  In the paper, some geometric properties of the plane interception curve
defined by a nonlinear ordinary differential equation are discussed. Its
parametric representation is used to find the limits of some triangle elements
associated with the curve. These limits have some connections with the
lemniscate constants A,B and Gauss's constant G, which were used to compare
with the classical pursuit curve. The analogous spherical geometry problem is
solved using a spherical curve defined by the Gudermannian function. It is
shown that the results agree with the angle-preserving property of Mercator and
Stereographic projections. The Mercator and Stereographic projections also
reveal the symmetry of this curve with respect to Spherical and Logarithmic
Spirals. The geometric properties of the spherical curve are proved in two
ways, analytically and using a lemma about spherical angles. A similar lemma
for the planar case is also mentioned. The paper shows symmetry/asymmetry
between the spherical and planar cases and the derivation of the properties of
these curves as limiting cases of some plane and spherical geometry results.
