A Golden Spiral Mystery Cache

A puzzle cache. The cache is NOT at the posted coordinates. Some elementary calculations may be required to determine its locations. As is usual in this series the VK notation is used and distances are in meters.

Here the spirals will be expressed in polar coordinates as was the case for other spirals in this series. In polar coordinates θ is the counterclockwise angle from the positive horizontal and r is the distance from the origin to the point on the curve. To convert from polar coordinates to the local rectangular system use
x = r cos θ, y = r sin θ.

Log spirals. The golden spiral is a special case of a logarithmic spiral. The simplest form of a log spiral is
r = e^θ. . Expressing the angle in terms of the distance we have θ = log r, where log is the natural log. Hence the name log spiral.

More generally a log spiral is of the form r = A e^bθ, where A and b are constants. The value of A determines the size of the spiral and b its shape, that is how tightly wound it is.

The golden spiral uses a special value of b, viz. b =ln( φ)/(π/2), where φ is the golden ratio, (1+√5)/2 and ln is the natural logarithm. Thus b = 0.30635, approximately. For each quarter turn of the angle the distance will grow exponentially by a factor of φ. This rate of growth occurs commonly in nature; thus the golden spiral can be observed in sunflowers, daisies, snails, etc..

Figure 1 on the left above shows one loop of the golden spiral for θ from 0 to 2π. The same shape would result for θ from 2π to 4π, or from -2π to 0 for that matter. Figure 2 on the left above shows these three loops from -2π to 4π. This self-replication of the golden spiral makes the mathematical formulas for its features relatively simple and easy to obtain. Some are given below.

r = A e^bθ

arclength = ( r / b) √(1+b²)

area =r²/(4b)

slope of tangent line, m = (Cos θ + b Sin θ )/(b Cos θ - Sin θ )

slope of normal line = -1/m

curvature = 1/R

radius of curvature, R = r √(1+b²)

The formulas for the arclength and area are for the angle going from -infinity up to θ. Note that the arclength and area are propotional to r and r², resp.

The slopes of the tangent line and the normal line (which is perpendicular to it) depend only on the shape of the curve and not its size.

Curvature Curvature is a measure of how fast a curve is turning. For a straight line the curvature is zero - i.e. there is no turning. For a circle the curvature is constant. Curvature is measured by the change in angle per unit length. Going entirely around a circle of radius R, the angle changes by 2π and the length by 2πR; thus the curvature is simply 1/R. This gives rise to the concept of the radius of curvature. For any sufficiently smooth curve, the radius of curvature is defined to be the reciprocal of the curvature. For the golden spiral,
r = A e^bθ, the radius of curvature R is simply √(1+b²) times r itself, i.e. R = r √(1+b²) . Then the curvature is 1/R. Consider Figure 3 on the left below

The circle of curvature, or osculating circle, for a point P on a curve, is the circle of radius R which matches the tangent and curvature of the curve at P. Its center is known as the center of curvature for P. In Figure 3 on the left below, the spiral is bold, the circle of curvature dashed.

So where is the cache? Consider Figure 3 again. The origin O(0,0) of the local coordinate system is at the posted coordinates, viz VK7261122459. The cache is located at the point P on the golden spiral. At that P the center of curvature is at C(222,555). That's enough information to determine the solution P.

But Wait! This problem may seem complicated, but as is often is the case in geometry something beautiful and simplifying happens. Here, it turns out, that for the golden spiral as well as in general for log spirals, the three points C, O and P form a right triangle. Given C(222 ,555 ), one can easily determine θ and r, and hence, P. (There's no need to find A.)

VerificationTo allow for some slight round-off for different ways to solve, in the checker the last digit of the easting is 3 and of the northing is 4. Express P as VKeeee3nnnn4 and enter it into the coordinate checker. If correct you will be given the lat/lon coords and a hint on the hide.

You can validate your puzzle solution with certitude.

>More than you need to know.

• Figure 4on the right above shows the radii of curvature for θ from 0 to 4π in increments of π/20. Note that the centers of curvature form a golden spiral themselves.
• The Fibonacci Spiral is an approximation to the golden spiral; it can be drawn as a series of quarter circles. Maybe it will be the subject for a future cache in this series (which is spiraling out of control.)

Keywords Logarithmic spiral, Golden Spiral, Fibonacci Spiral, curvature, osculating circle.

The first ten solvers:
1.      pfalstad        Mon, 27 Nov 2017 19:08:20
2.      foundinthewild  Tue, 28 Nov 2017 9:26:26
3.      rickrich        Tue, 28 Nov 2017 14:02:25
4.      salsman Thu, 30 Nov 2017 19:57:44
5.      ctc128  Thu, 30 Nov 2017 20:00:16
6.      PackADad        Fri, 1 Dec 2017 21:34:21
7.      pcc322  Mon, 1 Jan 2018 11:44:30
8.      rohrerhj        Thu, 12 Apr 2018 13:37:30
9.      kkmk    Thu, 30 Aug 2018 20:38:35
10.     Bigrock95       Mon, 10 Sep 2018 22:06:17