Your logic is correct; I was confusing the problem with something else. Now, I can't remember just what that something else was! What made it harder to visualize is that the range would run from 0 to 1 either way, as the angle changes from 0 to 90 degrees.
I did find a nice little java applet that illustrates the situation: http://andrewmarsh.com/blog/2010/02/01/cosine-law-and-surface-incidence Dave -----Original Message----- From: Roger Bailey [mailto:rtbai...@telus.net] Sent: Friday, January 03, 2014 8:41 PM To: David Bell Cc: Marcelo; Sundial List Subject: Re: Garden planning problem Hi Dave, My initial recollection was of a cosine effect. So I drew a little sketch to clarify the situation. I specified the angle as the elevation, the altitude measured from the horizontal. It would be the cosine for the angle measured from the zenith. The area of the direct projection is Pi R squared. The area of the ellipse on the projected on the ground is Pi R times the semi-major axis. This is R/ Sin Alt so the area is Pi R squared/Sin Alt. I considered the illumination inversely proportional to the area so directly related to the sine of the elevation, the altitude angle. QED. or is my logic wrong? At sunrise the elevation is zero, sine =0, the intensity is zero. Directly above the elevation is 90° and Sine 90 = 1. The intensity is full, undiminished by spreading over a larger area. Radiation from the sun follows the inverse square law. Twice the distance from the sun gets 1/4 the intensity but that is not the effect being discussed in this "gray posting". Regards, Roger -------------------------------------------------- From: "David Bell" <db...@thebells.net> Sent: Friday, January 03, 2014 2:06 PM To: "Roger Bailey" <rtbai...@telus.net> Cc: "Marcelo" <mmanil...@gmail.com>; "Sundial List" <sundial@uni-koeln.de> Subject: Re: Garden planning problem > One thought on that gray posting, Roger: > I may remember incorrectly, but I thought illuminance on a surface was > proportional to the square of the cosine of the incidence angle. > > Dave > > Sent from my iPhone > > On Jan 2, 2014, at 8:07 PM, "Roger Bailey" <rtbai...@telus.net> wrote: > >> Hello Marcelo, >> >> Many on this list empathize with your problem. We know what we want to do >> but the math is unfamiliar. In reality the trigonometry here is very >> simple, as you have laid out the problem. The ratio of the height of the >> shadow caster, G to the shadow length, L is the Tangent of the altitude, >> H. Tan H = G/L. Or rearranging L = G/Tan H. The shadow length is equal >> to the height of the shadow caster divided by a simple number. the >> Tangent of the solar altitude angle H. >> >> This assumes you know the altitude angle. At solar noon when the sun is >> on the meridian, this is an easy calculation as the Noon altitude equals >> the co-latitude minus the solar declination or H = 90-Lat-Dec. >> >> This assumes you know your latitude and solar declination. Latitude is >> easy from maps, websites, GPS etc. Solar declination is not as quite as >> easy but many tables, almanacs, programs and websites can give it to you. >> Google solar declination. >> >> What if it is not noon? The altitude and azimuth are still relatively >> easy to calculate using the classical formulae of spherical trigonometry >> used by navigators with sextants. Sin Sin Sin Cos Cos Cos is the first >> equation to know. Sin H = Sin Dec x Sin Lat + Cos Dec x Cos Lat x Cos t. >> Input your latitude, declination and time as an angle from noon to >> calculate H, the altitude angle that determines the shadow length. These >> intimidating trig expressions are just numbers, simple numbers that you >> can add, subtract, multiply and divide. >> >> But have you considered the Sine effect of the incident light? Light >> straight down on a surface such as a flowers leaves is fully effective. >> As the angle tilts from straight down to a lower angle, the effective >> incident light is diminishes. How much? By the Sine of the altitude. >> Straight on the altitude is 90° and Sin 90° = 1. At altitude = 45°, Sin >> 45 = 0.707, so the light is 70% as intense. At 30° altitude, the >> intensity is halved as Sin 30 = 0.5. >> >> Many on this mailing list have found the a little geometry, trigonometry >> and even spherical tri can be very useful in solving problems like yours. >> >> Regards, Roger Bailey >> -------------------------------------------------- >> From: "Marcelo" <mmanil...@gmail.com> >> Sent: Thursday, January 02, 2014 11:37 AM >> To: "Sundial List" <sundial@uni-koeln.de> >> Subject: Garden planning problem >> >>> Hello fellow dialists, how are you? >>> I'm with a problem here which doesn't concern exactly to sundials, but >>> since it deals with sun's position and his shadows, I couldn't think >>> of anyone better than you to help me. >>> I have a little garden here at home, a walled area where I grow some >>> plants in pots. I've found that, depending on the place, teher's a >>> difference greater than 2.5 hours in the sunlight a plant receives, >>> and that affects greatly its development. >>> I've measured the shadows cast before and after true noon during >>> summer solstice (I live slightly south to the Tropic of Capricorn). I >>> could repeat the process during equinox and winter solstice, but >>> that's a boring task, and above all, if the weather is cloudy I'll >>> miss the chance. >>> So, can you tell me some trigonometrical method for calculating the >>> shadows, using sun's altitude and azimuth? I couldn't devise one by >>> myself. >>> Thanks in advance, and Happy New Year! >>> >>> Marcial >>> --------------------------------------------------- >>> https://lists.uni-koeln.de/mailman/listinfo/sundial >>> >>> >>> >>> ----- >>> No virus found in this message. >>> Checked by AVG - www.avg.com >>> Version: 2014.0.4259 / Virus Database: 3658/6969 - Release Date: >>> 01/02/14 >> >> >> ----- >> No virus found in this message. >> Checked by AVG - www.avg.com >> Version: 2014.0.4259 / Virus Database: 3658/6971 - Release Date: 01/02/14 >> >> --------------------------------------------------- >> https://lists.uni-koeln.de/mailman/listinfo/sundial >> > > > ----- > No virus found in this message. > Checked by AVG - www.avg.com > Version: 2014.0.4259 / Virus Database: 3658/6973 - Release Date: 01/03/14 > > ----- No virus found in this message. 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