Graphing Gateway Arch Shadow

We, in Midwest America, in Missouri, in Saint Louis are in the midst of a heat wave. This week in July has had a sequence of very hot days — the mornings are hot and humid, the afternoons are like ovens, and the temperature doesn’t let up at night. I got an exposure to this kind of weather when I lived in Nevada. You can  perhaps begin to see my preoccupation with the Sun. And my mission to find some shade, a shadow for some relief. And also, they put those amps and big speakers where I used to play my guitar. So, I have been playing my guitar in the shadow of the Gateway Arch.

Some people developed an elementary mathematical equation for the shape of the Gateway Arch in St Louis. I found a formula at this website.  

I don’t know if this formula was used in the design and construction of the Arch, or was it developed later.

In Maxima, one can type in,

f(x):=211.49 – 20.96*cosh(0.03291765*x);

The units for x are meters, and x ranges from -91.2 to +91.2.

I can calculate a plot of the shadow, if I consider an X-Y-Z coordinate system, with Z up, Y positive pointing East, and X positive pointing South. At the Park, East is towards Illinois across the Mississippi river, and South is downstream of the Mississippi.

(%i5) h(x):=f(x)/tan((%pi/180)*el)

(%i6) sx(x):=x-h(x)*sin((%pi/180)*az)

(%i7) sy(x):=h(x)*cos((%pi/180)*az)

The units for the Sun’s azimuth az and elevation el are in degrees.

Here I used az = -10, and el = 45

north leg of gateway arch
north leg of gateway arch
South Leg Gateway Arch
South Leg Gateway Arch July 2023

How fast does the shadow move across the sidewalk? From when I start to play at about 10:30 to when I finish playing at about 11:30, the shadow on the south leg has moved about one tree. I stand at the sidewalk border next to the trees.

Another plot I made with az = -30 and el = 65.

gateway arch shadow
Gateway Arch shadow

This chart might be more accurate as I am using the Sun position charts I calculated

Saint Louis Sun Chart

Azimuth az = 0 would correspond I think perhaps to az = 90 in the Saint Louis Sun chart. Here’s a plan map of the grassy knoll and the old Court House and the Memorial Ave freeway overpass. I stand at number 13 when I play my guitar.

Gateway Arch Map
Gateway Arch Map

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