Editing Geometric mean (section)

===Aspect ratios===
[[File:Dr. Kerns Powers, SMPTE derivation of 16-9 aspect ratio.svg|thumb|right|Equal area comparison of the aspect ratios used by Kerns Powers to derive the [[SMPTE]] [[16:9]] standard.<ref name="Cinemasource" /> {{color box|red}}{{nbsp}}TV 4:3/1.33 in red, {{color box|orange}}{{nbsp}}1.66 in orange, {{color box|blue}}{{nbsp}}'''16:9/1.7{{overline|7}} in blue''', {{color box|#aaaa00}}{{nbsp}}1.85 in yellow, {{color box|mauve}}{{nbsp}}[[Panavision]]/2.2 in mauve and {{color box|purple}}{{nbsp}}[[CinemaScope]]/2.35 in purple.]]
The geometric mean has been used in choosing a compromise [[aspect ratio (image)|aspect ratio]] in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally. Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has the aspect ratio of their geometric mean.

In [[16:9 aspect ratio#History|the choice of 16:9]] aspect ratio by the [[SMPTE]], balancing 2.35 and 4:3, the geometric mean is <math display="inline">\sqrt{2.35 \times \frac{4}{3}} \approx 1.7701</math>, and thus <math display="inline">16:9 = 1.77\overline{7}</math>... was chosen. This was discovered [[Empirical evidence|empirically]] by Kerns Powers, who cut out rectangles with equal areas and shaped them to match each of the popular aspect ratios.  When overlapped with their center points aligned, he found that all of those aspect ratio rectangles fit within an outer rectangle with an aspect ratio of 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1.<ref name="Cinemasource">{{cite web |url=http://www.cinemasource.com/articles/aspect_ratios.pdf#page=8 |title=TECHNICAL BULLETIN: Understanding Aspect Ratios |publisher=The CinemaSource Press |year=2001 |access-date=2009-10-24 |url-status=live |archive-url=https://web.archive.org/web/20090909132530/http://www.cinemasource.com/articles/aspect_ratios.pdf#page=8 |archive-date=2009-09-09 }}</ref> The value found by Powers is exactly the geometric mean of the extreme aspect ratios, [[4:3]]{{nbsp}}(1.33:1) and [[CinemaScope]]{{nbsp}}(2.35:1), which is coincidentally close to <math display="inline">16:9</math> (<math display="inline">1.77\overline{7}:1</math>). The intermediate ratios have no effect on the result, only the two extreme ratios.

Applying the same geometric mean technique to 16:9 and 4:3 approximately yields the [[14:9]] (<math display="inline">1.55\overline{5}</math>...) aspect ratio, which is likewise used as a compromise between these ratios.<ref>{{cite patent | title = Method of showing 16:9 pictures on 4:3 displays | country = US | number = 5956091 | gdate = September 21, 1999 }}</ref> In this case 14:9 is exactly the ''[[arithmetic mean]]'' of <math display="inline">16:9</math> and <math display="inline">4:3 = 12:9</math>, since 14 is the average of 16 and 12, while the precise ''geometric mean'' is <math display="inline">\sqrt{\frac{16}{9}\times\frac{4}{3}} \approx 1.5396 \approx 13.8:9,</math> but the two different ''means'', arithmetic and geometric, are approximately equal because both numbers are sufficiently close to each other (a difference of less than 2%).