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In classes like Calc 3 or Computer Graphics, I want my students to be comfortable describing common curves and surfaces parametrically (such as lines, triangles, circles, or surfaces of revolution). What are some good resources for teaching these skills?

Edit: Specifically, if I had a particular shape in mind (like a Mobius strip, a helix, etc.) I could sit down and work out a parameterization for that shape. But I can't articulate my general process. I can describe how I got any specific parameterization, but I'm looking for a resource that describes the general skill. Most of the resources I've found (including calculus texts) tend to just give examples and (if one is lucky) describe how those specific examples work.

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    $\begingroup$ Lines, circles, and surfaces of revolution are pretty well covered by most standard calculus texts, such as Stewart. (Unless you want something non-obvious.) Polygons can be a bit more tricky, depending on what you want out of the parameterization. eg. you can be differentiable or arc length parameterized, but not both. $\endgroup$
    – Adam
    Commented Apr 11, 2022 at 13:18
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    $\begingroup$ Iñigo Quílez's Painting with Maths explains CG in terms of Calc 3 It's amazing! but it doesn't explain parametrized surfaces. It's more like $z=f(x,y)$. Watching the videos makes you happy you took Calc 3. $\endgroup$
    – user1815
    Commented Apr 14, 2022 at 2:24
  • $\begingroup$ That painting with maths video is fantastic! $\endgroup$
    – TomKern
    Commented Apr 16, 2022 at 21:50

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An introduction to parametrized surfaces, from the "Math Insight" website, is short but quite clear, with some nice 3D rotatable images.

     

     A parametrized helicoid.

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