Spline: tangency to another line.
Splines in AutoCAD are mathematically called non-uniform rational B-spline or NURBS for short. In general when you create a spline in AutoCAD using the Fit or Control Vertex method the spline can be considered just a B-spline (leaving out the non-uniform aspect). The more complicated NURBS spline are produced when you start editing the weights of the CVs. Since the spline is defined mathematically the slope of the spline can be determined at any point along the spline by doing a calculus operation known as a derivative. The derivative of a curve is the slope of the curve. The math of a NURBS or B-spline is beyond the scope of a discussion here but if you would like a taste of it check out this link.
An easy way to determine the slope of a spline in AutoCAD is to trim the spline a the point you want to examine and then look at the location of the first Control Vertex after the trim. A line going from the end of the trimmed splined to the next CV defines the slope of the spline at its end.
For example: the spline in the picture below was defined with 7 Fit points.
Let's say we would like to know the slope of the spline where the red line intersects the spline. We could trim the spline to the red line and then display the Control Vertices which would yield the following:
A line from CV point A to CV point B defines the slope of the spline at A.
Unfortunately I do not believe there is an OSNAP for CVs. You can however get their location with the list command.
A program could be written to examine a B-spline and find points on the spline that have a specified slope. AutoLISP has a function that outputs the first derivative which could be used to find the point.
@Anonymous wrote:
... a point called X belongs to a spline, [draw a] tangent of that spline in X?
You can draw a line tangent on a Spline in a given point X as shown in >>this video<<.
- Draw a small circle with its centerpoint at X (here the intersection between a line and the spline)
- Draw a line from the one intersection spline/circle to the other intersection spline/circle.
- move the line from its midpoint to the centerpoint of the circle.
- (if needed increase the length of the line)
It is "only" an approximated solution. But you can increase the precision of the solution, the smaller the diameter of the circle. I think in most practical cases it an acceptable solution. (And it is not necessary to break the spline. And you can grab the tangent line using osnaps for further purposes).
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Jürgen Palme
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@Anonymous wrote:
.... I have a spline, and I want to find its points of tangency to the horizontal. ....
If it were any other kind of curved object, such as a Circle or Arc or [non-spline-curved] Polyline, its bounding box would provide a horizontal tangent edge at the uppermost or lowermost extents, and INTersection Osnap would give you the point of tangency. Unfortunately, a Spline's bounding box often extends beyond the actual shape of it, reaching out to include at least some Control Vertices [though it's not as simple as that -- not always all of them, and I don't know what determines it]. Here the dashed blue is the bounding box:
And if you need a tangency-to-horizontal that is not at the extremes, such as at the bottom of the rightward swoop above, the bounding box can't help you.
But a routine could be made to step along and test the first-derivative angle at intervals, and when it gets close to horizontal, reduce the size of the intervals, and back up if it went too far, until it gets to a point where the direction is horizontal, but within some tolerance, not with absolute precision. You could make the tolerance very tight to get very close. Is that worth pursuing?
Kent Cooper, AIA 
@j.palmeL29YX I like your approach for finding the approximate slope of a spline at a point. Too bad AutoCAD's parametric features do not include splines for constraints or you could make the solution more automatic.
The following lisp program create points at all the locations on a spline where the slope is approximately zero. I arbitrarily set a tolerance of 0.01 for the vertical component of the slope vector p determined by vlax-curve-getFirstDeriv. This could be tweaked but the step size for the B-spline's independent variable tt if too big may miss a solution.
(defun c:zeroslope (/)
; adds points on a spline where the slope is approxmately 0.
