Crop spirals are a form of geoart in which patterns are drawn in fields of grain. They are created by flattening some of the grain stalks. Many people believe that very intelligent aliens are using the earth as some sort of etch-a-sketch employing magnetic fields, or gravitational fields. Their alleged purposes are said to be for their own amusement and to impress one another and us by their clever designs. We now know that is not all. An ulterior purpose is to hide geocaches.
Take this cache for example. A few nights ago, Involute from the planet Circle (pronounced "Sir Clay") in the Spiral Galaxy landed his flying saucer at the given coordinates. The circular spacecraft was centered on the posted coordinates and had a radius of M = 108 meters. A very thin, very long, very strong filament was wrapped clockwise around the edge of the craft such that its free end terminated at the eastern edge of the craft by the entrance/exit hatch. He planted his cache exactly s = 5877 meters away from this point unaware that the actual cache must be within 3200 meters of the posted coordinates . Fortunately, Circleans have a unique way of measuring distance which is along spirals - in this case the distance traveled to place the cache. Involute proceeded to the cache site by unwinding the filament counterclockwise always keeping the filament taunt and flattening the grain with his tri-peds as he crept along. That's all you really need to know to determine the location of the cache. Got the picture? Well, here it is.
Let O denote the point at the origin (0, 0), i.e. the center of the circle; E the entrance/exit where the end point is originally located ; C a point on the circle where the filament is always tangent; and P(x,y) a corresponding point on the spiral. The lengths of OE and OC are both M. Similarly the lengths of the arc CE as well as the tangent CP are both t M, where t is the angle between OE and OC. With a little bit of math (see keywords to search below), one finds that in terms of M (in meters) and the angle t (naturally in radians), the point (x,y) and the length of the spiral are given by the formulae:
x = M ( cos t + t sin t )
y = M (sin t - t cos t )
s = (M/2) t2
The figure at the left below shows the situation after one revolution, i.e. when t reaches 2 Pi (360°); and the figure on the right shows what the spiral looks like after four revolutions of t.
The figure below shows what the field will look like after Involute genetically modifies the grains.
Keywords: crop spirals, involute of a circle.
Verification: If rounded, the local rectangular coordinates x and y will both end in 0. Thus when (x, y) is offset from the posted VK coordinates, 80975, 20590, the easting will still end in 5 and the northing in 0. Thus your solution to be entered into the geochecker is of the form VKabcd5fghi0. If successful you will be given the lat/lon coordinates as well as a hint to the hide. Please note that to reach the cache you do not need to cross any fields or fences, not even any ditches. You will only trample a few weeds.
You can validate your puzzle solution with certitude.
Congrats to the first ten solvers:
1. ctc128 Mon, 12 Jun 2017 7:25:47
2. mnboyscout Mon, 12 Jun 2017 11:01:30
3. foundinthewild Mon, 12 Jun 2017 12:09:44
4. rickrich Mon, 12 Jun 2017 12:31:50
5. pfalstad Mon, 12 Jun 2017 13:29:17
6. salsman Mon, 12 Jun 2017 15:08:55
7. PackADad Fri, 15 Sep 2017 21:56:48
8. rohrerhj Thu, 12 Oct 2017 19:24:33
9. pcc322 Thu, 2 Nov 2017 5:59:54
10. PA-20 Tue, 28 Nov 2017 15:26:58