The window "Domain" displays a domain (as a
wireframe) and a point (u0, v0) in the domain.
The window "Surface
X(u,v)" shows the plane X(u,v) = (u, v, 0), the point X(u0,
v0), and the
tangent vectors Xu(u0,v0) and Xv(u0,v0)
their tails at that point.
The window "Projection" displays the projection of
the surface X(u,v) onto the tangent plane at X(u0, v0).
window displays the four elements of the matrix g associated to the
first fundamental form. Note that g11 gives the square of
the length of
Xu(u0,v0) (the red vector), g22
the square of the
Xv(u0,v0) (the green vector), and g12
and g21 give the dot
Xu(u0,v0) and Xv(u0,v0),
which equals the product
of the lengths of the
two vectors times the cosine of the angle between them. It is possible
to change u0 and v0 in the control window.