This is a series about creating mathematical curves in BricsCAD (and AutoCAD).
Part 0, just hackish with a spreadsheet, result driven.
Part 1, the math, understanding ways to iterate.
Part 2, the coding, techniques available.
THIS ARTICLE IS UNDER CONSTRUCTION. YOUR POSITIVE INPUT IS APPRECIATED.
In part 1 the mathematics are explained. It turns out to be a solid base with a very functional algorithm in order to draw approximated curves.
Status Quo
This pages offers clues for coding. I’ve tried several concepts as described in part 1. After ending up writing a few hundred lines of code I decided to stop because a proper solution is simply a lot of work in unpaid time that lacks.
That is not the complete story, after tinkering, I also came to the conclusion that using a non uniform rational basis spline, or NURBS , or spline is a proper alternative after all and, combined with the theories in part 1, it offers best possibilities (and consumes a lot of my time).
Does it mean that this page is worthless? No, it offers additional insights that you will not find on the net in a structured way.
Making a NURBS
Ways in Lisp
In Lisp there are several ways to create a NURBS or spline:
- Very easy with (command “._spline” …), not recommended for production environments.
- The object-way with (vla-AddSpline …), not discussed further.
- The old fashioned way, using (entmake …), discussed here.
Example
“Entmake” that spline
Consider the following code:
(setq point-list '((0 0) (1 1) (2 0) (3 -2) (4 0)))
(entmake
(append
(list
'(0 . "SPLINE")
'(100 . "AcDbEntity")
'(100 . "AcDbSpline")
'(70 . 4)
'(71 . 3)
(cons 74 (length point-list))
)
(mapcar '(lambda (xy) (cons 11 xy)) point-list)
)
)
The first line is paste-able on the CAD command line as:
(setq point-list '( (0 0) (1 1) (2 0) (3 -2) (4 0)))
The second part as:
(entmake (append (list '(0 . "SPLINE") '(100 . "AcDbEntity") '(100 . "AcDbSpline") '(70 . 4) '(71 . 3) (cons 74 (length point-list))) (mapcar '(lambda (xy) (cons 11 xy)) point-list)))
Going through the code:
- “70 . 4” is DXF code 70 for spline type 4, i.e. “Rational spline”.
- “71 . 3” means degree 3.
- There are 5 points, so: “74 . 5”.
- “(mapcar “(lambda (xy) (cons 11 xy)) point-list)” returns a list with code “11” sublists: “((11 0 0) (11 1 1) (11 2 0) (11 3 -2) (11 4 0))”
That is all it takes to create a spline. Code “11” points are fit-points, going through the curve, what we want. This could also be “10” for control-points.
There is much more to be aware of, see DXF-codes for Splines. and more on NURBS at Wikipedia.
How to feed mathematical functions to the program?
We’ll start with a few examples.
Function files
Let’s say, we have a function . What we do is put it in a file with extension fun, for example: curve.fun. The filename should represent what you define in the file.
Please, do yourself a favour, use a suitable text editor like notepad++ and mark your code as Lisp during editing.
Syntax explained
What is in the file? First of all: Valid Lisp code, more on that later. What is needed is:
- A safety factor to make sure the curve is within the deviation limits: fuzzy
- A maximum deviation: dev
- A tolerance as a part of dev: tol
- A start value for x: xmin
- An end value for x: xmax
- Divide calculations: xdiv
- The function: fx
A valid file contains:
(tol 0.2)
(fuzzy 0.000)
(dev 0.05)
(xmin 0)
(xmax 2)
(xdiv 1)
(fx (* x x))
That should be clear, setq assigns a value to a variable. What can go wrong? The number of opening parenthesis should be equal to the number of closing parenthesis. That is the number 1 error.
