The piecewise plot below displays a sphere truncated by the plane z = 2 - y.

f := proc (x, y, z) options operator, arrow; piecewise(z <= 2-y, x^2+y^2+z^2-16, z-2+y) end proc; implicitplot3d(f, -4 .. 4, -4 .. 4, -4 .. 4, style = surface, numpoints = 50000);

The 3 transforms below when executed in display(T(sphere([0, 0, 0], 4, numpoints = 50000)), scaling = constrained) display the truncated sphere differently:

1) the truncating plane only partly conforms to the boundary of the truncated sphere

2) the truncated sphere is correct provided that the else condition coordinate is in the truncating plane and truncated sphere

3) the truncated sphere is correct but hollow

1) T := transform(proc (x, y, z) options operator, arrow; `if`(z <= 2-y, [x, y, z], [x, y, 2-y]) end proc)

2) T := transform(proc (x, y, z) options operator, arrow; `if`(z <= 2-y, [x, y, z], [0, 0, 2]) end proc)

3) T := transform(proc (x, y, z) options operator, arrow; `if`(z <= 2-y, [x, y, z], [`&+-`(sqrt(16-y^2-z^2)), y, 2-y]) end proc)

Please explain the different behavior of the three transforms.           

dsolve(f(x,y),y(x),parametric) in Maple 9.5 outputs an expression for y(_T) and for x(_T).

(from the book "Computer Algebra Recipes for Mathematical Physics by Richard H. Enns, section 7.1.1, finding the parametric equation of the brachistochrone)

but the same command in Maple 2016 outputs an expression for y(x) only.

Why do Maple 9.5 and Maple 2016 produce different output?

Where can I find a help page explaining or giving examples of dsolve, parametric?

I am trying to achieve an intuitive (non-mathematical) understanding of parallel transport on the unit sphere.
A vector parallel transported along a unit sphere's geodesic clearly maintains as constant its initial orientation vis-a-vis the geodesic i.e. with regard to its moving bases vectors.

Dr. Lopez's application "Visualizing a Parallel Field in a Curved Manifold" displays animation of the parallel transport of a vector along a latitude, which is not a geodesic. As seen by an observer moving with it, the vector rotates clockwise with regard to its moving bases vectors. The rate of rotation increases with higher latitudes.

Then to what is the transported vector maintaining parallelism?

I have uploaded a Maple 2016 worksheet Parallel_transport_on_the_unit_sphere.mw which mostly copies Dr. Lopez's application and, for additional clarity, adds the unit sphere, the moving bases vectors and the moving tangent plane to his display.

In a Maple Primes reply by Joel Riel on Sept. 14, 2011 he included the following command referring to warnings issued from a dsolve, numeric command having events containing a halt action:

_Env_in_maplet := true:  # incantation to suppress integrator warnings

Where can if find explanatory help for his command and any others of a similar nature? 

The help text for dsolve,numeric,events describes many kinds of triggers and actions and other event specifications, but shows only a very limited selection of examples of these possibilities.

Please tell me of any sources containing extensive examples of the wide variety of available specifications and the problems they are useful in solving.