Here's a first version, the meeting point in the middle lokks still bad, but I'm working on it.
\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}
\usepackage{ifthen}
\newcommand{\cucubr}[7]{%
%origin point, circle radius, start angle, end angle, distance c-b, brace radius, brace options
\pgfmathsetmacro{\helpangleedge}{atan(#6/(#2+#5+#6))}%
\pgfmathsetmacro{\helpanglemid}{atan(#6/(#2+#5+2*#6))}%
\pgfmathsetmacro{\halfangle}{(#4-#3)/2+#3}%
\pgfmathsetmacro{\innerradius}{#2+#5}%
\pgfmathsetmacro{\midradius}{#2+#5+#6}%
\pgfmathsetmacro{\outerradius}{#2+#5+2*#6}%
\pgfmathsetmacro{\firstmidanglestart}{mod(\halfangle-\helpanglemid+180,360)}%
\pgfmathsetmacro{\secondmidanglestart}{mod(\halfangle+\helpanglemid+180,360)}%
\pgfmathsetmacro{\firstmidanglestop}{mod(\halfangle-\helpanglemid+180,360)-90}%
\pgfmathsetmacro{\secondmidanglestop}{mod(\halfangle+\helpanglemid+180,360)+90}%
%
\draw[#7] (#1) ++ (#3:\innerradius) arc (#3+270:#3+360:#6) arc (#3+\helpangleedge:\halfangle-\helpanglemid:\midradius) arc (\firstmidanglestart:\firstmidanglestop:#6);%
\draw[#7] (#1) ++ (#4:\innerradius) arc (#4+90:#4:#6) arc (#4-\helpangleedge:\halfangle+\helpanglemid:\midradius) arc (\secondmidanglestart:\secondmidanglestop:#6);% node[black, text width=2.5cm,draw] {mid:\firstmidanglestart, one:\firstmidanglestop, two:\secondmidanglestop, half:\halfangle};%
}
\begin{document}
\begin{tikzpicture}
\draw (0,0) -- (0:4) arc (0:90:4) -- cycle;
\draw (0,0) -- (90:4) arc (90:180:4) -- cycle;
\draw (0,0) -- (180:4) arc (180:270:4) -- cycle;
\draw (0,0) -- (270:4) arc (270:360:4) -- cycle;
\cucubr{0,0}{4}{0}{90}{0.4}{0.5}{thick,red}
\cucubr{0,0}{4}{90}{180}{0.9}{1.0}{thick,green}
\cucubr{0,0}{4}{180}{270}{0.2}{0.3}{thick,blue}
\cucubr{0,0}{4}{270}{360}{0.6}{0.6}{thick,teal}
\end{tikzpicture}
\end{document}

Edit 1: I got a much better variant than before, it involves more math now. While this should be correct, I still got variations, so the brace would't always close in the middle. I think my math is correct, and I ran into some computation deviations in TikZ: the angles computed by TikZ vary by up to 0.35 degrees from the real. While this does does not seem much, it was up to 10% of the angle, or the equivalent of 1.5 linewidths. As I found no way around, I decided to start drawing from the center. That way the brace ends meet in the middle, and I only needed to adapt one value due to deviations. Here's the final code:
\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}
\newcommand{\cucubr}[7]{%
%origin point, circle radius, start angle, end angle, distance c-b, brace radius, brace options
\pgfmathsetmacro{\helpangleedge}{acos(1-pow(#6,2)/2/pow(#2+#5,2))}%
\pgfmathsetmacro{\turnangleedge}{90+(\helpangleedge/2)}%
\pgfmathsetmacro{\helpanglemid}{acos(1-pow(#6,2)/2/pow(#2+#5+2*#6,2))}%
\pgfmathsetmacro{\turnanglemid}{90-(\helpanglemid/2)}%
\pgfmathsetmacro{\halfangle}{(#4-#3)/2+#3}%
\pgfmathsetmacro{\midradius}{#2+#5+#6}%
\pgfmathsetmacro{\outerradius}{#2+#5+1.88*#6}%
\pgfmathsetmacro{\firstmidanglestart}{mod(\halfangle-\helpanglemid+180,360)}%
\pgfmathsetmacro{\secondmidanglestart}{mod(\halfangle+\helpanglemid+180,360)}%
\pgfmathsetmacro{\firstmidanglestop}{mod(\halfangle-\helpanglemid/2+180,360)-\turnanglemid}%
\pgfmathsetmacro{\secondmidanglestop}{mod(\halfangle+\helpanglemid/2+180,360)++\turnanglemid}%
%
\draw[#7] (#1) ++ (\halfangle:\outerradius) arc (\firstmidanglestop:\firstmidanglestart:#6) arc (\halfangle-\helpanglemid:#3+\helpangleedge:\midradius) arc (#3+270+\turnangleedge+\helpangleedge/2:#3+270+\helpangleedge/2:#6) ;%
%
\draw[#7] (#1) ++ (\halfangle:\outerradius) arc (\secondmidanglestop:\secondmidanglestart:#6) arc (\halfangle+\helpanglemid:#4-\helpangleedge:\midradius) arc (#4+90-\turnangleedge-\helpangleedge/2:#4+90-\helpangleedge/2:#6);%
}
\begin{document}
\begin{tikzpicture}
\draw (0,0) -- (00:4) arc (0:90:4) -- cycle;
\draw (0,0) -- (90:4) arc (90:180:4) -- cycle;
\draw (0,0) -- (180:4) arc (180:270:4) -- cycle;
\draw (0,0) -- (270:4) arc (270:360:4) -- cycle;
\cucubr{0,0}{4}{0}{90}{0.4}{0.5}{red,densely dotted}
\cucubr{0,0}{4}{90}{180}{0.9}{1.0}{thin,green}
\cucubr{0,0}{4}{180}{270}{0.2}{0.3}{ultra thick,blue,densely dashed}
\cucubr{0,0}{4}{270}{360}{0.6}{0.6}{thick,teal,-latex}
\end{tikzpicture}
\end{document}

