Editing Appendix/Ramblings/VirtualReality (section)

==In-Tangents and Out-Tangents==

According to pp. 474-476 of the [https://books.google.com/books?id=i_4S0tBcraAC&pg=PA476&lpg=PA476&dq=what+is+an+outtangent&source=bl&ots=BJbUjcRoQL&sig=ACfU3U1nUBFfB4TeEizBInvJ5sOp-Gz7Sw&hl=en&sa=X&ved=2ahUKEwjQ_vbKsZ_lAhUGC6wKHTejAm8Q6AEwCHoECAkQAQ#v=onepage&q=what%20is%20an%20outtangent&f=false Complete Maya Programming: An Extensive Guide to MEL and C++ API]:
* ''In-tangent'' defines the speed at which the animation curve approaches a key.
* ''Out-tangent'' defines the speed at which the animation curve leaves a key.

And, beginning on p. 211 of [https://books.google.com/books?id=vR2vCgAAQBAJ&pg=PA211&lpg=PA211&dq=what+is+an+outtangent&source=bl&ots=l7VgWQxfjP&sig=ACfU3U0TOF0_LsLayFoaGLc-HFe13AI2cQ&hl=en&sa=X&ved=2ahUKEwjQ_vbKsZ_lAhUGC6wKHTejAm8Q6AEwBnoECAcQAQ#v=onepage&q=what%20is%20an%20outtangent&f=false Digital Character Development:  Theory and Practice, Second]:
* The in- and out-tangents define how the curve enters and leaves the point and are usually depicted graphically as two vectors originating at the point.  The angle of these tangents will modify the direction of the curve based on interpolation.

Let's study in more detail Chapter 4 (the ''Programming Guide'') of the PDF-formatted document titled, [https://www.khronos.org/files/collada_spec_1_4.pdf COLLADA &#8212; Digital Asset Schema Release 1.4.1] &#8212; Specification (2<sup>nd</sup> Edition), March 2008.  Within COLLADA, both <geometry>/<spline> and <animation>/<sampler> define curves.  The first represents curves that can be displayed; the second represents curves that are used to create animations; for now, we will focus primarily on the latter, as we are dealing with animations.

COLLADA defines a semantic attribute (e.g., '''INPUT''', '''OUTPUT''', '''IN_TANGENT''', and '''OUT_TANGENT''') for the <input> element that identifies the data needed for interpolating curves.  In addition, the <Name_array> within a source allows an application to specify the type of curve to be processed; the common profile defines the values '''BEZIER''', '''LINEAR''', '''BSPLINE''', and '''HERMITE'''.  I think that '''HERMITE''' will prove to be the most useful for our efforts to follow the motion of a group of Lagrangian fluid particles in Riemann ellipsoids.

A curve is defined in segments.  Each segment is defined by two endpoints.  Each endpoint of a segment is also the beginning point of the next segment.  The endpoints for segment[''i''] are given by '''POSITION'''[''i''] and '''POSITION'''[''i+1''].  Therefore, a curve with ''n'' segments will have ''n+1'' positions.  Points can be defined in two or in three dimensions (2-D or 3-D).

The behavior of a curve between its endpoints is given by a specific interpolation method and additional coefficients.  Each segment can be interpolated with a different method.  By convention, the interpolation method for a segment is attached to the first point, so the interpolation method for segment[''i''] is stored in '''INTERPOLATION'''[''i''].  If an n-segment curve is open, then '''INTERPOLATION'''[''n+1''] is not used, but if the curve is closed (the endpoint of the last segment is connected to the beginning point of the first segment) then '''INTERPOLATION'''[''n+1''] is the interpolation method for this extra segment.  The closed attribute of the <spline> element indicates whether the curve is closed (true) or open (false; this is the default).