But, in general, a Vertical Line doesn’t intersect this Center.
Right?
Right.
The angular difference between the direction from a point on the surface to the center and the direction normal to the best fit ellipsoidal approximation to the geoid (i.e. local vertical using the best fit ellipsoidal model of gravitaiton) at that place defines the difference between two major distinct definitions of the concept of latitude. The angle between the equatorial plane and the line through both the local point and the center is defined as the geocentric latitude, and the angle between the equatorial plane and the perpendicular to the best fit to the ellipsoidal geoid is called the geodetic latitude. The magnitude of the geocentric latitude is always less than or equal to the magnitude of the geodetic latitude (because the geoid is oblate rather than prolate). The minimum difference between them is zero at both the equator and poles and the maximum difference between these two definitions of latitude occurs near +/- 45 degrees latitude and or Earth the maximum deviation between them is 0.1924243 deg. Actually, the largest discrepancy occurs for a geodetic latitude of +/- 45.09621215 deg corresponding to a geocentric latitude of +/- 44.90378785 deg.
There are a number of other common definitions of latitude that have their uses. Most important for everyday life is the astronomical latitude. It is defined as the angle between the exact local vertical as defined by a plumb bob (averaged over tidal effects and devoid of air currents) and the equatorial plane. The only difference between the astronomical latitude and the geodetic latitude is tiny net gravitational effect of the various small imperfections in the planet (due to it actually not quite being a fluid in static equilibrium in the rotating frame) making the actual gravitational field more complicated than can be represented by a ellipsoidal geoid. Needless to say there isn't much practical difference between the astronomical latitude and the geodetic latitude. For all practical purposes they are the same. The usual meaning of the plain unmodified term latitude is typically either the astronomical latitude or the geodetic latitude.
Another important kind of latitude is the so-called parametric or reduced latitude. This latitude is most useful in solving the mathematics of the problem of the ideal equilibrium shape because it is the natural latitude angle that occurs in the orthogonal coordinates of the 3-dimensional confocal spheroidal coordinate system for which the ellipsoidal geoid is naturally an equipotential surface. One can understand the parametric/reduced latitude as being the latitude one would get *if* one took the best fit ellipsoid (oblate spheroid) and uniformly scaled the polar Cartesian coordinates of the oblate surface points by a common factor and inflated them so the oblate spheroid became a nice round sphere whose radius was the equatorial radius everywhere.
There is a fairly simple mathematical relationship between the geodetic, parametric/reduced, and geocentric latitudes. The value of the parametric/reduced latitude is always in between those of the geodetic and geocentric latitudes because the tangent of the parametric/reduced latitude is the geometric mean of the tangents of the geodetic and geocentric latitudes, respectively. In particular, let C = the geocentric latitude, D = the geodetic latitude, and let P = the parametric latitude of some particular surface point. Also let b = sqrt(1 - e*2) = 1 - f = (polar radius/equatorial radius) where e is the eccentricity of the best fit ellipsoid and f is the corresponding flattening factor or oblateness. Then the relationship between these various kinds of latitudes is:
b*tan(G) = tan(P) = tan(C)/b .
There is another kind of latitude which is useful when calculating surface paths along the surface. That latitude is the rectifying latitude. The rectifying latitude is based on the surface meridian through a given surface point. In this case one takes the geodesic length of the meridian through the point of interest from the equator to the hemispheric pole for the hemisphere the point resides in and scales it between 0 and 90 degrees (or 0 & [pi]/2 radians), and whatever scaled angle the surface point has is the rectifying latitude of that point. The rectifying latitude has the property that equal north/south geodesic surface distances correspond to equal latitude differences. The mathematical relationship between the rectifying latitude and the other latitudes above is quite complicated because the fact that the perimeter/circumference of an ellipse is related to an elliptic integral of the 2nd kind of an argument defined in terms of the ellipse's semi-axes & eccentricity. This makes the any formula for the rectifying latitude involve elliptic integrals involving any of the other above latitudes.
There are also other specialty kinds of latitudes that cartographers may use on occasion related to the details of how they make a projection of a spheroidal surface onto a sphere, and then from a sphere onto a plane.