OrthopticSurfaceCylinder::usage="The orthoptic surface of a surface is the geometric locus of the points through which pass 3 planes orthogonal to each other and tangents to the original surface. References: AI Assistant; Personal.";
Development:
Obtaining the orthoptic surface of the cylinder
2
x
+
2
y
=
2
r
:
Normal vector of the cylinder:
In[]:=
σ[θ_,v_]:={rCos[θ],rSin[θ],v};​​
σ
θ
[θ_,v_]=D[σ[θ,v],θ];​​
σ
v
[θ_,v_]=D[σ[θ,v],v];​​n=
σ
θ
[θ,v]
σ
v
[θ,v]
Norm[
σ
θ
[θ,v]
σ
v
[θ,v]]
//FullSimplify

rCos[θ]
r(
2
Abs[Cos[θ]]
+
2
Abs[Sin[θ]]
)Conjugate[r]
,
rSin[θ]
r(
2
Abs[Cos[θ]]
+
2
Abs[Sin[θ]]
)Conjugate[r]
,0;​​{Cos[θ],Sin[θ],0};
Then, any plane tangent to the cylinder will be (Remember the plane definition P.

n
= d):
In[]:=
{x,y,z}.{Cos[θ],Sin[θ],0}==r​​
xCos[θ]+ySin[θ]r;
Due to the perpendicularity condition

n
2
.

n
1
= Cos[
θ
2
]
Cos[
θ
1
]
+ Sin[
θ
2
]
Sin[
θ
1
]
= Cos[
θ
2
-
θ
1
]=0,if
and only if
θ
2
=
θ
1
+
π
2
.
A second perpendicular plane would be:
Cosθ+
π
2
x+Sinθ+
π
2
y==r;​​-Sin[θ]x+Cos[θ]y==r;
The third plane has to be parallel to the XY plane (and it is not possible that it is tangent to the cylinder):
z==v;
Cartesian equation of the orthoptic surface of the cylinder (Removing the θ angle. See the similarity with the orthoptic curve of the circumference):
In[]:=
Expand[
2
(Cos[θ]x+Sin[θ]y)
-
2
r
+
2
(-Sin[θ]x+Cos[θ]y)
-
2
r
==0]//FullSimplify
2
2
r

2
x
+
2
y
;​​
2
x
+
2
y
==2
2
r
Point of intersection of the three planes:
In[]:=
Solve[{Cos[θ]x+Sin[θ]y==r,-Sin[θ]x+Cos[θ]y==r,z==v},{x,y,z}]//FullSimplify
{{xr(Cos[θ]-Sin[θ]),yr(Cos[θ]+Sin[θ]),zv}};
Animation:
In[]:=
Manipulate[r=3;​​u1[θ_]:={Cos[θ],Sin[θ],0};​​u2[θ_]:={-Sin[θ],Cos[θ],0};​​u3={0,0,1};​​Q[θ_,v_]:=ru1[θ]+ru2[θ]+{0,0,v};​​Show[​​​​ParametricPlot3D[{rCos[θ1],rSin[θ1],v1},{θ1,0,2π},{v1,-5,5},ColorFunction->"SolarColors",MeshNone,MaxRecursion->3],​​ParametricPlot3D[
2
rCos[θ1],
2
rSin[θ1],v1,{θ1,0,2π},{v1,-5,5},​​ColorFunction->"BlueGreenYellow",PlotStyle->Opacity[0.6],MeshNone,MaxRecursion->3],​​Graphics3D[{Black,Thick,Sphere[Q[θ,v],0.1],Line[{Q[θ,v],Q[θ,v]+2u1[θ]}],Line[{Q[θ,v],Q[θ,v]+2u2[θ]}],Line[{Q[θ,v],Q[θ,v]+2u3}],{Gray,Opacity[0.4],Hyperplane[u1[θ],r],Hyperplane[u2[θ],r],Hyperplane[u3,Q[θ,v]]}}]​​,​​BoxRatios->Automatic,​​Axes->True,BoxedFalse,​​AxesOrigin->{0,0,0},​​AxesLabel->{x,y,z},​​PlotRange->6,​​ImageSize->900,ViewPoint{1.3,-2.4,3.}​​],Style["Orthoptic surface: Concentric cylinder",Bold,Large],​​{{θ,0.00002,"θ angle (Turn H)"},0,2π,0.00001},Delimiter,{{v,-5+0.00002,"v (Turn V)"},-5,5,0.00001},ControlPlacement->Top,AutorunSequencing->{1,2}​​]
Out[]=
​
Orthoptic surface: Concentric cylinder
θ angle (Turn H)
v (Turn V)