Edit 2: I double and tripple checked the math, it really, really should be correct. So I started investigating the deviations when calculating. The TikZ manual states that the power and accuracy of it's engine are limited due to limitations of TeX. Sadly, this is the first time I ran into them. I calculated the values of the angles I needed for different values of circle radius and brace radius:

The values should decrease left to right and top down, but for small radii they don't. As this limits the accuracy achievable, I decided to choose yet another approach: starting the brace from inside and outside, and then connecting the pieces via to[out=<degrees>,in=<degrees>]
. This looks well enough, at least if you don't choose a lage brace radius for a small circle radius.
\documentclass[parskip]{scrartcl}
\usepackage[margin=15mm]{geometry}
\usepackage{tikz}
\usetikzlibrary{calc}
\pgfkeys{/tikz/savenumber/.code 2 args={\global\edef#1{#2}}}
\newcommand{\cucubr}[7]{%
%origin point, circle radius, start angle, end angle, distance c-b, brace radius, brace options
\pgfmathsetmacro{\helpangleedge}{acos(1-pow(#6,2)/2/pow(#2+#5,2))}%
\pgfmathsetmacro{\turnangleedge}{90+(\helpangleedge/2)}%
\pgfmathsetmacro{\helpanglemid}{acos(1-pow(#6,2)/2/pow(#2+#5+2*#6,2))}%
\pgfmathsetmacro{\turnanglemid}{90-(\helpanglemid/2)}%
\pgfmathsetmacro{\halfangle}{(#4-#3)/2+#3}%
\pgfmathsetmacro{\innerradius}{#2+#5}%
\pgfmathsetmacro{\midradius}{#2+#5+#6}%
\pgfmathsetmacro{\outerradius}{#2+#5+2*#6-sqrt(1/#2)*0.01}%
\pgfmathsetmacro{\firstmidanglestart}{mod(\halfangle-\helpanglemid+180,360)}%
\pgfmathsetmacro{\secondmidanglestart}{mod(\halfangle+\helpanglemid+180,360)}%
\pgfmathsetmacro{\firstmidanglestop}{mod(\halfangle-\helpanglemid/2+180,360)-\turnanglemid}%
\pgfmathsetmacro{\secondmidanglestop}{mod(\halfangle+\helpanglemid/2+180,360)+\turnanglemid}%
%
\draw[#7] (#1) ++ (\halfangle:\outerradius) arc (\firstmidanglestop:\firstmidanglestart:#6) node (A) {};%
\draw[#7] (#1) ++ (#3:\innerradius) arc (#3+270+\helpangleedge/2:#3+270+\turnangleedge+\helpangleedge/2:#6) node (B) {};%
%
\draw[#7] (#1) ++ (\halfangle:\outerradius) arc (\secondmidanglestop:\secondmidanglestart:#6) node (C) {};%
\draw[#7] (#1) ++ (#4:\innerradius) arc (#4+90-\helpangleedge/2:#4+90-\turnangleedge-\helpangleedge/2:#6) node (D) {};%
%
\draw[#7] (A.center) to[out=\firstmidanglestop+180,in=#3+90+\helpangleedge/2] (B.center);%
\draw[#7] (C.center) to[out=\secondmidanglestop-180,in=#4-90-\helpangleedge/2](D.center);%
}
\begin{document}
\begin{tikzpicture}
\draw[very thin,gray,densely dotted] (0,0) -- (20:7);
\draw[very thin,gray,densely dotted] (0,0) -- (80:7);
\draw[very thin,gray,densely dotted] (0,0) -- (150:7);
\foreach \r in {1,...,6}
{ \cucubr{0,0}{\r}{20}{80}{0.2}{0.3}{blue}
\cucubr{0,0}{\r}{80}{150}{0.5}{0.5}{red}
}
\end{tikzpicture}
\end{document}

As you can see, the smaller braces look "staircasey", but this vanishes with higher ratios of circle radius to brace radius.
