Astronomical Applications Department, U.S. Naval Observatory Curves 3 D Page 6
=
3
+
+
1
9
t
4
9
t
2
>
kappa=kappa( Twicubic(t), t );
=
2
+
+
1
9
t
4
9
t
2
(
)
+
+
1
4
t
2
9
t
4
/
3 2
>
b=Binormal( Twicubic(t), t );
=
b
1
2
[
]
,
,
6
t
2
-
6
t 2
+
+
1
9
t
4
9
t
2
>
n=UnitNormal( Twicubic(t), t );
=
n
1
2
[
]
,
,
- -
4
t 18 t
3
-
+
18
t
4
2
+
6
t 12 t
3
+
+
1
9
t
4
9
t
2
+
+
1
4
t
2
9
t
4
>
T=Tangent( Twicubic(t), t );
=
T
[
]
,
,
1 2
t 3 t
2
+
+
1
4
t
2
9
t
4
>
foo:=linalg[crossprod]( Tangent(Twicubic(t),t),
UnitNormal(Twicubic(t),t) ):
>
foo:=map( factor, foo );
:=
foo
,
,
3
t
2
+
+
1
9
t
4
9
t
2
-
3
t
+
+
1
9
t
4
9
t
2
1
+
+
1
9
t
4
9
t
2
>
evalm( foo-Binormal( Twicubic(t), t ) );
[
]
, ,
0 0 0
>
curveplot3D( Helix(Lemniscate(1,t),0.1,t), t=0..10,
shading=none, axes=box );
Page 6
