CAD: An approach to mathematical curves for engineering, part 1, the math.

THIS ARTICLE IS UNDER CONSTRUCTION. YOUR POSITIVE INPUT IS APPRECIATED.

Introduction

Mechanical engineers sometimes need mathematical curves. For example:

  • A satellite dish is a basic parabola.
  • A gear has the involute of a circle as its base.

These kind of curves are usually not directly supported in CAD systems. They must therefore be drawn as approximate. Read on.

A blunt approach is always possible, make sure the curve is made up of many line segments so that the CAD curve is little different from the theoretical curve. However, this has a number of disadvantages:

  • Files can become large and therefore slow.
  • CNC devices may have problems in the processing of very large amounts of segments.
  • Intensive monitoring, calculating, of the curve is necessary to ensure the deviation is not too large.

I have to say that I’ve seen many implementations on the net and none was very clear when it came to using a proper algorithm. Without a suitable algorithm, the result will always be sloppy. This does not mean these solutions are not valuable, I just think there is something to win when we first start with the mathematics. Coding afterwards becomes like filling in a form. Feels good!

What is required?

Programmers without mechanical engineering experience can make mistakes in the approach to the problem. To start the discussion, take notice of the following arguments:

  • The curve should be smooth, sharp corners are not allowed, the change of direction (angle) determines the amount of nodes. And no polylines but splines or arcs. Counter arguments:
  • There is no objection to use straight segments as long as there are enough. More straight lines, less calculations.
    • If a radius of 1/1000 mm is composed of three line segments in steps of 30 degrees, then it will usually not be a problem. Defining one node per degree in that case is not smart.

It is no answer to the question, but it illustrates the complexity of the problem. Similarly, there are approaches whereby the length of the line segments of a curve is taken as a starting point. So what do we really want?

What an engineer wants

The solution can be made incredibly complex, but a mechanical engineer wants one thing primarily: a controlled tolerance of shape. In other words, a maximum deviation – or less.

What a CNC device wants

Not too much segments. Something to take into account when iterating points

An example

A shaft having a diameter of 10 mm (R5, blue) has a tolerance of -0.1 mm (magenta). The circumference is not circular, but drawn as polyline with straight segments (red). The distance of the polyline to the circle can not be more than 10% of the tolerance. This is 0.01 mm. There are only 50 segments needed to define the circle.

If the tolerance is reduced further, there are more segments. If the value is not 0.01 mm but 0.001 mm, the circle can be described by 223 straight segments. Considering the precision, this is not bad. These are values which both CAD systems and CNC machines can handle.

Jumping into one of many solutions

Newton Raphson iterates for example by taking decisions based on the resulting y values calculated by using the x3 value between x1 and x2. Result: x3=(x1+x2)/2. We could use this approach.

As an alternative, for angular equations it is similar: \angle3=(\angle1+\angle2)/2 (within limits).

So let’s see what it looks like.

Red: The theoretical curve. Left-bottom of triangle: Our point to check: x3.

The convergence of error margin e to 0

When x3 is entered in our function, the answer for y is y3 + b. Nice, but we want to know if d is within an acceptable range, in order to approve point x3, or not.

Now here comes an interesting thing: f + e = d and the more we iterate, the smaller part e becomes. This is easy to explain, the more we “zoom in”, the straighter the red line. So:

Important conclusion 1:

(1)   \begin{equation*}f_n + e_n \to f_n\end{equation*}

Focus is now on f. Knowing a and b, the value of f is:

(2)   \begin{equation*}\frac{1}{f^2}=\frac{1}{a^2}+\frac{1}{b^2}\end{equation*}

How do we get a?

Little bummer, we need to transform all used functions from f(x) to f(y) manually in order to retrieve a. Or, use a numerical solution. Both solutions add a lot of complexity. Can we work around that?

The fastest ingredient for calculating f

I would not have spent time writing this page if the answer was “no”. Read on. The point is, I like to “see” things in order to “better understand” things.

And that is not all, BricsCAD does have a reasonable proper SVG export facility, so you can enjoy the show too. So what is it? Look at the following:

  • I’ve iterated an arc manually. Centre point 0,0 (1) start 10,0 (2) and end 0,10 (3).
  • The magenta part:
    • Line from 2 to 3. From middle 4 to 5 and from 4 to 6.
    • Line from 5 to 6 and from 4 perpendicular to 7.
    • Point 8 on intersecting of arc en line 4-7
    • Conclusion: distance 4-8 is too large, so I iterate again starting at half x, between 3 and 4, being point 9.
  • The blue part: Same procedure as magenta, not good enough, iterate the red part.
  • The red part: Same procedure as blue. Hey! The result is fine, within limits..

You can’t see the details of blue and red. Here they are:

So again, what we want to know is the distance from point 4 to point 7. We do have the distance from 4 to 5. If we could say something about an angle like angle 4 5 7 or angle 5 4 7, we could solve.

And we can say something about it. Look at the baseline and the hypotenuse. Except for the first iteration, the angle between the baseline and the hypotenuse gets closer to 0^\circ with each iteration. At the blue iteration, this angle is 2.78^\circ, at the red iteration, it is reduced to 1.46^\circ.

Important conclusion 2:

(3)   \begin{equation*}\angle { baseline-hypotenuse} \to 0\end{equation*}

We can calculate the angle of the baseline based on start and end point. As a consequence of this theorem, we know the angle of the hypotenuse. Knowing b plus knowing the angle means we know f.

YESS!!! PROBLEM SOLVED!!!

And… I plan to implement this in a beautiful way. To be continued 😉

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