Defining your functions
Arithmetic functions
Here is a list of some arithmetic functions…
(+ [number number] …) | Add |
(- [number number] …) | Subtract |
(* [number number] …) | Multiply |
(/ [number number] …) | Divide |
(~ int) | Bitwise NOT |
(1+ number) | Increment |
(1- number) | Decrement |
(abs number) | Absolute value |
(atan num1 [num2] ) | Arctangent of num1 or num1/num2 |
(cos ang) | Cosine |
(exp number) | “e” raised to power |
(expt base power) | Base raised to power |
(fix number) | Nearest smaller integer |
(float number) | Number to real |
(gcd int1 int2) | Greatest common denominator |
(log number) | Natural log of number |
(logand int int …) | Logical bitwise AND |
(logior int int …) | Logical bitwise inclusive OR |
(lsh int numbers) | Logical bitwise shift by numbers |
(max number number …) | Largest of numbers |
(min number number …) | Smallest of numbers |
(sin ang) | Sine |
(sqrt number) | Square root |
Examples
- Note that 6 is an integer and 6.0 is a real.
- (!)
- Angles are in radians, 2*pi radians is 360 degrees.
- Expressions are evaluated inside out: (than-this (first-this))
The line containing a function, (fx (* x x)), is a list in a list. The examples below are like (fx (example)).
- (* x x)
- (expt x 2)
- (+ (* x x) (* 6.0 x) -2.5)
- (- (sin x) (* x (cos x)))
- (abs x)
- (int x)
- (sqrt (expt x 2))
How configuration data is handled
What curve does is very basic: It reads lines from the files an interpretes them when they are in list format. A bit more in detail: the line “(xmax 3.2)” is a string. By using the read function, the line is converted to a list. From that point, further processing in Lisp is possible with a set statement (not setq).
Additional considerations
The math part describes a way to calculate valid fit points by iterating checks based on core CAD-Lisp functions. In addition, it is possible to use a set of curve measurement Lisp functions – link broken, cache:
Function | Description |
---|---|
(vlax-curve-getArea curve-obj) | Returns the area inside the curve |
(vlax-curve-getDistAtParam curve-objparam) | Returns the length of the curve’s segment from the curve’s beginning to the specified point |
(vlax-curve-getDistAtPoint curve-objpoint) | Returns the length of the curve’s segment between the curve’s start point and the specified point |
(vlax-curve-getEndParam curve-obj) | Returns the parameter of the endpoint of the curve |
(vlax-curve-getEndPoint curve-obj) | Returns the endpoint (in WCS coordinates) of the curve |
(vlax-curve-getParamAtPoint curve-objparam) | Returns the distance along the curve from the beginning of the curve to the location of the specified parameter |
(vlax-curve-getParamAtPoint curve-objpoint) | Returns the parameter of the curve at the point |
(vlax-curve-getPointAtDist curve-objdist) | Returns the point (in WCS coordinates) along a curve at the distance specified by the user |
(vlax-curve-getPointAtParam curve-objparam) | Determines the point on the curve that corresponds to the param parameter and returns the point |
(vlax-curve-getStartParam curve-obj) | Returns the start parameter on the curve |
(vlax-curve-getStartPoint curve-obj) | Returns the start point (in WCS coordinates) of the curve |
(vlax-curve-isClosed curve-obj) | Determines if the specified curve is closed (i.e., start point is same as endpoint) |
(vlax-curve-isPeriodic curve-obj) | Determines if the specified curve has an infinite range in both directions and there is a period value dT, such that there is a point on curve at (u + dT) = point on curve (u), for any parameter u |
(vlax-curve-isPlanarcurve-obj) | Determines if there is a plane that contains the curve |
(vlax-curve-getClosestPointTo curve-obj givenPnt [extend]) | Returns the point (in WCS coordinates) on a curve that is nearest to the specified point |
(vlax-curve-getClosestPointToProjectioncurve-obj givenPnt normal [extend]) | Returns the point (in WCS coordinates) on a curve that is nearest to the specified point |
(vlax-curve-getFirstDeriv curve-obj param) | Returns the first derivative (in WCS coordinates) of a curve at the specified location |
(vlax-curve-getSecondDerivcurve-obj param) | Returns the second derivative (in WCS coordinates) of a curve at the specified location |
In BricsCAD these functions are also available on OS-X and Linux.
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