RadiusCurvatureSpiral::usage="Demonstrate that the radius of curvature of a conical spiral a
hφ

Cos[φ ]​,a
hφ

Sin[φ ]​,b
hφ

​ isproportional to the distance between a point of the spiral and the axis of the cone.References: A.S. Fedenko, Problemas de Geometría Diferencial. Editorial Mir: Moscú, p.68, 1981.";
Development:
In[]:=
r[φ_]:=a
hφ

Cos[φ],a
hφ

Sin[φ],b
hφ

;​​rc=
3
(Norm[r'[φ]])
Norm[r'[φ]r''[φ]]
In[]:=
Simplify
3/2

2Re[hφ]

2
(bh)
+
2
a
hφ

hCos[φ]-a
hφ

Sin[φ]
+
2
a
hφ

Cos[φ]+a
hφ

hSin[φ]


2
-ab
2hφ

2
h
Cos[φ]+ab
2hφ

hSin[φ]
+
2
-ab
2hφ

hCos[φ]-ab
2hφ

2
h
Sin[φ]
+
2

2
a
2hφ

2
Cos[φ]
+
2
a
2hφ

2
h
2
Cos[φ]
+
2
a
2hφ

2
Sin[φ]
+
2
a
2hφ

2
h
2
Sin[φ]


3/2
(
2
b
2Re[hφ]

2
h
+
2
a
2hφ

(1+
2
h
))
2
a
4hφ

(1+
2
h
)(
2
b
2
h
+
2
a
(1+
2
h
))
==
3hφ

3/2
(
2
b
2
h
+
2
a
(1+
2
h
))
a
2hφ

1+
2
h
==
hφ

(
2
b
2
h
+
2
a
(1+
2
h
))
a
1+
2
h
Distance between a point of the spiral and the axis of the cone:
In[]:=
d=
Norm[a
hφ

Cos[φ],a
hφ

Sin[φ],b
hφ

-{0,0,0}{0,0,1}]
Norm[{0,0,1}]
2Re[hφ]

2
(aCos[φ])
+
2Re[hφ]

2
(aSin[φ])
==a
hφ

If we divide both quantities:
In[]:=
hφ


2
b
2
h
+
2
a
(1+
2
h
)
a
1+
2
h
​
a
hφ

Out[]=
(
2
b
2
h
+
2
a
(1+
2
h
))
​
2
a
1+
2
h
It is a constant amount, which means that they are proportional amounts.
Gráfica, sean a = 1, b =
1
, h =
1
4
:
In[]:=
ManipulateClearAll;r[φ_]:=
1
4
φ

Cos[φ],
1
4
φ

Sin[φ],
1
4
φ

;a=1;b=1;h=
1
4
;​​d1[t1_]:=
h(4Log[t1])

(
2
b
2
h
+
2
a
(1+
2
h
))
a
1+
2
h
;d2[t1_]:=a
h(4Log[t1])

;​​p=Grid[{{CurvatureRadius,ConeAxisSpiralDistance,DistancesRatio},{d1[t1],d2[t1],d1[t1]/d2[t1]}},FrameAll,ItemSize->15,ItemStyle->15];​​Show[{ParametricPlot3D[r[φ],{φ,-4Log[t1],4Log[t1]},PlotStyle{Thick,Blue}],​​ParametricPlot3D[-r[φ],{φ,-4Log[t1],4Log[t1]},PlotStyle{Thick,Blue}],ParametricPlot3D[{vCos[u],vSin[u],v},{u,0,2π},{v,-20,20},PlotStyleDirective[Green,Specularity[White,20],Opacity[0.1]],MeshFalse,PerformanceGoal"Quality"],Graphics3D[{Black,Dashed,Thick,Line[{{0,0,t1},r[4Log[t1]]}]}],​​Graphics3D[{Black,Dashed,Thick,Line[{{0,0,-t1},-r[4Log[t1]]}]}],Graphics3D[{Red,Ball[{0,0,t1},0.43]}],Graphics3D[{Red,Ball[r[4Log[t1]],0.43]}],Graphics3D[{Red,Ball[-r[4Log[t1]],0.43]}],Graphics3D[{Red,Ball[{0,0,-t1},0.43]}]}​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ViewPoint{-3,3,2},ImageSize500,PerformanceGoal"Quality"],Style["Conical Spiral:",Bold,Large],{{t1,0.1,"Value (t1)"},0.01,20,0.01},Delimiter,{{p,2,"Measurable parameters"}},ControlPlacementTop
Out[]=
​
Conical Spiral:
Value (t1)
Measurable parameters
CurvatureRadius
ConeAxisSpiralDistance
DistancesRatio
16.0656
14.72
1.09141