 0 / 2
PREDICT BEFORE MEASURING
Which surface route is shorter?
Two points share a latitude. The cyan parallel looks straight on a flat map; the yellow great circle is straight according to the sphere's metric.
Two points share a latitude. The cyan parallel looks straight on a flat map; the yellow great circle is straight according to the sphere's metric.
This lab compares two routes on a unit sphere between points at the same latitude. Hold latitude at forty degrees north. Start with twenty degrees of longitude separation. The latitude route length is cosine of latitude times the longitude gap. The great circle central angle is inverse cosine of sine squared latitude plus cosine squared latitude times cosine of the gap. Here the lengths are about fifteen point two nine degrees and fifteen point three two degrees. They are close, but they are not equal. Yellow tracks the great circle; cyan tracks the latitude route. Now increase only the longitude gap from twenty to one hundred twenty degrees. The upper marker follows that actual input. Both lower markers follow their respective spherical formulas on shared axes. They move upward as the endpoints become farther apart. The curves use angular length, not the flat screen length of the globe animation. This comparison changes endpoint separation; it is not a race between two moving travelers. Hold longitude gap at one hundred twenty degrees. The great circle is about eighty three point one two degrees long, while the latitude route is about ninety one point nine three degrees. The difference is about eight point eight zero degrees. The markers stay fixed while we read these numbers. The shorter great circle comes from a plane through the sphere center. Multiply angular length in radians by sphere radius to convert either route to physical distance. The display is a spherical teaching model, not an ellipsoidal Earth flight planner. Keep the longitude gap at one hundred twenty degrees. Begin again at forty degrees latitude. The yellow great circle length is about eighty three point one two angular degrees, and the cyan latitude route length is about ninety one point nine three degrees. The horizontal coordinate now represents latitude rather than longitude gap. Both output curves still use the same angular length scale. Away from the equator, a latitude circle turns sideways within the spherical surface; the great circle does not. Decrease only latitude from forty degrees toward zero, the equator. Watch the input marker move to the left. The two lower markers follow the exact length formulas as the latitude changes. Both route lengths increase in this comparison, and their difference closes. These markers represent new endpoint locations and calculated lengths at each latitude; they do not represent a spacecraft trajectory or a flat map metric. Hold latitude at zero. Both lengths are now one hundred twenty angular degrees. At the equator, the latitude circle itself is a great circle, so the two formulas agree. The plotted markers coincide and remain there during this explanation. Locally straight on a sphere means there is no sideways bending within the surface. Three dimensional curvature normal to the surface can still exist. Equality here is a geometric special case, not evidence that every latitude route is a geodesic. For a transfer comparison, fix latitude at sixty degrees north. Start with one hundred twenty degrees of longitude gap. The great circle is about fifty one point three two angular degrees long. The latitude route is sixty degrees, because cosine of sixty degrees is one half. The input coordinate once again means longitude gap. We compare both route formulas on the same vertical scale, while keeping latitude fixed throughout this scene. Increase only the longitude gap from one hundred twenty to one hundred fifty degrees. The cyan latitude route grows in direct proportion to that gap. The yellow great circle follows the inverse cosine metric and grows more slowly here. Both animated markers are recomputed from their actual changing input. Camera perspective and the screen length of a rendered line do not enter either formula. Hold longitude gap at one hundred fifty degrees. The transfer great circle length is about fifty seven point seven six degrees, and the latitude route is seventy five degrees. The difference is about seventeen point two four degrees. The source Transfer button chooses precisely this sixty degree latitude and one hundred fifty degree gap case. Commit to the great circle and reveal the result to finish the exercise. Download learning evidence exports the actual worked and transfer receipt. Reset clears its local progress. Those original controls and the globe remain available above the video.