Movements on a sphere

Movements of some 2-D shapes on a spherical surface follow different rules than those on a planar surface (e.g. one end of a line can meet the other). The below attempts to demonstrate some of the associated geometries on a flat paper. It does so by using hemispheric projections, which are plots of points as positioned on a spherical surface, but then projected and represented on a flay piece of paper. If you imagine how lines of latitude (small-circles) and longitude (great-circles) look on a paper, that is a start.

In this image the modern day trace of North America was plotted in red outline on an equal-angle hemisphere using the Dr. Allmendinger's stereonet program (available at http://www.geo.cornell.edu/geology/faculty/RWA/programs.html) and accompanying continental outline trace files. Dr. Allmendinger's generosity in making this easily available and usable is much appreciated. We are plotting on an equal-angle hemisphere plot (instead of an equal area plot) because angular relationships are undistorted and small-circle geometries are undeformed. The points creating the outline of North America where then rotated around a pole of rotation (rotation axis) in 10 degree increments creating the series of differently colored outlines you see above. Note that distintive points on the outline of North America follow small circle paths - basically lines of 'latitude'. This is, however, a very special case - one where the rotation axis of the earth which fundamentally defines where the latitude (small circle) and longitude (great circle) arcs are is the same as the pole of rotation describing how a plate moves. There is no reason that a platepole of rotation could not be in a very different position, with corresponding different small-circles and great-circles as is explored below. You can imagine it as if there are a second set of small and great circles centered on an axis that is oblique to the vertical axis in the standard plot (see the last image on this page for an example).

This image has been produced in a very similar way as the above image, except that North America was rotated around a quite different pole of rotation (labeled here as the rotation axis). If you follow a distinctive point from outline to rotated outline it is clear that they follow circular paths - these are small-circle paths centered on the rotation axis (not the rotation axis of the earth, but holding lines of latitude and longitude fixed, the pole of rotation of the plate on the sphere). In order to define a pole of rotation you have to hold something fixed (the reference frame). Here, for convenience, it is lines of latitude and longitude. More usually you are describing the relative motions of two or more plates, and so you hold one fixed, and describe the pole of rotation for the other relative to a fixed plate. It is important to remember what your reference frame is.

This image is the same as above, but where we continued to move the outline of North America in 10 degree increments (the new positions are in color), but with a new pole of rotation. Using the partial small circle path formed by distinctive points on the outline for this later movement period, can you visually estimate where the new pole of rotation is?

This stereonet plot is similar to the first one, but with 20 degree rotation increments, and a pole of rotation for the plate motion that is not in the plane of the page. The lines of latitude in this particular view would be the small circle paths that would be taken by any points moving about this pole of rotation. If geographic N is to the top (as is a common convention in viewing world projections) then the plate pole of rotation would be 60 degrees away from it.

The red outline is an artifical plate boundary geometry consisting of small circle transform faults segments joining great circle spreading ridge segments. If we hold the left side of the spreading ridge fixed, then the plate boundary moves relative to it as new ocean crust is added to the fixed plate, and the boundary must shift to the right. The yellow through purple outlines show increments of 15 degree rotation, which can be considered increments of oceanic basin growth. For convenience here we have made the N-S axis the pole of rotation for these plate movements, as we did with the first example above of moving North America around on a sphere. The area between each plate boundary position would be the amount of new oceanic crust added to the fixed left plate during each growth/rotation episode. Note again how points on the transform follow the transform as a path because in an ideal geometry transforms are small circles. Points on the great-circle ridge segments move at an angle of 90 to the ridge - pure divergence. Of course a question is how closely do real ridges and transforms follow this geometry.

There are a lot of lines on this diagram. It is basically the same view as above, but with an off-axis pole of rotation. The thin red lines represent modern day latitude and longitude lines that are centered on geographic north and the rotation axis of the earth. The thin grey lines are the small-circles and great circles associated with the pole of rotation describing the plate motion. This may help you see more clearly the differenceand similarity between the present geographic framework, and the framework provided by the pole of rotation that describes how a plate on the spherical earth moves relative to another plate or some other reference frame.