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# Helical coordinate on a cone

I could not find the equations for a helix on a cone, so I derive them below. An example of results is Figure 2.

trycart
Figure 2
Helix on a cone. Top shows the embedded helix. Bottom hides it. Apex angle .

Define to be the angle from the axis of the cone to its surface. I call this the apex angle. I discovered by accident that certain apex angles give interesting patterns in the crossline directions while most do not. I found these patterns to be insensitive to the choice of N and show them here for about N=1400. Another value of apex angle with an interesting pattern is .It gives the charming pattern in Figure 3. (I've seen this pattern before on a party hat. I attributed it to an ingenious artist. Now I realize that like all mathematics, this art existed before the big bang.)

YinYang
Figure 3
Helix on a cone. Top shows the embedded helix. Bottom hides it. Apex angle equals one third radian.

The radius r of the cone divided by its altitude z has a ratio where is the angle from the axis of the cone to its surface. The area of a circle is .The surface area of a cone is .Dividing the area into N cells, the surface area per cell is .Taking any cell area to be square, the length of a side is
 (3)
The number of points running one cycle around a rim is
 (4)
On one cycle around the rim, the radius must change by where the hypotenuse of the triangle lies on the surface of the cone. Thus, for each mesh point going around the rim
 (5)
Since we want square'' mesh points, must equal , hence
 (6)
The algorithm starts on the rim at and r=R. At each step, update r and with and .Stop before becomes negative.

Next: FILLING VOLUMES Up: Claerbout: Helical meshes on Previous: Helical coordinate on a
Stanford Exploration Project
4/20/1999