; lrm 5/27/2020
(vl-load-com)
(command "_pdmode" 35)
(setvar "osmode" 0)
(setq ent (car (entsel)) ; select curve
curve-Obj (vlax-ename->vla-object ent)
startSpline (vlax-curve-getStartParam curve-Obj)
endSpline (vlax-curve-getEndParam curve-Obj)
tt 0.0
dt (/ (- endSpline startSpline) 1000.0)
)
(while (< tt endSpline)
(setq p (vlax-curve-getFirstDeriv curve-Obj tt))
(setq y (abs (cadr p)))
(if (< y 0.01)
(progn
(setq tsolution tt)
(princ "\nt solution = ")
(princ tsolution)
(setq pp (vlax-curve-getPointAtParam curve-obj tsolution))
(command "_point" pp)
)
)
(setq tt (+ tt dt))
) ;end while
(princ "\Done")
(princ)
)
Thank you for your tips, and for the explanations.
Honestly, I couldn't imagine that a problem of such (apparent) simplicity wouldn't have had a click-and-go solution: I have a spline, and I want to contain it within a rectangle.
As we said, tangency is a matter of derivative, and if we know the formula of our curve, we also should be able to know the tangent (slope) for every point.
Using trim @leeminardi or little circles @j.palmeL29YX to get to the tangent of the spline in one point (the slope of the spline in X) is clever, but in fact my aim was rather to know where the spline has a certain slope (zero, horizontal): it's the other side of the problem.
Unfortunately, my knowledge is not so deep as yours, so I looked at your talk woth sort of embarassment. Thank you @Kent1Cooper for introduce me to "bound", even though not so good for splines: I didn't know of such a tool, where can I find it in AutoCAD?
I'm not familiar with LISP: how can I make use of spline-slope.lsp?
Extract the file Spline-slope.lsp from the zipped file in my last post. In AutoCAD go to the Manage tab and click Load Application and select the file Spline-slope.lsp and click Load.
YOu now have a new command to use called splineslope! Just enter it at the command prompt then select a spline and enter an angle. Points will appear on the spline at the specified slope. If the spline is fairly flat you may get more than one point. If so, just use the midpoint between them.
Note the 2 points were created in the area within the red box. This is because there's a tolerance on the angle of +/- 0.5°. A smaller tolerance could have been used but this could keep a solution from being calculated. The command changes the point style to ensure that the points can be seen.
@Anonymous wrote:
….
.... Thank you @Kent1Cooper for introduce me to "bound", even though not so good for splines: I didn't know of such a tool, where can I find it in AutoCAD?
….
>Here< is an example of the use of the (vla-get-BoundingBox) function.
Kent Cooper, AIA
@SEANT61 thank you for the compliment.
I was not familiar with Hodographs but studied your screencast and I think I understand except that when I went to the 5th degree spline example at the link I posted earlier I could not see how the Hodograph was constructed.
I got one that looks like this:
The hook at the end does not not accurately show the change in slope as you traverse from 8 to 10. Any idea what I am doing wrong?
Thanks.
Having looked at the situation again I see that the depiction on the link is correct. Laying out my own Bezier decomposition creates the Hodograph like that link predicted.
Clearly, my interpretation of the math needs refinement. This investigation might offer a nice peek under the knickers of Spline mechanics.
Nice work @SEANT61
One thing I know for sure is I am still learning about non-rational and rational B-splines. Your understanding of them is much better than mine. I compared your Hodograph curve created from the piecemeal Bezier to the “wrong assumption” curve and could determine no procedure to go directly from the B-spline to the Hodograph. It is interesting to note that the second and next to the last CV for the curve are located at the midpoint of their corresponding vectors.
To better understand B-splines I created a custom B-spline function in Excel/VBA. It nicely reproduces any degree non-rational B-splines that are created in AutoCAD. I used the CVs from your AutoCAD file to duplicate the 5th degree B-spline. For graphing purposes I choose a delta u of 0.15.
Looking at the similarities and differences between the B-spline and B-spline derivative functions
I thought I could modify my Excel Bspline function to make a Bspline-derivative function by making slight changes to the subscripts and range and to reference an array of Q points rather than the original CVs. Keeping track of subscript nomenclature was a real challenge when I created the Bspline program and would be a challenge for me to create the derivative function. A project I’ll put on the back burner unless you want to give it a go!