[Tom McC]

Originally Posted by didier.brest

Just a bit of trigonometry:



thumbsup

Didier, we risk changing this into Geekslutz <g>.
It's late in the day to discuss your maths line by line, but let me instead return to your post #49 and make some points:

In that post you had:

a) M pattern = 1 + cosine(angle) <= agreed
b) S pattern = sine(angle) <= agreed
c) X pattern = 1 + cosine(angle + 45°) <= No. 1 + cos(angle - 45°)
d) Y pattern = 1 + cosine(angle - 45°) <= No. 1 + cos(angle - 315°) = 1 + cos(angle + 45°)

Assuming the usual convention that the angle starts at straight ahead ("North", say), increases in counterclockwise direction, and that the X mic represents the "North-West" pointing one. But, what is the point of tussling with the mapping formula for the 1 + cos cardioid patterns for the angled X and Y mics, for we have already found that a virtual mic at 45° won't be inherting a cardioid pattern, anyway?.

Look, "the proof of the pudding is in the eating thereof" <g>, so I would urge you to try the following: take equations (a) and (b) - that we all agree are OK - and compute "M + S" on a spreadsheet for a range of angle values. I would suggest displaying angle values in columns at 5 deg. intervals (thus +90, +85, 80, 75, 70....5, 0, -5, -10, -15°..etc.).

By inspection you can locate the maximum value that M+S takes. This represents the angle that the main axis of the virtual mic lies on, or in other words the half-included-angle. You'll discover that it lies between 60 and 65°, and indeed is the 63.43° as calculated by Dooley, Streicher and lil 'ol me <g>. At the same time, you could convert the obtained M+S values from voltage into dB, to get a handle on the new polar pattern.

And by then I'll have gone through your maths.

regards,