Better Parametric Smoothing for Video

video needs elliptic curve-fitting because wheels must spin without bumps, and steer without changing size

The visually intuitive way to draw a metrically precise curvilinear diagram, is to mark (rovot-sample) extremal points, localized maxima and minima as viewed from locally lobe-centric perspectives, and mathematically fit, interpolate a simple equation curve - however, this leaves midway points to flap freely, widely, and we should likewise mark these as secondary or sub-extremal: Thus a digitized sketch is a sequence of alternately extremal and midway sub-extremal points, explicating and depicting both stature and shapeliness every 90 degrees. [3-D space needs tertiary sub-extremal points, on a quadrivial surface-web]

However, the shape of an ellipse is a linearly stretched circle, and the slope at any midway point on a subtended arc, equals the slope of the subtending chord: which is not so easily located manually: the sub-extremal points must be selected first to determine their slope.

Also, an ellipse is determined by 5 coefficients, while we would rather use four points to determine the curve between the central two. However, we might determine by consecutive 5's, then average adjacent pairs of determinations.

And while ellipses, and circles, parabolas, hyperbolas, are real conic sections, sinusoidal curves connecting alternating turns to successive points, are not real but complex sections.

The locally intuitive way to interpolate between consecutive marks, is linear between points, plus diminishingly extrapolated portions of the adjacent segments: If the euclidean distance is used as the parameter (base measure) the derivative at extremal points is very stable - so stable, that any four points on a circle will guide to redraw a tangent-correct circle: which is not useful for controlling the steering or flipping aspects of the sketched object. And, we'd rather have weight applied to each successive point nonlinearly and symmetrically balanced - like a sinc [sin x/x] function distribution:

The sinc function tends very non-locally, and is spectrumly well-contained, and produces a line segment between two points, ellipses for equally dividing, equally angling, demarked points, more than two, and sine function for alternating points. We can also accelerate the computation of the sinc function for cycles, by counting the number of points in the cycle, and using a pre-simplified sinc-summation function.

[under reconstruction]

The more direct approach, estimating extremal points as the zero-angle position of zero slope perpendicular to the local lobe central line, derives parametric base measure on the angular positions of the adjacent two midway points as points along the same ellipse-end - this more nearly utilizes euclidean distance, but also their actual vectors from their locally central extremal points: If the next-adjacent (extremal) points remain on the same ellipse-end forming an "O" through the central three points, then the angles subtended by the central two vectors are some portion of that ellipse-end, about 90 degrees each, given our early depiction; But if the next-adjacent points are folded back to form a "W" of 5 points, the angles subtended by the central two vectors reaches 180 degrees, of a cosinusoid: it adjusts next-order shaping around parabolic, as well as increasing angle subtended - and further folding further diminishes or exceeds, but for our early intuition on selecting marking points, extremal and subextremal: meaning, long curves should be marked with points about every 90 degrees. We then fit and interpolate overlapping elliptic sections, such that curves closely match elliptic, parabolic, or cosinusoidal, but it won't and shouldn't be a linear interpolation because the end-points of the adjacent sections should not greatly affect, neither the position nor the derivative at a central point. Four points suffice to designate a simple curve and its end-point derivatives between the central two, and slope-continuous. [We can also use the several near points to estimate the angle of the central two, rather than exacting the selection of points be separated by multiples of 90 deg -- as would be useful for parametrizing a parabola, for example]

Grand-Admiral Petry
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