While looking for a solution of computing the angle that corresponds to a certain arc length, I stumbled upon this question and the current answer. Unfortunately, neither this answer nor any other resource that I found on the web could directly be used for an implementation.
Obviously, computing the inverse of the arc length function (that was also provided in the question) is very difficult. But an approximation of this inverse using Newtons Iterative Method is possible. The following is a class that mainly offers two methods:
computeArcLength(double alpha, double angleRad): Computes the arc length of a point on the Archimedean Spiral where alpha is the distance between successive turnings, and angleRad is the angle in radianscomputeAngle(double alpha, double arcLength, double epsilon): Computes the angle at which the point for the given arc length is located on the Archimedean Spiral, where alpha is the distance between successive turnings, and epsilon is the approximation threshold for the Newton Iteration
The code is implemented here in Java, but these core methods should be fairly language-agnostic:
import java.awt.geom.Point2D;
/**
* A class for computations related to an Archimedean Spiral
*/
class ArchimedeanSpiral
{
/**
* Computes an approximation of the angle at which an Archimedean Spiral
* with the given distance between successive turnings has the given
* arc length.<br>
* <br>
* Note that the result is computed using an approximation, and not
* analytically.
*
* @param alpha The distance between successive turnings
* @param arcLength The desired arc length
* @param epsilon A value greater than 0 indicating the precision
* of the approximation
* @return The angle at which the desired arc length is achieved
* @throws IllegalArgumentException If the given arc length is negative
* or the given epsilon is not positive
*/
static double computeAngle(
double alpha, double arcLength, double epsilon)
{
if (arcLength < 0)
{
throw new IllegalArgumentException(
"Arc length may not be negative, but is "+arcLength);
}
if (epsilon <= 0)
{
throw new IllegalArgumentException(
"Epsilon must be positive, but is "+epsilon);
}
double angleRad = Math.PI + Math.PI;
while (true)
{
double d = computeArcLength(alpha, angleRad) - arcLength;
if (Math.abs(d) <= epsilon)
{
return angleRad;
}
double da = alpha * Math.sqrt(angleRad * angleRad + 1);
angleRad -= d / da;
}
}
/**
* Computes the arc length of an Archimedean Spiral with the given
* parameters
*
* @param alpha The distance between successive turnings
* @param angleRad The angle, in radians
* @return The arc length
* @throws IllegalArgumentException If the given alpha is negative
*/
static double computeArcLength(
double alpha, double angleRad)
{
if (alpha < 0)
{
throw new IllegalArgumentException(
"Alpha may not be negative, but is "+alpha);
}
double u = Math.sqrt(1 + angleRad * angleRad);
double v = Math.log(angleRad + u);
return 0.5 * alpha * (angleRad * u + v);
}
/**
* Compute the point on the Archimedean Spiral for the given parameters.<br>
* <br>
* If the given result point is <code>null</code>, then a new point will
* be created and returned.
*
* @param alpha The distance between successive turnings
* @param angleRad The angle, in radians
* @param result The result point
* @return The result point
* @throws IllegalArgumentException If the given alpha is negative
*/
static Point2D computePoint(
double alpha, double angleRad, Point2D result)
{
if (alpha < 0)
{
throw new IllegalArgumentException(
"Alpha may not be negative, but is "+alpha);
}
double distance = angleRad * alpha;
double x = Math.sin(angleRad) * distance;
double y = Math.cos(angleRad) * distance;
if (result == null)
{
result = new Point2D.Double();
}
result.setLocation(x, y);
return result;
}
/**
* Private constructor to prevent instantiation
*/
private ArchimedeanSpiral()
{
// Private constructor to prevent instantiation
}
}
An example of how to use this for the goal described in the question is given in this snippet: It generates a certain number of points on the spiral, with a desired (arc length!) distance between the points:
import java.awt.geom.Point2D;
import java.util.Locale;
public class ArchimedeanSpiralExample
{
public static void main(String[] args)
{
final int numPoints = 50;
final double pointArcDistance = 0.1;
final double alpha = 0.5;
final double epsilon = 1e-5;
double totalArcLength = 0.0;
double previousAngleRad = 0.0;
for (int i=0; i<numPoints; i++)
{
double angleRad =
ArchimedeanSpiral.computeAngle(alpha, totalArcLength, epsilon);
Point2D point =
ArchimedeanSpiral.computePoint(alpha, angleRad, null);
totalArcLength += pointArcDistance;
// Compute and print the arc lengths, for validation:
double currentArcLength =
ArchimedeanSpiral.computeArcLength(alpha, angleRad);
double previousArcLength =
ArchimedeanSpiral.computeArcLength(alpha, previousAngleRad);
double arcDistance = (currentArcLength - previousArcLength);
System.out.printf(Locale.ENGLISH,
"Point (%6.2f, %6.2f distance in arc "
+ "length from previous is %6.2f\n",
point.getX(), point.getY(), arcDistance);
previousAngleRad = angleRad;
}
}
}
The actual arc length distance of the computed points is printed, and one can see that they are in fact equidistant, with the desired arc length distance.
, where 
