Extending a curve of points under rotation while matching the curve’s profile to an originating plane. Let's say I had a given fixed end point for a curve (which is on a given intersecting plane) and that I wanted to rotate the curve but also extend the curve to meet a fixed plane while matching an originating curve's planar profile for a fixed projection. While I could rotate the curve of points, the end point of such curve, however, will no longer touch such plane given that a rotation of points will only yield (under rotation) any points of the same vector length. We can parameterize this rotation in a different way, however, providing that the rotated curve of points also touches the plane after rotation, but also for a given projection maintains the curve’s profile relative a given fixed projection. This is actually very simple since all we technically need do is use a parametric transformation for the curve under its normal coordinate plane. In this case a linear transformation of points of the curve along such normal coordinate plane should where a scale factor for such transformation is chosen in keeping to a given rotational plane, a little bit of trigonometry should yield this scale factor. Interestingly enough one need not under parametric transformation necessarily be restricted to linear transformations matching the incident curve of point's profile.
Applications: The simplest example that I could think of such use is found in finishing carpentry where the profile of say a given molding curve is projection drawn onto a intersecting mold. Thus instead of mitering such joint intersection (allowing for the coupling of both molds), the 'coped' joint means that one mold literally overlaps and intersects the other matching the molds intersecting curve profile at such angle of intersection. Recently examples that I have been examining include the Exeter Cathedral interior, at least in the rudimentary sense.
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