- #1
jdinatale
- 155
- 0
OK, this looks like a differential geometry problem, which it is, but at the end of the day I am trying to figure out why the unit normal and unit tangent vectors to a curve aren't orthogonal, so even if you don't know about DG, please respond.
Obviously the two choices for E_1 and E_2 are the unit normal and unit tangent vectors to the curve.
Using Mathematica...
Or by hand...
[itex]\alpha(t) = {Cos(t), 2Sin(t)}[/itex]
[itex]\alpha'(t) = {-Sin(t), 2Cos(t)}[/itex]
[itex]\alpha''(t) = {-Cos(t), -2Sin(t)}[/itex]
However, graphically, the unit tangent and unit normal vectors are far from perpendicular on this curve!
Here is my mathematica code
Obviously the two choices for E_1 and E_2 are the unit normal and unit tangent vectors to the curve.
Using Mathematica...
alpha[t_] := {Cos[t], 2 Sin[t]};
alphaprime[t_] := {-Sin[t], 2 Cos[t]};
alphaprimeprime[t_] := {-Cos[t], -2 Sin[t]};
unittangentvector[t_] := alphaprime[t] / Norm[alphaprime[t]];
unitnormalvector[t_] := alphaprimeprime[t] / Norm[alphaprimeprime[t]];
Or by hand...
[itex]\alpha(t) = {Cos(t), 2Sin(t)}[/itex]
[itex]\alpha'(t) = {-Sin(t), 2Cos(t)}[/itex]
[itex]\alpha''(t) = {-Cos(t), -2Sin(t)}[/itex]
However, graphically, the unit tangent and unit normal vectors are far from perpendicular on this curve!
Here is my mathematica code
VFieldOnCurve2D[dominterval_, CurveEq_, FrameField_, CodomainBox_,
Size_] :=
Module[{a2, b2, Content, IS, DomainPieces, DomainPiece1,
DomainPiece2, CodomainCenter, CodomainWidth, len, EE1, EE2,
ImagePieces, ImagePiece0, ImagePiece1, ImagePiece2},
{a2, b2} = dominterval;
IS = 300;
Content = Mapping12Content[dominterval, CurveEq];
DomainPiece1 = Content[[1]];
DomainPiece2[t_] := Points2D[{{0, t}}, .3];
DomainPieces[t_] :=
Show[DomainPiece1, DomainPiece2[t], ImageSize -> IS/4];
{CodomainCenter, CodomainWidth} = CodomainBox;
len = Length[FrameField];
If[len == 2, EE1 = FrameField[[1]];
EE2 = FrameField[[2]], {EE1} = FrameField];
ImagePiece0 = EmptySpace2DXCenter[CodomainCenter, CodomainWidth];
ImagePiece1 = Content[[2]];
ImagePiece2[t_] :=
If[len == 2, {Vec[CurveEq[t], EE1[t]], Vec[CurveEq[t], EE2[t]]},
Vec[CurveEq[t], EE1[t]] ];
ImagePieces[t_] :=
Show[ImagePiece0, ImagePiece1, ImagePiece2[t], ImageSize -> Size];
t0 = (a2 + b2)/2;
Manipulate[
Row[{DomainPieces[t], ImagePieces[t]}], {{t, t0, "t"}, a2, b2},
SaveDefinitions -> True]
]
alpha[t_] := {Cos[t], 2 Sin[t]};
alphaprime[t_] := {-Sin[t], 2 Cos[t]};
alphaprimeprime[t_] := {-Cos[t], -2 Sin[t]};
unittangentvector[t_] := alphaprime[t] / Norm[alphaprime[t]];
unitnormalvector[t_] := alphaprimeprime[t] / Norm[alphaprimeprime[t]];
E1[t_] = unitnormalvector[t];
E2[t_] = unittangentvector[t];
DomainInterval = {0, 2 \[Pi]};
initvalue = 0;
CodomainBox = {Origin2D, 2};
Size = 400;
VFieldOnCurve2D[DomainInterval, alpha, {E1, E2}, CodomainBox, Size]