Gdtyjr

I am trying to replace the black plan with a segment of a sphere using tikz :

The sphere would be of centre H(Yi) and radius ||H(Yi) , H(Yi+1_0)|| on which will lie all the points from k=0 (green point) to k (blue point) (=> corresponding to an arc of a circle in the plan designated by the axis system in the upper left corner).

Any propositions using tikz to add a part of a sphere would be greatly appreciated. Thank you.

The purpose of this post is to define a style that draws a part of a sphere. More specifically, it will draw a segment in which the latitude and longitude are in a certain range. Drawing this segment is then as simple as saying

draw[thin,fill=white,fill opacity=0.6,
 sphere segment=phi from 60 to 120 and theta from -10 to 50 and radius 4];

where phi and theta are the longitude and latitude parameters.

documentclass[tikz,border=3.14mm]standalone 
usepackagetikz-3dplot 
usetikzlibrary3d 
begindocument 
tdplotsetmaincoords7030 
begintikzpicture[tdplot_main_coords,bullet/.style=fill,circle,inner
sep=1pt,sphere segment/.style args=%
phi from #1 to #2 and theta from #3 to #4 and radius #5insert path=%
 plot[variable=x,smooth,domain=#2:#1] 
 (xyz spherical cs:radius=#5,longitude=x,latitude=#3)
 -- plot[variable=x,smooth,domain=#3:#4] 
 (xyz spherical cs:radius=#5,longitude=#1,latitude=x)
 --plot[variable=x,smooth,domain=#1:#2] 
 (xyz spherical cs:radius=#5,longitude=x,latitude=#4)
 -- plot[variable=x,smooth,domain=#4:#3] 
 (xyz spherical cs:radius=#5,longitude=#2,latitude=x),
>=stealth,declare function=f(x)=exp(-2+0.5*x);] 
 draw[thick] (0,0,0) -- (4,0,0);
 draw[thick,orange] plot[variable=x,smooth,domain=0:4] (x,0,f(x));
 % note: this domain is an approximation, it should really be 
 % from 0 to xmax where xmax is the solution of x^2+f(x)^2=4^2
 draw[thin,fill=white,fill opacity=0.6,
 sphere segment=phi from 60 to 120 and theta from -10 to 50 and radius 4] ; 
 draw[thick,->] (4,0,0) -- (6,0,0);
 draw[thick,orange] plot[variable=x,smooth,domain=4:6] (x,0,f(x));
 % should be xmax:6
endtikzpicture
enddocument