There's a problem with using GeoPath[{domeOfTheRock, versaillesCross, oakIsland},"Geodesic"]} to verify if a path from the Dome of the Rock to the Versailles Cross is connected to Oak Island. The path created by GeoPath represents a path formed by joining paths between consecutive locations, so that each segment connects consecutive locations with a geodesic path. This means we have a path connecting the Dome to Versailles, and a second path connecting Versailles to Oak Island, so the Versailles to Oak Island segment will always connect to the island!
Update:
One way to illustrate if the path from the Dome to Versailles connects to the island is to extend the path to the island. Jagar’s comment suggests a second method using a plane through the Dome and Versailles.
Here’s my original method using the true shape of the Earth, and extending the path. I’ve added a demonstration using a plane that assumes the Earth is a perfect sphere. This plane-and-sphere method shows a solution without correcting for the Earth’s true shape.
Extend path using geographic coordinates
To check if a path from the Dome to Versailles really does connect to Oak Island, find the direction from the Dome to Versailles, and then follow the path past Versailles to see if the path connects to the island.
We need the direction from the Dome to Versailles and a rough estimate of the distance from the Dome to the island. The direction is -45.9623°, and the distance is about 5,200 miles, which we get from GeoDistance[domeOfTheRock, oakIsland].
If we compare the direction from the Dome to Versailles (-45.9623°) to the direction from the Dome to the island (-46.9061°), which we get from GeoDirection[domeOfTheRock,oakIsland], we see that the path may overshoot the island.
Let’s check the path from the Dome for 6,000 miles starting with direction a... thanks to Carl’s test.
versaillesCross = GeoPosition[{48.81008221499617, 2.100137383293789}];
oakIsland = GeoPosition[{44.5167, -64.2992}];
domeOfTheRock = GeoPosition[{31.778063322333196, 35.23541700515525}];
a = GeoDirection[domeOfTheRock, versaillesCross];
Table[GeoGraphics[{Red,
GeoPath[{domeOfTheRock, Quantity[6000,"Miles"], a},"Geodesic"],
Blue, PointSize[Large],
Point /@ {versaillesCross, oakIsland, domeOfTheRock}},
GeoRange -> p,
GeoRangePadding ->
Quantity[0.01, "AngularDegrees"]], {p, {oakIsland, versaillesCross,
domeOfTheRock}}]
However close the path appears on the world map, the path misses the island! How far is the path from the island? This map shows the path and the GeoPosition of the island.
GeoGraphics[{
Blue, PointSize[Large], Point[oakIsland],
Red, Thick, GeoPath[{domeOfTheRock, Quantity[6000, "Miles"], a},"Geodesic"]},
GeoRange -> Quantity[250,"Miles"],
GeoCenter -> oakIsland, GeoScaleBar->"Imperial"]
We can get a close estimate of the distance from the path to the island by splitting the path into points and finding the nearest point on the path to the island.
pathPoints = GeoDestination[domeOfTheRock,
GeoDisplacement[
{Range[Quantity[5000, "Miles"], Quantity[5200, "Miles"]], a}]];
geoPosition = First@Nearest[pathPoints, oakIsland];(*nearest to island*)
GeoDistance[oakIsland, geoPosition]
62.8529 miles
Another map shows the distance.
GeoGraphics[{
Blue, PointSize[Large], Point[oakIsland], Point[geoPosition],
Blue, Thick, GeoPath[{geoPosition, oakIsland}, "Geodesic"],
Red, Thick,
GeoPath[{domeOfTheRock, Quantity[6000, "Miles"], a}, "Geodesic"]},
GeoRange -> Quantity[100, "Miles"],
GeoCenter -> oakIsland,
GeoScaleBar -> "Imperial"]
Note:
Mathematica uses a model for the Earth’s surface that is not a sphere. If we assume a sphere and use spherical trigonometry, the result would be different. Someone doing a paper-and-pencil solution might not have corrected for the true shape of the Earth, and maybe the path would pass closer to Oak Island.
Illustrate a plane and the Earth as a sphere
A paper-and-pencil solution would probably not correct for true shape of the Earth. I’ll use the same coordinates, but assume the Earth is a sphere.
We need a plane through the center of the Earth that includes the Dome and Versailles, a sphere with grid lines for orientation, and the three points on the sphere's surface.
versaillesCross = GeoPosition[{48.81008221499617, 2.100137383293789}];
oakIsland = GeoPosition[{44.5167, -64.2992}];
domeOfTheRock = GeoPosition[{31.778063322333196, 35.23541700515525}];
earthRadius =
QuantityMagnitude[
Entity["Planet", "Earth"][EntityProperty["Planet", "Radius"]]];
(*plane through the Dome, Versailles, and the center of the Earth*)
pCenter = {0., 0., 0.};
domeVersaillesPlane = Graphics3D[{
Blue, Opacity[.5], InfinitePlane[{p1, p2, pCenter}]},
PlotRange -> 2 earthRadius];
(*sphere with grid lines*)
sphereGrid = SphericalPlot3D[
earthRadius, {θ, 0, Pi}, {ϕ, 0, 2 Pi},
PlotRange -> 1.2 earthRadius,
Axes -> False, Boxed -> False];
(*points on the surface of the sphere*)
{p1, p2, p3} = FromSphericalCoordinates[
{earthRadius, π/2 - #1, #2}] & @@@
Normal@LatitudeLongitude[{domeOfTheRock, versaillesCross, oakIsland}];
This view shows the Dome and Versailles at the right and the island near the middle, with the plane that connects the Dome and Versailles.
Show[
sphereGrid,
pointsOnSphere,
domeVersaillesPlane
]
A close-up view of the island shows the plane overshoots the island by a small amount.
Show[
sphereGrid,
pointsOnSphere,
domeVersaillesPlane,
ViewAngle -> π/50
]
This next view shows the red plane (connecting the Dome, Versailles and Oak Island), does not match the blue plane through the Dome, Versailles and Earth's center.
Show[
sphereGrid,
pointsOnSphere,
Graphics3D[{
Red, Opacity[.8], InfinitePlane[{p1, p2, p3}],
Blue, Opacity[1], InfinitePlane[{p1, p2, pCenter}]}]
]
In each of these views, it appears that the distance between the plane and the island is small. How close is the plane to the island?
UnitConvert[
Quantity[RegionDistance[InfinitePlane[{p1, p2, pCenter}], p3],
"Meters"], "Miles"]
0.0425934 miles
So assuming the Earth is a sphere, and that the geographic locations are points on the sphere, the error is much smaller than the result we get when we use the geodesic path on the Earth's true surface.
Summary
By extending the geodesic path from the Dome to Versailles, the path misses the island by about 60 miles on the Earth's true surface. However, assuming the Earth is a sphere, the error is negligible. The plane-and-sphere method illustrates a paper-and-pencil solution without correction for the Earth's true shape.
So the we can assume historians in the The Curse of Oak Island episode also assumed a spherical Earth, and with this approximation, they found their presumed location of the Knights Templar treasure.
