Intuitively, what is the general overlap/difference between conformal vs orthogonal transformations, or the terms in general?

Question

I've been having difficulty finding a clear definition on the differences between the two in practical/geometric terms. Orthogonal transformations being those which the coordinate surfaces or trajectories meet at right angles, and conformal transformations being those which preserve angles.

I can see how the notions overlap, and have a vague intuition on how they are different, but I am having trouble clarifying their exact distinction, specifically in the context of differential/vector calculus with respect to concepts like the Jacobian and it's area preserving properties, differential equations for orthogonal trajectories, integral transforms, etc.

Or in more direct terms, when is something orthogonal but not conformal, and vice versa, and when are they both?

Answer 1

A conformal linear map is the composition of a homothety (stretch) and an orthogonal linear map.

Answer 2

The most important part of the intuition is this: Special orthogonal transformations are rotations. Orthogonal transformations are rotations plus reflections. Conformal transformations are rotations plus dilations. Conformal and anticonformal transformations are rotations plus dilations plus reflections.

Mathematically speaking, this means: Orthogonal transformations preserve the scalar product. Special orthogonal transformations also preserve orientation (positive determinant). Conformal and anticonformal transformations preserve angles. Conformal transformations also preserve orientation (positive determinant). More precisely, orthogonal transformations $T$ satisfy

$$\langle Tv,Tw\rangle=\langle v,w\rangle,$$

while special orthogonal transformations additionally satisfy

$$\det T>0.$$

It can even be shown that orthogonal transformations already satisfy $\det T=\pm1$, making $\det T=1$ for special orthogonal transformations. Conformal and anticonformal transformations $S$ satisfy

$$\frac{\langle Sv,Sw\rangle}{\Vert Sv\Vert\Vert Sw\Vert}=\frac{\langle v,w\rangle}{\Vert v\Vert\Vert w\Vert},$$

(for $v,w\neq0$) while conformal maps additionally satisfy $\det S>0$. It can be shown that this makes the (anti)conformal transformations equal to orthogonal maps multiplied by a nonzero constant. (Anti)conformal transformations are thus orthogonal transformations with an added dilation. If we call the various groups containing these transformations $\operatorname{O},\operatorname{SO}$ (orthogonal and special orthogonal), $\operatorname{CO}$ (conformal plus anticonformal), and $\operatorname{CSO}$ (just conformal), then we have the following relations:

$$ \operatorname{SO}\subsetneq\operatorname{O}\subsetneq\operatorname{CO}\\ \operatorname{SO}\subsetneq\operatorname{CSO}\subsetneq\operatorname{CO}\\ \operatorname{CO}=I\cdot\operatorname{O}\\ \operatorname{CSO}=I\cdot\operatorname{SO},$$

where $I$ is the group of dilations.