spline length
The only thing you'd be able to calculate manually from control points [or fit points] would be the sum of the straight-line distances between them, which of course is not the length of the Spline. I couldn't say how AutoCAD does it, but I don't think there's anything you could use from ordinarily-available information like entity data. [May I ask why you want to do this?]
Kent Cooper, AIA 
This formula and Autocad return same value for 4 control point but for higher control point we have some mistake
This would be a complete different approach..
but if CAD is imported in REVIT software ( Autodesk ),
Revit gives the length of the Spline directly...
@Corsten.Au wrote:
.... if CAD is imported in REVIT software ( Autodesk ), Revit gives the length of the Spline directly...
If all they needed was to get the length of an already-drawn Spline, there's certainly no need to go through importing it into Revit! Either the LIST or AREA command will report it, and put the value in the PERIMETER System Variable for use in any processing or calculation desired. Or it can be extracted with AutoLisp:
(vlax-curve-getDistAtParam spl (vlax-curve-getEndParam spl))
where 'spl' is either the Spline's entity name or a VLA-object conversion of that.
But they don't want to do it that way [re-read Message 1].
Kent Cooper, AIA
@Anonymous wrote:
... A simple example is given control poni P0=1,1 P1=2,0 P3=3,1 P4+4,0 and P5=5,1 Can you calculate y parameter in X=2?
So maybe that's something of an answer to my question in Message 2 [at least it's some kind of reason for wanting to calculate something based on control points], except that it has nothing to do with the Subject line or the original question in Message 1, both being about length, which seems irrelevant to this example. Though the Bezier-curve formulas are beyond my mathematical experience, I'd still be interested in the "why" of it all....
Kent Cooper, AIA
@leeminardi wrote:. . . . At what point must the 5th CV be considered?
The way the splines are evaluated is by stepping through the parametric Range, t, a small increase at a time. The incremental t value is plugged into the equations which spit out an xyz value. Much like the spreadsheet setup I posted earlier, at any given t the influence of each of the 4 CVs (in this case) varies.
The Range, though, is split into different zones with Knots at the border. When a Knot value in reached, the influence of the earliest CV of the 4 goes to zero, and no longer play a part, and the next CV (4th from the CV just removed) in line starts to contribute to the equation. Minimally at first, but increasing as the t values progress.
See the attached drawing and chart. This 8 CV AutoCAD spline has a Knot Vector (Nurbs term) as shown in the diagram. The Points in the drawing correspond to the Knot values shown in that grouping. The term Full Multiplicity refers to when a value is repeated Degree number of times. That indicates to the algorithm to uses the corresponding CV for %100 of the influence. It is why the AutoCAD curve lands directly on the start and end CV. As t increases from 0, the next 3 CVs start contributing to the equation. Notice how the curves on the chart start rising from the initial 0 influence state.
At any point along that horizontal (X) of the Chart, a vertical line will intersect several curves, all of the Y values will total 1.0 (%100) . Notice, also, that at each knot value, one curve ends while another starts.
As noted in post #3 by @SEANT61, if you created a spline with 5 control points the AutoCAD default would be a 3rd degree (4th order) B-spline. For the points you specified in post #8 the spline should look like the green curve in the image below. The equation for a B-spline can be found in the link of post #4. Check it out if you have not yet done so. The math is more complex than a Bezier spline encouraging the use of recursion (deBoors algorithm) to calculate points on the spline.
You asked for the value of y when x = 2 (in Euclidean and not parametric space I assume). Set luprec to 8 and then add a point at the intersection of the line and the original spline using object snap int (intersection). The B-spline will be accurately evaluated. You should find that the coordinates of the point are: x = 2.00000000, y = 0.39061808.
As I noted in post #5 the splinedit command can be used to convert a spline to a polyline. This will give you a series of points along the spline.