OsculatingSpherePlaneCircleSlinky::usage="In this animation osculating sphere, circle and plane are observed moving along the Slinky curve.Referencia: Wolfram MathWorld: Osculating Sphere; Personal.";
Development :
In[]:=
r
1
[t_]:=1+
2
5
Cos[40t]Cos[t],1+
2
5
Cos[40t]Sin[t],
3
10
t+
2
5
Sin[40t];​​k1=
Norm[
r
1
'[t]
r
1
''[t]]
3
(Norm[
r
1
'[t]])
In[]:=
Simplify
2
-
3Cos[t]
10
-
5203
25
Cos[t]Cos[40t]-
51232
5
Cos[t]
2
Cos[40t]
-
3152
5
Sin[t]Sin[40t]+256Cos[40t]Sin[t]Sin[40t]-10240Cos[t]
2
Sin[40t]
+
2
3Sin[t]
10
+
5203
25
Cos[40t]Sin[t]+
51232
5
2
Cos[40t]
Sin[t]-
3152
5
Cos[t]Sin[40t]+256Cos[t]Cos[40t]Sin[40t]+10240Sin[t]
2
Sin[40t]
+
2
2
Cos[t]
+
3204
5
2
Cos[t]
Cos[40t]+
6404
25
2
Cos[t]
2
Cos[40t]
+
2
Sin[t]
+
3204
5
Cos[40t]
2
Sin[t]
+
6404
25
2
Cos[40t]
2
Sin[t]
+512
2
Cos[t]
2
Sin[40t]
+512
2
Sin[t]
2
Sin[40t]

3/2
2
3
10
+16Cos[40t]
+
2
-1+
2
5
Cos[40t]Sin[t]-16Cos[t]Sin[40t]
+
2
AbsCos[t]1+
2
5
Cos[40t]-16Sin[t]Sin[40t]

Out[]=
20
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]

3/2
19263+100
2
AbsCos[t]+
2
5
Cos[t]Cos[40t]-16Sin[t]Sin[40t]
+6346Cos[2t]+780Cos[38t]+1000Cos[40t]-820Cos[42t]-3042Cos[78t]+6404Cos[80t]-3362Cos[82t]
In[]:=
​​
τ
1
=
Det[{
r
1
'[t],
r
1
''[t],
r
1
'''[t]}]
2
(Norm[
r
1
'[t]
r
1
''[t]])
In[]:=
Simplify
3
2
Cos[t]
10
-
620394
25
2
Cos[t]
Cos[40t]-
1999319994
125
2
Cos[t]
2
Cos[40t]
+
327987264
25
2
Cos[t]
3
Cos[40t]
+
3
2
Sin[t]
10
-
620394
25
Cos[40t]
2
Sin[t]
-
1999319994
125
2
Cos[40t]
2
Sin[t]
+
327987264
25
3
Cos[40t]
2
Sin[t]
-
80791296
5
2
Cos[t]
2
Sin[40t]
+13123584
2
Cos[t]
Cos[40t]
2
Sin[40t]
-
80791296
5
2
Sin[t]
2
Sin[40t]
+13123584Cos[40t]
2
Sin[t]
2
Sin[40t]

2
Abs-
3Cos[t]
10
-
5203
25
Cos[t]Cos[40t]-
51232
5
Cos[t]
2
Cos[40t]
-
3152
5
Sin[t]Sin[40t]+256Cos[40t]Sin[t]Sin[40t]-10240Cos[t]
2
Sin[40t]

+
2
Abs
3Sin[t]
10
+
5203
25
Cos[40t]Sin[t]+
51232
5
2
Cos[40t]
Sin[t]-
3152
5
Cos[t]Sin[40t]+256Cos[t]Cos[40t]Sin[40t]+10240Sin[t]
2
Sin[40t]

+
2
Abs
2
Cos[t]
+
3204
5
2
Cos[t]
Cos[40t]+
6404
25
2
Cos[t]
2
Cos[40t]
+
2
Sin[t]
+
3204
5
Cos[40t]
2
Sin[t]
+
6404
25
2
Cos[40t]
2
Sin[t]
+512
2
Cos[t]
2
Sin[40t]
+512
2
Sin[t]
2
Sin[40t]


