OrthopticSurfaceParaboloid::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; Personal.";
Development:
Obtaining the orthoptic surface of the paraboloid
2
x
9
+
2
y
4
=2z
:
Let the tangent plane (where the normal vector is an unitary vector):
Ax+By+Cz==D;
Condition of tangent between a plane and a paraboloid with vertex at the origin:
Tangent plane:
x
0
x
2
a
+
y
0
y
2
b
-z==
z
0
;​​-
A
C
x-
B
C
y-z==-
D
C
;​​
x
0
==-
A
2
a
C
;​​
y
0
==-
B
2
b
C
;​​
z
0
==-
D
C
;​​
Sustituting the point {
x
0
,
y
0
,
z
0
}
in the paraboloid’s equation and simplifying:
1
2
a
2
-
A
2
a
C
+
1
2
b
2
-
B
2
b
C
==2-
D
C
;​​
2
A
2
a
+
2
B
2
b
==-2CD;
So for the paraboloid that concerns us:
9
2
A
+4
2
B
==-2CD;
So for the three planes we have:
9
2
A
1
+4
2
B
1
==-2
C
1
D
1
;​​9
2
A
2
+4
2
B
2
==-2
C
2
D
2
;​​9
2
A
3
+4
2
B
3
==-2
C
3
D
3
;
Adding member to member the three equations and considering that the rows of an orthogonal matrix are also unit vectors:
-2(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==9
2
A
1
+
2
A
2
+
2
A
3
+4
2
B
1
+
2
B
2
+
2
B
3
;​​-2(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==9(1)+4(1)==13;
Since the three normal vectors are mutually orthogonal and unitary, (and by properties of orthogonal matrices) it is simplified to:
-2(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-2
C
1
(
A
1
x+
B
1
y+
C
1
z)-2
C
2
(
A
2
x+
B
2
y+
C
2
z)-2
C
3
(
A
3
x+
B
3
y+
C
3
z);
-2(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-2x(
C
1
A
1
+
C
2
A
2
+
C
3
A
3
)-2y(
C
1
B
1
+
C
2
B
2
+
C
3
B
3
)-2Z
2
C
1
+
2
C
2
+
2
C
3
;​​-2(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-2x(0)-2y(0)-2z(1);​​-2(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-2z;
So it turns out how we expected Monge’s plane:
z
== -
2
a
+
2
b
2
which is the orthoptic surface of the paraboloid. There are many similarities in 2D with the parabola and its orthoptic curve, the directrix. Finally, the equation sought is:
z==-
13
2
;
Animation:
In[]:=
Manipulatea2=9;b2=4;​​u1[α_,β_]:={Sin[α]Sin[β],Cos[β],Cos[α]Sin[β]};​​u2[α_]:={Cos[α],0,-Sin[α]};​​u3[α_,β_]:={Sin[α]Cos[β],-Sin[β],Cos[α]Cos[β]};​​d1[α_,β_]:=-
a2
2
u1[α,β][[1]]
+b2
2
u1[α,β][[2]]
2u1[α,β][[3]]
;​​d2[α_]:=-
a2
2
u2[α][[1]]
+b2
2
u2[α][[2]]
2u2[α][[3]]
;​​d3[α_,β_]:=-
a2
2
u3[α,β][[1]]
+b2
2
u3[α,β][[2]]
2u3[α,β][[3]]
;​​q[α_,β_]:=d1[α,β]u1[α,β]+d2[α]u2[α]+d3[α,β]u3[α,β];​​​​​​Show​​​​ParametricPlot3D[3
2
uCos[v],2
2
uSin[v],
2
u
,{u,0,6},{v,0,2π},ColorFunction->"SolarColors",MeshNone,PlotPoints->60],​​ContourPlot3Dz==-
13
2
,{x,-16,16},{y,-16,16},{z,-8,6},​​ContourStyle{Green,Opacity[0.5]},MeshNone,​​If[​​u1[α,β][[3]]≠0,​​Graphics3D[{Gray,Opacity[0.5],Hyperplane[u1[α,β],d1[α,β]]}],Nothing],​​​​If[​​u2[α][[3]]≠0,​​Graphics3D[{Gray,Opacity[0.5],Hyperplane[u2[α],d2[α]]}],Nothing],​​​​​​If[​​u3[α,β][[3]]≠0,​​Graphics3D[{Gray,Opacity[0.5],Hyperplane[u3[α,β],d3[α,β]]}],Nothing],​​​​If[​​u1[α,β][[3]]≠0&&u2[α][[3]]≠0&&u3[α,β][[3]]≠0,​​Graphics3D[​​{​​Black,Thick,​​Sphere[q[α,β],0.3],​​Line[{q[α,β],q[α,β]+3u1[α,β]}],​​Line[{q[α,β],q[α,β]+3u2[α]}],​​Line[{q[α,β],q[α,β]+3u3[α,β]}]​​}​​],​​Nothing​​]​​,​​BoxRatios->Automatic,​​Axes->True,BoxedFalse,​​AxesOrigin->{0,0,0},​​AxesLabel->{x,y,z},​​PlotRange->16,​​ImageSize->900,ViewPoint{5,-2.4,2.}​​,Style["Orthoptic surface: Monge's plane",Bold,Large]​​,​​{{α,0.00002,"α angle (Turn H)"},0.000001,2π,0.00001},Delimiter,β,
π
4
,"β angle (Turn V)",-
π
2
,
π
2
,0.00001,ControlPlacement->Top,AutorunSequencing->{1,2}​​
Out[]=
​
Orthoptic surface: Monge's plane
α angle (Turn H)
β angle (Turn V)