关于平移迪潘圆纹面（Dupin cyclide）圆形孔洞且维持曲线无变形的技术咨询

Hey there! I totally get your frustration with trying to shift the circular hole of a Dupin cyclide without ending up with those annoying elliptical deformations—tweaking parameters only to mess up the shape is such a headache. Let’s walk through the right way to do this without breaking the geometry.

First, let’s clear up why adjusting the original parameters (like a, b, c, d) leads to deformed holes: those values control the intrinsic shape of the cyclide (hole size, thickness of the torus-like ring, etc.), not its position. Messing with them changes the curvature of the surface’s cross-sections, which turns your perfect circular hole into an ellipse.

The solution here is to use a rigid translation transformation—this moves the entire cyclide (and its hole) to your desired position without altering any of its geometric properties. Here’s how to implement it:

• Decide your translation vector: let’s say you want to shift along the x-axis by h units, y-axis by k units, and z-axis by m units. If you only need to shift along x or y, set the other values to 0.
• Apply this shift directly to each component of the original parametric equations. Instead of modifying the shape parameters, you simply add your translation offset to each coordinate.

Your adjusted parametric equations (for shifting x by h and y by k, for example) would look like this:

\begin{align}
x'(u,v) &= \frac{d(c - a\cos u \cos v) + b^2 \cos u}{a - c \cos u \cos v} + h, \
y'(u,v) &= \frac{b\sin u (a-d\cos v)}{a - c \cos u \cos v} + k,\
z'(u,v) &= \frac{b\sin v (c\cos u - d)}{a - c \cos u \cos v}.
\end{align}

Quick Tips to Avoid Deformation: #

• Always stick to the standard Dupin cyclide parameter relationship: (c^2 = a^2 - b^2). If this doesn’t hold, your surface isn’t a true Dupin cyclide, and the hole might be elliptical even before translation.
• Keep your parameter ranges for u and v as ([0, 2\pi]) to ensure you’re generating the full, closed surface.

This method will keep your hole perfectly circular while moving it exactly where you want it along the x or y axis—no more tedious adjustments to fix unwanted shape changes!

备注：内容来源于stack exchange，提问作者kuu huu