# Re: Circle of Confusion Question

From: Loris Medici ^lt;[email protected]>
Date: 08/08/04-07:05:30 PM Z
Message-id: <001001c47dac\$f85dcbc0\$bd02500a@Loris>

1 degree = 60 minutes, when I check for tan(1 / 60) using the calculator
provided with my operating system, indeed the result is 0.00029... I
crosschecked the result by finding arctan(0.00029) * 60 = 0.9960... :) (it's
not exactly 1 because tan(1 / 60) is not exactly 0.00029) So there's no
error in the claim "tangent of one minute of arc is 0.00029".

BUT:

linear measurement 70 microns = 0.07mm = 0.007cm
distance = 25cm
l = d * tan(a)
0.007 = 25 * tan(a)
tan(a) = 0.007cm / 25cm = 0.00028 (<-- I guess this is exactly what raises
the confusion!?)
arctan(0.00028) = 0.016042... * 60 = 0.9625

This is not 1. I guess Williams' definition of "rough" is 3.75% tolerance
range :)

FWIW, I personally use a CoC value according to the planned enlargement
factor. I aim for a final resolution of 8 lpmm on the print because I have
read somewhere (don't remember) that this is the approx. resolution limit
"for most of us" ("at minimum viewing distance"). This is quite
contradictory to the result from Williams (~14 lpmm @ 25cm viewing
distance)! No problem, I'm fine with this figure. So, if I want to make a
20x30cm print from a 35mm film frame, I first find the enlargement factor:
30 / 3.6 = 8.3, then I multiply this with 8 (because I want 8 lpmm in the
print) 8.3 x 8 = 66.4 lpmm. Therefore, the CoC that will use for 8.3
enlargement factor would be 1 / 66.4 = 15microns.

----- Original Message -----
From: "Sandy King" <sanking@clemson.edu>
Sent: Monday, August 09, 2004 2:53 AM
Subject: Circle of Confusion Question

> Let me send this again with a proper subject line.
>
> I am at something of a impasse with the following problem and hope
> that someone with a better understanding of mathematics can provide
> some assistance.
>
> The issues is this. I am trying to figure out how to convert angular
> measurements of point discrimination to linear measurements for the
> purpose of creating custom Circle of Confusion tables for minimum and
> critical thresholds of human vision for 1 minute of arc and 25
> seconds of arc. I would like to create the tables for a common
> observing distance of 25 cm, or 10 inches.
>
> John B. Williams, in Image Clarity: Theory of High-Resolution
> Photography, gives the following formula for the conversion of
> angular to linear measurements.
>
> l = Dtana
>
> Where l is the linear measurement, D is the observing distance and a
> is the angular size.
>
> He provides the answer for this equation for a distance of 25 cm as a
> linear measurement of 0.07mm, or 70 microns, and continues by noting
> that the tangent of one minute of arc is roughly 0.00029. I have
> worked this problem backwards several ways and still have not been
> able to figure out how he determined that the tangent of one minute
> of arc is 0.00029.
>
> Help appreciated from all competent persons.
>
> Sandy King
Received on Mon Aug 9 12:16:28 2004

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