Oscillating Plane and Z axis - Energy Balance

[Questionable Thought]

Hello,

I have a few questions regarding the osculating plane, which I understand to be the plane generated by the Principal unit tangent vector and principal unit normal vector (both are orthogonal), and the inertial Cartesian z-axis.

Ultimately, I plan to understand the geometric/kinematic equations necessary to derive an energy balance of an element starting from a force balance. Also, elementary fluid mechanics textbooks often show this derivation and it seems the z-axis is in the osculating plane.

Here are the equations (image attached). Let the angle between the z-axis in the normal vector be 'theta', let the differential distance in the direction of the motion (tangent vector) be 'ds' and let the differential distance in the direction of normal be 'dn'.

ds*sin(theta) = dz <--- this is the one needed to convert force balance in the z direction into an energy balance.

dn*cos(theta) = dz <---- this one indicates to me that the osculating plane has the z-axis within itI don't really need help with the force balance I just need justification for the geometry.

For instance, is this always/generally true? I feel like the z-axis is not always in the osculating plane. Thus, wouldn't Direction cosines be more general? Thank you

 

[Questionable Thought]

I will say that generally the z axis is not in the osculating plane. However, the angle, theta, the tangent direction, s, makes with a arbitrary horizontal axis, h1, (that is orthogonal to z axis) is the same angle the normal direction makes with the z axis or equivalently it makes a 90-theta angle with another horizontal axis, h2, (that too is orthogonal to z axis).

These two horizontal axes h1 and h2 are not necessarily x or y, but may coincide with them.