Fast Hypotenuse Algorithm for Embedded Processor?

Question

Is there a clever/efficient algorithm for determining the hypotenuse of an angle (i.e. sqrt(a² + b²)), using fixed point math on an embedded processor without hardware multiply?

Answer 1

If the result doesn't have to be particularly accurate, you can get a crude approximation quite simply:

Take absolute values of a and b, and swap if necessary so that you have a <= b. Then:

h = ((sqrt(2) - 1) * a) + b

To see intuitively how this works, consider the way that a shallow angled line is plotted on a pixel display (e.g. using Bresenham's algorithm). It looks something like this:

+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| | | | | | | | | | | | | | | | |*|*|*|    ^
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+    |
| | | | | | | | | | | | |*|*|*|*| | | |    |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+    |
| | | | | | | | |*|*|*|*| | | | | | | | a pixels
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+    |
| | | | |*|*|*|*| | | | | | | | | | | |    |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+    |
|*|*|*|*| | | | | | | | | | | | | | | |    v
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 <-------------- b pixels ----------->

For each step in the b direction, the next pixel to be plotted is either immediately to the right, or one pixel up and to the right.

The ideal line from one end to the other can be approximated by the path which joins the centre of each pixel to the centre of the adjacent one. This is a series of a segments of length sqrt(2), and b-a segments of length 1 (taking a pixel to be the unit of measurement). Hence the above formula.

This clearly gives an accurate answer for a == 0 and a == b; but gives an over-estimate for values in between.

The error depends on the ratio b/a; the maximum error occurs when b = (1 + sqrt(2)) * a and turns out to be 2/sqrt(2+sqrt(2)), or about 8.24% over the true value. That's not great, but if it's good enough for your application, this method has the advantage of being simple and fast. (The multiplication by a constant can be written as a sequence of shifts and adds.)