Out[]=
(30(-1339700773+1091308180Cos[40t]+6820802Cos[80t]-85280Cos[120t]))(36
2
Abs[3209+5340Cos[40t]-1066Cos[80t]]
+
2
Abs[512175Cos[t]+20963Cos[39t]-10557Cos[41t]-3120Cos[79t]+3280Cos[81t]]
+
2
Abs[512175Sin[t]-20963Sin[39t]-10557Sin[41t]+3120Sin[79t]+3280Sin[81t]]
)
In[]:=
n1[t_]:=ResourceFunction["NormalVector"][
r
1
[t],t];​​n1[t]​​
In[]:=
Out[]=

-2625625Cos[t]-39183441Cos[39t]-43193201Cos[41t]-386140Cos[79t]-446940Cos[81t]+156Cos[119t]-164Cos[121t]
25717+1040Cos[40t]+8Cos[80t]
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]
,
-2625625Sin[t]+39183441Sin[39t]-43193201Sin[41t]+386140Sin[79t]-446940Sin[81t]-156Sin[119t]-164Sin[121t]
25717+1040Cos[40t]+8Cos[80t]
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]
,-
160(513985Sin[40t]+5197Sin[80t])
25717+1040Cos[40t]+8Cos[80t]
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]

In[]:=
b1[t_]:=ResourceFunction["BinormalVector"][
r
1
[t],t];​​b1[t]
In[]:=
Simplify
50
3Sin[t]
10
+
5203
25
Cos[40t]Sin[t]+
51232
5
2
Cos[40t]
Sin[t]-
3152
5
Cos[t]Sin[40t]+256Cos[t]Cos[40t]Sin[40t]+10240Sin[t]
2
Sin[40t]
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]
,
50-
3Cos[t]
10
-
5203
25
Cos[t]Cos[40t]-
51232
5
Cos[t]
2
Cos[40t]
-
3152
5
Sin[t]Sin[40t]+256Cos[40t]Sin[t]Sin[40t]-10240Cos[t]
2
Sin[40t]
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]
,50
2
Cos[t]
+
3204
5
2
Cos[t]
Cos[40t]+
6404
25
2
Cos[t]
2
Cos[40t]
+
2
Sin[t]
+
3204
5
Cos[40t]
2
Sin[t]
+
6404
25
2
Cos[40t]
2
Sin[t]
+512
2
Cos[t]
2
Sin[40t]
+512
2
Sin[t]
2
Sin[40t]

263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]

Out[]=

512175Sin[t]-20963Sin[39t]-10557Sin[41t]+3120Sin[79t]+3280Sin[81t]
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]
,
-512175Cos[t]-20963Cos[39t]+10557Cos[41t]+3120Cos[79t]-3280Cos[81t]
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]
,
6(3209+5340Cos[40t]-1066Cos[80t])
263799072767+11488191540Cos[40t]-11733150Cos[80t]-1534880Cos[120t]-12792Cos[160t]

In[]:=
tt1[_]:=ResourceFunction["TangentVector"][
r
1
[t],t];​​tt1[t]
Out[]=

-((1+0.4Cos[40t])Sin[t])-16.Cos[t]Sin[40t]
257.17+10.4Cos[40t]+0.08Cos[80t]
,
Cos[t](1+0.4Cos[40t])-16.Sin[t]Sin[40t]
257.17+10.4Cos[40t]+0.08Cos[80t]
,
0.3+16.Cos[40t]
257.17+10.4Cos[40t]+0.08Cos[80t]

In[]:=
Out[]=
​
Osculating sphere:
Valor (t1)
Center and radius of the osculating sphere
Center
Radius
{-0.590243,-0.790889,1.20397}
0.390142
Center and radius of the osculating circle
Center
Radius
{-0.62867,-0.760665,1.20247}
0.387064
The intersection of the plane and the osculating sphere represent the osculating circle that appears in black.
The tangent vector in blue, the normal vector in green, the binormal vector in purple. The point in red, the osculating plane in gray, the curve in yellow and the sphere in red.
the contact order is the second order
The contact order is the third order.