OrthopticSurfaceHyperbolicParaboloid::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: Personal.";
Development:
Obtaining the orthoptic surface of the hyperbolic paraboloid
2
x
9
-
2
y
4
=z
:
Let the tangent plane (where the normal vector is an unitary vector):
Ax+By+Cz==D;
Condition of tangent between a plane and a hyperbolic paraboloid with saddle point at the origin:
Tangent plane:
2
x
0
x
2
a
-
2
y
0
y
2
b
-z==
z
0
;​​-
A
C
x-
B
C
y-z==-
D
C
;​​
x
0
==-
A
2
a
2C
;​​
y
0
==
B
2
b
2C
;​​
z
0
==-
D
C
;​​
Sustituting the point {
x
0
,
y
0
,
z
0
}
in the hyperbolic paraboloid’s equation and simplifying:
1
2
a
2
-
A
2
a
2C
-
1
2
b
2
B
2
b
2C
==-
D
C
;​​
2
A
2
a
-
2
B
2
b
==-4CD;
So for the hyperbolic paraboloid that concerns us:
9
2
A
-4
2
B
==-4CD;
So for the three planes we have:
9
2
A
1
-4
2
B
1
==-4
C
1
D
1
;​​9
2
A
2
-4
2
B
2
==-4
C
2
D
2
;​​9
2
A
3
-4
2
B
3
==-4
C
3
D
3
;
Adding member to member the three equations and considering that the rows of an orthogonal matrix are also unit vectors:
-4(
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
;​​-4(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==9(1)-4(1)==5;
Since the three normal vectors are mutually orthogonal and unitary, (and by properties of orthogonal matrices) it is simplified to:
-4(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-4
C
1
(
A
1
x+
B
1
y+
C
1
z)-4
C
2
(
A
2
x+
B
2
y+
C
2
z)-4
C
3
(
A
3
x+
B
3
y+
C
3
z);
-4(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-4x(
C
1
A
1
+
C
2
A
2
+
C
3
A
3
)-4y(
C
1
B
1
+
C
2
B
2
+
C
3
B
3
)-4z
2
C
1
+
2
C
2
+
2
C
3
;​​-4(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-4x(0)-4y(0)-4z(1);​​-4(
C
1
D
1
+
C
2
D
2
+
C
3
D
3
)==-4z;
So it turns out how we expected Monge’s plane:
z
== -
2
a
-
2
b
4
which is the orthoptic surface of the hyperbolic paraboloid. There are many similarities in 2D with the parabola and its orthoptic curve, the directrix. Finally, the equation sought is:
z==-
5
4
;
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]]
4u1[α,β][[3]]
;​​d2[α_]:=-
a2
2
u2[α][[1]]
-b2
2
u2[α][[2]]
4u2[α][[3]]
;​​d3[α_,β_]:=-
a2
2
u3[α,β][[1]]
-b2
2
u3[α,β][[2]]
4u3[α,β][[3]]
;​​q[α_,β_]:=d1[α,β]u1[α,β]+d2[α]u2[α]+d3[α,β]u3[α,β];​​​​​​Show​​​​ContourPlot3D
2
x
9
-
2
y
4
==z,{x,-16,16},{y,-16,16},{z,-16,16},ColorFunction->"SolarColors",MeshNone,MaxRecursion->3,​​ContourPlot3Dz==-
5
4
,{x,-16,16},{y,-16,16},{z,-16,16},​​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)
Note: It should be remembered that the tangent plane at any point of a hyperbolic paraboloid intersects two generatrix lines (it is a double-ruled surface) that intersect at that point, unlike an ellipsoid or a sphere where the tangent point is an isolated point.
These points on a hyperbolic paraboloid are called hyperbolic points (negative Gauss curvature).