Improve BSpline related files and docs

[saeeedtorabi]

No description provided.

[tcobbs-bentley]

I don't understand bsplines well enough to provide a useful review of the algorithmic changes or of BSpline.md. However, I tried to review the code and the comment.

[dassaf4]

Mostly I have suggestions to improve overall consistency, clarity, and notation.

[dassaf4]

Suggested change

	* Bspline knots and poles for 1d-to-Nd.
	* Knots and poles for a B-spline function mapping R to R^n.

[dassaf4]

Suggested change

	- If the knots are strictly increasing (no duplicates) the curve has `order-2` (i.e., `degree-1`) continuous derivatives.
	- Introducing repeated knots reduces the continuity (at the joints between spans). In particular, with `order-1` repeated knots, there is a cusp (abrupt slope change) at that knot.
	- If the knots are strictly increasing (i.e., "simple" knots, no duplicates) the curve has `order-2` continuous derivatives everywhere.
	- Introducing repeated knots reduces the continuity of the B-spline curve at the knot. In particular, at a knot with multiplicity `order-1`, there is a cusp (abrupt slope change) at that knot.

[dassaf4]

Suggested change

	## Example: Order 3 (quadratic) bspline curve
	- An order 3 bspline curve has degree 2, i.e., is piecewise quadratic.
	- The curve does _not_ pass through the control points (dark).
	- There are 4 interior knots and therefore, 4 span breaks (circles).
	- Span breaks (circles) are exactly at the midpoints of interior edges.
	- For a parametric curve, derivatives are vectors, so to be continuous, they have to have both direction and magnitude continuity.
	- Direction (first derivative) is continuous at each span change (circles).
	- The concavity (second derivative) changes abruptly at each span change (circles).
	- This concavity change is not always visually obvious.
	- These curves are not as smooth as your eye thinks.
	- There are no concavity changes within any single span.
	- Clamping (2 identical knots at start, 2 identical knots at end) makes the curve pass through the end control points and point at neighbors.
	## Example: Order 3 (quadratic) B-spline curve
	- Degree is 2, i.e., the curve is piecewise quadratic.
	- The black dots in the figure refer to control points; the circles refer to points where the spans join.
	- The curve does _not_ pass through the control points.
	- The spans join at the midpoints of interior edges of the control polygon.
	- The curve has discontinuous second derivative where the spans join; either the second derivative direction (concavity) changes abruptly, as is seen here, or its magnitude changes abruptly, which is not always visibly obvious.
	- There are no concavity changes within any single span.
	- Specifics for this example:
	- Knot vector (clamped, normalized, uniform): `[0,0, 0.2, 0.4, 0.6, 0.8, 1,1]`.
	- This curve interpolates the first and last points and tangent directions of its control polygon because its knot vector is clamped.
	- This curve has 4 interior knots and therefore 5 spans.
	- This curve has continuous first derivative (tangent magnitude and direction) where the spans join because its interior knots are simple.

[dassaf4]

Suggested change

	## Example: Order 4 (cubic) bspline curve
	- An order 4 bspline curve has degree 3, i.e., is piecewise cubic.
	- The curve does _not_ pass through the control points (dark).
	- The curve does _not_ pass through particular points of the polygon edges.
	- Span changes (circles) are generally "off the polygon".
	- Direction and concavity are both continuous at span changes.
	- There can be one concavity change within a span.
	- Clamping (3 identical knots at start, 3 identical knots at end) makes the curve pass through the end control points and point at neighbors.
	## Example: Order 4 (cubic) B-spline curve
	- Degree is 3, i.e., the curve is piecewise cubic.
	- The black dots in the figure refer to control points; the circles refer to points where the spans join.
	- The curve does _not_ pass through the control points.
	- The curve does _not_ pass through particular points of the control polygon.
	- The spans join at points off the control polygon.
	- There can be at most one concavity change within a span.
	- Specifics for this example:
	- Knot vector (clamped, normalized, uniform): `[0,0,0, 0.25, 0.5, 0.75, 1,1,1]`.
	- This curve interpolates the first and last points and tangent directions of its control polygon because its knot vector is clamped.
	- This curve has 3 interior knots and therefore 4 spans.
	- This curve has continuous first and second derivatives (tangent and concavity magnitude and direction) where the spans join because its interior knots are simple.

[dassaf4]

Suggested change

	## Example: Order 5 (quartic) bspline curve
	- An order 5 bspline curve has degree 4, i.e., is piecewise quartic.
	- The curve does _not_ pass through the control points (dark).
	- The curve does _not_ pass through particular points of the polygon edges.
	- Span changes (circles) are generally "off the polygon".
	- Direction, concavity, and third derivative are all continuous at span changes.
	- There can be two concavity change within a span.
	- Clamping (4 identical knots at start, 4 identical knots at end) makes the curve pass the end control points and point at neighbors.
	## Example: Order 5 (quartic) B-spline curve
	- Degree is 4, i.e., the curve is piecewise quartic.
	- The black dots in the figure refer to control points; the circles refer to points where the spans join.
	- The curve does _not_ pass through the control points.
	- The curve does _not_ pass through particular points of the control polygon.
	- The spans join at points off the control polygon.
	- There can be at most two concavity changes within a span.
	- Specifics for this example:
	- Knot vector (clamped, normalized, uniform): `[0,0,0,0, 1/3, 2/3, 1,1,1,1]`.
	- This curve interpolates the first and last points and tangent directions of its control polygon because its knot vector is clamped.
	- This curve has 2 interior knots and therefore 3 spans.
	- This curve has continuous first, second, and third derivatives where the spans join because its interior knots are simple.

[dassaf4]

Suggested change

	There are innumerable books and web pages explaining splines. There is a high level of consistency of the concepts -- control points, basis functions, knots, and order. But be very careful about subtle details of indexing. Correct presentations may superficially appear to differ depending on whether the writer has considers `n` indices to run:
	- C-style, `0 <= i < n` (with index `n` _not_ part of the sequence)
	- Fortran style , `1 <= i <= n`
	- (rare) `0 <= i <= n`
	Some typical descriptions are:
	- <https://en.wikipedia.org/wiki/B-spline>
	- <http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node17.html>
	- <https://www.cs.unc.edu/~dm/UNC/COMP258/LECTURES/B-spline.pdf>
	Be especially careful about the number of knot counts, which can differ by 2 as described in the "over-clamping" section.
	There are innumerable books and web pages explaining B-spline curves. There is a high level of consistency of the concepts: control points, basis functions, knots, and order. But be very careful about subtle details of indexing. Correct presentations may superficially appear to differ depending on whether the writer considers `n` indices to run:
	- C-style, `0 <= i < n`
	- Fortran style , `1 <= i <= n`
	- (rare) `0 <= i <= n`
	Be especially careful about the number of knot counts, which can differ by 2 as described in the section "Overclamping" above.
	Some typical descriptions are:
	- <https://en.wikipedia.org/wiki/B-spline>
	- <http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node17.html>
	- <https://www.cs.unc.edu/~dm/UNC/COMP258/LECTURES/B-spline.pdf>