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Plotting 3D Surface Intersections as 3D Curves

Posted 10 years ago
As a project to brighten my retirement, I am trying to achieve the following and need some (a lot of) help. Given two 3D surfaces expressed as two 3D equations whose domains cause the 2 surfaces to intersect, plot the resulting 3D curve of intersection. My simplest example has one of the surfaces as a reference sphere and the other as a circular cylinder whose size and location can be adjusted for study. The final plot would show only the reference sphere with the curve of intersection on its surface.

Someone has probably done this already, but I can't seem to find it anywhere in the Community or the Virtual Book. There is something called a Viviani Curve in Wolfram Alpha which comes close, but I cannot understand their notation.

Any help would be greatly appreciated.
Thanks, Ed

4 Replies
Posted 5 months ago

Ed, I found your ten-year-old message by searching the forum archives on the curve name. I hope you're still active in the community discussions.

This French webpage has an investigation of Viviani's Curve. It shows a variety of intersections of 3D surfaces that can generate the curve. Here's a sample from the webpage: enter image description here

The page is in French; current MacOS Safari will translate it to English.

I also have generated the curve in the Wolfram Language with a parametric plot. This is part of a project to visualize and understand RMT Ropes (AKA "Flow Rope"): an interesting way to move (but not jump) with a jump rope:

@Arben Kalziqi : This is what your were helping me create during the Wolfram Language study group earlier in 2023. I used Manipulate[] to find appropriate offsets for the motions in x/y/z. Thank you!

POSTED BY: Phil Earnhardt

Very neat!! Glad I could help :)

POSTED BY: Arben Kalziqi
Here is a nice approach using ContourPlot3D (source):

ContourPlot3D[{x^2 + y^2 + z^2 - 4, (x - 1)^2 + y^2 - 7/8}, {x, -2, 2}, {y, -2, 2}, {z, -2, 2},
     ContourStyle -> {Automatic, Opacity[0]}, Mesh -> None,
     BoundaryStyle -> {2 -> None, {1, 2} -> {Green, Thick, Dashed}}]

POSTED BY: Ilian Gachevski
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