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5.4.5 The Geodesic
Definition of the Geodesic
The geodesic is defined as a curve on which the adjacent two arc elements of each
point lie in the same normal section plane of this point. We can use Fig. 5.21 to
make this definition clear. Let AB be a geodesic on the surface, P be an arbitrary
point on the geodesic, dS 1 and dS 2 be the adjacent arc elements of point P, and PK
be the normal to the surface at point P.dS 1 and dS 2 are both arc elements, i.e., the
points P 1 and P 2 are infinitely close to point P, so we can use chords PP 1 and PP 2 to
substitute dS 1 and dS 2 . Consequently, dS 1 lies in the normal section plane PKP 1 and
dS 2 lies in the normal section plane PKP 2 . According to the definition of geodesic,
the above two normal section planes will be in the same normal section plane at
point P. To put it another way, the infinitely near points, P 1 , and P 2 are all in the
same normal section plane at point P. If every point on the same curve shares this
property, then this curve will be the geodesic.
The geodesic can also be defined as a curve on a surface where at each point of
the curve the principal normal of the curve coincides with the normal to the surface
of the ellipsoid at this point. It is quite convenient to determine whether a curve on
the surface of the ellipsoid is a geodesic according to this definition. We will
elaborate on this below.
For space curves, the line perpendicular to the tangent line to a curve at the point
is called the normal to the curve at that point. Thus, the space curve at a given point
has a bunch of normals. The aggregation of the normals forms a plane called the
normal plane. The principal normal to a curve at a point is a special normal on the
normal plane that points towards the concave side of the curve. This definition is
consistent with the previous one since taking an arbitrary arc element means that the
concave side of the curve at this point, namely the principal normal to the curve at
this point, is determined. Moreover, the normal section is determined by the normal
to the surface. The arc element is required to lie in the normal section, and then the
principal normal to a curve coincides with the normal to the surface.
Properties of the Geodesic
The Geodesic Is the Line of Shortest Distance Between Two Points
on the Ellipsoid
In Fig. 5.21 , with the projection of the adjacent two arc elements of point P on the
geodesic orthogonally onto a plane tangent to the ellipsoid at this point, one obtains
P 0 1 PP 0 2 . Since the three points are on the same normal section plane, P 0 1 PP 0 2 is a
straight-line element. The shortest path between two points on any plane is a
straight line. The orthogonal projection of the adjacent two arc elements of any
point on the geodesic are the straight-line elements. Hence, the geodesic is the
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