Pattern  from triangle (not a euclidian)  and twincircles?

UHG_module_2D_-_6-9-2026.mw

a non-periodic function

@Kitonum 
Note that the planes are positioned perpendicular to each other, so this requires additional code.and other  things in a earlier example, so  i don't see your point  ? 
No textplots are possible with draw and display.commands.
 

restart:
with(geom3d):
plane(P1,x+y=1,[x,y,z]):
plane(P2,x-y=2,[x,y,z]):
point(A,1,0,0):
point(B,2,0,0):
intersection(L,P1,P2):
draw([P1(color=blue,transparency=0.55,style=surface),P2(color=yellow,transparency=0.55,style=surface),L(color=black,thickness=4),A(color=red,symbol=solidsphere,symbolsize=16),B(color=red,symbol=solidsphere,symbolsize=16)],axes=boxed,labels=[x,y,z],scaling=constrained,view=[-3..5,-4..4,-3..4],orientation=[55,65],title="Two Planes using geom3d");

Its a lot of work to do this with geom3d 

@erik10 
I looked into what you asked in even more detail and am using only the geom3d package here, but a different approach using a linear algebra package is also possible
My investigation..
twee_vlakken_volledig_via_geom3d_pakket_30-8-2026.mw

restart:
with(geom3d):
plane(P1,x+y=1,[x,y,z]):
plane(P2,x-y=2,[x,y,z]):
Equation(P1);
Equation(P2);
n1:=NormalVector(P1);
n2:=NormalVector(P2);
n1;
n2;
ArePerpendicular(P1,P2);
alpha_oriented:=FindAngle(P1,P2);
alpha:=abs(alpha_oriented);
alpha_deg:=simplify(alpha*180/Pi);
alpha_oriented;
alpha;
alpha_deg;
point(A,1,0,0):
point(B,2,0,0):
coordinates(A);
coordinates(B);
IsOnObject(A,P1);
IsOnObject(B,P2);
intersection(L,P1,P2):
detail(L);
point(Au,2,-1,0):
point(Av,1,0,1):
dsegment(U1,[A,Au]):
dsegment(V1,[A,Av]):
point(An,2,1,0):
dsegment(N1,[A,An]):
point(Bu,3,1,0):
point(Bv,2,0,1):
dsegment(U2,[B,Bu]):
dsegment(V2,[B,Bv]):
point(Bn,3,-1,0):
dsegment(N2,[B,Bn]):
point(Q,3/2,-1/2,3/2):
h:=1/3:
point(Q1,3/2+h,-1/2-h,3/2):
point(Q2,3/2+h,-1/2+h,3/2):
point(Q3,3/2+2*h,-1/2,3/2):
segment(RA1,[Q1,Q3]):
segment(RA2,[Q2,Q3]):
draw([P1(color=blue,transparency=0.55,style=surface),P2(color=yellow,transparency=0.55,style=surface),L(color=black,thickness=4),A(color=red,symbol=solidsphere,symbolsize=16),B(color=red,symbol=solidsphere,symbolsize=16),U1(color=red,thickness=4),V1(color=green,thickness=4),N1(color=black,thickness=4),U2(color=red,thickness=4),V2(color=green,thickness=4),N2(color=black,thickness=4),RA1(color=red,thickness=6),RA2(color=red,thickness=6)],axes=boxed,labels=[x,y,z],scaling=constrained,view=[-3..5,-4..4,-3..4],orientation=[55,65],title="Perpendicular Planes using geom3d");

special_first-order_resultaat_modigeficeerde_Besselverg_2_e_orde_afleiding_28-6-2026_def.mw

For n = 1, the formula has been numerically verified 
Now by complete induction ....

This should be the solution sequenze for the modified Bessel function at the bottom

@dharr 
More to find 

@dharr 
This is ..

@Ronan 
The code works with all the points you mentioned. 

In all these examples, the twin circle does not intersect the zero circle, but there are points of tangency with the zero circle Does this hold true in general? : It should be possible to explain theoretically why this might hold true in hyperbolic geometry.

Download twin_circles_en_driehoek_in_nulcircle-mapleprimes_24-8-2026.mw

@Ronan

It's still quite a challence to get those twin circles to work properly..
An ideal triangle in the zero circle does not produce twin circles, because they turn out to coincide.