Monday, September 7, 2015

Plotting a LWPOLYLINE in Maxima

In this post, I want to show how to produce a representation of LWPOLYLINEs in Maxima that is suitable for plotting to the plot2d function. The direct usefulness of such an operation is dubious. This is more in the way of a worked example to help build understanding of a few mathematical and computing tools.

Recap
The LWPOLYLINE is a 2d representation of polylines that represents each vertex as having x-value, y-value, and bulge value. The bulge value indicates the path of travel from the current point to the next one. Either it will go straight (bulge = 0), veer to the left and back as a CW arc (bulge < 0), or it will veer to the right and back as a CCW arc (bulge > 0).

So, it will
  • bulge left (< 0),
  • go straight (= 0), or
  • bulge right (> 0).
More details on the mathematics of the bulge factor can be found here and here.

The Pieces
First of all, we define our problem and break its solution up into pieces. A polyline is of such a nature that is does not make a good "\(f(x)\)" type function (plotting \(y\) versus \(x\)). However, it lends itself to parametric expression well enough. So, we will have an \(x(s)\) function and a \(y(s)\) function. Ideally, I would like to specify a length along the polyline (measured from the start) and get a point \((x(s), y(s))\). (If you have a surveying background, think \(s = \) station number.)

So, I need to have parametric equations for lines and for arcs that are parameterized for arc length and initial value of the parameter. I then need to stitch these together to make both an \(x\) and a \(y\) piecewise function.

Linear

Parametric equations for lines involve a direction and a starting point. Simply normalize the direction vector to parameterize for length. This looks like
\[(x(s), y(s)) = A + \frac{(B-A)}{||B-A||}s,\]
where \(A\) is the starting point and \(B\) is the final point. We will need to adjust the initial value to account for all of the length of the polyline prior to the initial point on this segment, say, \(s_i\) which changes our equations to
\[(x(s), y(s)) = A + \frac{(B-A)}{||B-A||}(s-s_i).\]

Arc

Parametric equations for arcs use a center, radius, initial angle, direction of rotation, and total angle subtended (some variation of feasible input parameters admitted). Parameterizing for arc length isn't as obvious as it was for lines, so let's start with a basic circle:
\[x(s) = c_x + r \cos{s}\]
\[y(s) = c_y + r \sin{s},\]
where \((c_x, c_y)\) is the center and \(r\) is the radius of the circle. For our reparameterization to work we require that radian measure be used (or else require degrees and do the conversion in the function, but why make work for ourselves and introduce inefficiency?). Recall that the definition of radians is the ratio of arc length to radius (\(\theta = s/r\)). So, our parameter becomes \(s/r\) and so we have

\[x(s) = c_x + r \cos{\left(\frac{s}{r}\right)}\]
\[y(s) = c_y + r \sin{\left(\frac{s}{r}\right)}.\]

To get the right starting point for our function we need the parameter inside the trig functions to be equal to the correct initial angle when the input value of \(s_i\) (length of polyline so up to the start of the arc) is given. We need, then, the initial angle, say \(\theta_i\), plus the angle past initial that the given \(s\) corresponds to. Thus,

\[x(s) = c_x + r \cos{\left(\theta_i + \sigma \frac{s-s_i}{r}\right)}\]
\[y(s) = c_y + r \sin{\left(\theta_i + \sigma \frac{s-s_i}{r}\right)},\]
where \(\sigma\) is 1 or -1 as per the sign of the bulge factor.

Piecewise

Making a piecewise function parameterized for length requires that we keep track of the length of the polyline up to the segment we are about to address. Implementing it in Maxima also requires a bit of acquaintance with the methods available to us for doing so. It is possible to use if-then-else logic to achieve this and obtain something that works fine for plotting purposes. As an example of this, consider a really simple example:

f(x) := if x < 1 then x^2 else x^3;
plot2d(f(x),[x,-1,2]);

Fig 1. If-then-else logic in functions plots fine.
There's another, somewhat handier approach to this problem, namely, the characteristic function. The characteristic function function is a way of expressing conditions as multiplication by 1 or 0 in order to include and exclude the appropriate expression. charfun2(x,a,b) returns 1 if \(x\in [a,b)\), else 0. So, we can modify the above function to be as follows and obtain the same result. 

load(interpol);
f(x) := charfun2(x,-inf,1)*x^2 + charfun2(x,1,inf)*x^3;
plot2d(f(x),[x,-1,2]);

One nice thing about these expressions is that it is possible to create them separately and add them together later and get a relatively normal Maxima expression. (There are some hiccups if you try to differentiate or integrate them.) I can't comment on the performance of the characteristic function approach versus the if-then-else approach, but the characteristic function approach may be more manageable to produce.

The Program
There are a number of details that don't specifically relate to this problem that I will not go into detail about and other details I have given in the two previously mentioned posts on bulge factors. Here is the complete code for producing the parametric equations and plotting them followed by the result for a specific example polyline.



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