Open any textbook on computer graphics [yeah, like those are common any 
more...] and there will be at least an entire chapter if not an entire 
part dedicated solely to splines. You would not believe how many 
different kinds of spline technology exist!

So anyway, recently I've decided to revisit this topic using the 
ever-correct and well-written Wikipedia.


everything there is to know about these now. Wikipedia has some lovely 
animations which visually illustrate exactly how to plot such a curve:


Suppose you want to plot the point 3/5th of the way along the curve.

- Draw a straight line between each consecutive pair of points, and find 
the point 3/5th of the way along each line. (There will be N of these.)

- We have now constructed a new set of N points from the existing set of 
N+1 points. Each time we repeat the procedure, we end up with a smaller 
set of points. Repeat the process until only 1 point removes. This is 
the point 3/5th of the way along the curve.


into two parts of unequal length. If you keep the first point of every 

spline. Ditto for the last point in every set. The two new splines 
coincide with the old one exactly (aside from the obvious change of 
parameter values).

Alternatively, you can represent the spline as the sum of the control 
points multiplied by the Bernstein polynomials of the appropriate order. 
But this is [apparently] less numerically-stable than the above, and 
just more complicated to program.


move any single control point, THE ENTIRE SPLINE CHANGES. It must be 
pretty damned annoying to spend hours getting one part of the curve just 
right, and then trying to edit another part and seeing the part you just 
did start moving around. (!)


100 control points, you use them to control several dozen independent 

effort, I have yet to determine how you can construct such a set of 
splines from the control points so that the final curve appears smooth 
and joinless.


transform the curve by just transforming the control points, or being 
able to cut the curve in half. But aside from the problem of editing 

circle exactly, only in approximation. (But then, straight lines can 
describe a circle *in approximation*! Computer games try it all the time...)


control point has a "weight" associated with it. As best as I can tell, 
the algorithm works like this: Let's assume we want a 2D spline. You 
draw the spline in 3D, with the weight being the distance from the 
camera. You then project the spline back into 2D using a perspective 
projection. The control points have their coordinates adjusted so that 
no matter what the distance from the camera is, they still appear at the 
same position in 2D space (from the camera's viewpoint). But this still 
affects the resulting spline curve.

(Most weirdly, a *negative* weight in effect calculates some kind of 
360-degree frog-eye camera weirdness. But the end result is useful.)


sections ("although the representation is not unique").

And then we come to B-splines, and that's where it all starts to get a 
bit fuzzy. A B-spline has a "knot vector" which controls the blending 
functions. If the number of control points equals the degree, the 

equidistant, the B-spline has the property that moving one control point 
only affects a limited part of the curve.

(This already suggests a way to have 100 control points define a single, 
continuous curve. If only, say, 4 of the points are active at any time, 
but the blending functions go smoothly to zero, you can calculate each 
segment of curve using only 4 control points, but the resulting curve is 
guaranteed to be smooth.)

The blending functions appear to be recursively-defined, so that the 
zero-order functions are piece-wise constant, the first-order functions 
are piece-wise linear, and higher functions are linear combinations of 
the next order down, so the functions get smoother and smoother as you 
go up.

Still not really comprehending this knot vector business...

Wikipedia has a huge, huge article on NURBS. But NURBS starts for 
"non-uniform rational B-spline". It's a B-spline which is rational 

projecting into the distance) and has a non-uniform knot vector. So if 
you can figure out B-splines... basically, you know NURBS.

...my head hurts! >_<

Post a reply to this message

Invisible <voi### [at] devnull> wrote:

> 100 control points, you use them to control several dozen independent

> effort, I have yet to determine how you can construct such a set of
> splines from the control points so that the final curve appears smooth
> and joinless.

If you wanted to smoothly join spline1 and spline2:
1)the endpoint of spline1 must be in the same location as the start point of
spline2
2) the last control point of spline1, the endpoint of spline1/start point of
spline2, and the first control point of spline2 must be co-linear.

The discontinuity is caused by the change in slope between the control points
and their respective endpoint.  Note that the line between a control point and
its nearest endpoint form a line that has the slope of the curve at the
endpoint.  If you make the slopes of the two adjacent splines different, then
they do not connect smoothly.

-Reactor

Post a reply to this message

Reactor wrote:

> If you wanted to smoothly join spline1 and spline2:
> 1)the endpoint of spline1 must be in the same location as the start point of
> spline2
> 2) the last control point of spline1, the endpoint of spline1/start point of
> spline2, and the first control point of spline2 must be co-linear.
> 
> The discontinuity is caused by the change in slope between the control points
> and their respective endpoint.  Note that the line between a control point and
> its nearest endpoint form a line that has the slope of the curve at the
> endpoint.  If you make the slopes of the two adjacent splines different, then
> they do not connect smoothly.

As I discovered further down, using a B-spline means that the degree of 
the curve is unrelated to the number of control-points, and smoothness 
is guaranteed (depending on the order of the curve).

Now all I have to do is figure out how B-splines actually work. >_<

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