Truly Bizarre - The unit tangent and unit normal vectors aren't orthogonal

In summary, the conversation discusses a differential geometry problem involving unit normal and unit tangent vectors to a curve. The participants use Mathematica code to explore the topic and determine that the chosen E1 and E2 vectors provide a frame field on the curve.
  • #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.

Joseph-2.png


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!

642.png


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]
 
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  • #2
The unit normal isn't alphaprimeprime[t] / Norm[alphaprimeprime[t]]. It's the derivative of the UNIT tangent vector divided by the norm of the derivative of the UNIT tangent vector. alphaprime isn't UNIT.
 
Last edited:
  • #3
Dick said:
The unit normal isn't alphaprimeprime[t] / Norm[alphaprimeprime[t]]. It's the derivative of the UNIT tangent vector divided by the norm of the derivative of the UNIT tangent vector. alphaprime isn't UNIT.

Thanks, and that would seem to do the trick, but mathematica is hating me right now. Any ideas what's going on? Here is the newly defined unit normal vector

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_] :=
unittangentvector'[t] / Norm[unittangentvector'[t] ];

E1[t_] = unitnormalvector[t];
E2[t_] = unittangentvector[t];


failure.png



Implementation

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_] :=
unittangentvector'[t] / Norm[unittangentvector'[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]
 
  • #4
Not really, sorry. It looks right. But I haven't done Mathematica since other people stopped paying for it for me. It's really expensive and unfree. So I can't test that in any detail.
 
  • #5
Dick said:
Not really, sorry. It looks right. But I haven't done Mathematica since other people stopped paying for it for me. It's really expensive and unfree. So I can't test that in any detail.

Well, at least theoretically, would you agree that my choice of E1 and E2 provide a frame field on the curve?
 
  • #6
Sure. The unit tangent T points along your curve. T' must be perpendicular to that. Just differentiate T.T=1. It's got to work, right?
 

1. What is the difference between a unit tangent and unit normal vector?

A unit tangent vector is a vector that is tangent to a curve at a specific point, with a length of 1. It represents the direction of the curve at that point. On the other hand, a unit normal vector is perpendicular to the tangent vector and points towards the center of curvature of the curve at that point.

2. Why are the unit tangent and unit normal vectors not always orthogonal?

The unit tangent and unit normal vectors are only guaranteed to be orthogonal in the case of a circle or a straight line. In other cases, the curvature of the curve causes the direction of the tangent and normal vectors to change, resulting in them not being perpendicular to each other.

3. Can the unit tangent and unit normal vectors ever be parallel?

Yes, in the case of a straight line, the unit tangent and unit normal vectors will be parallel. This is because the direction of the curve does not change, so the tangent vector is always pointing in the same direction as the normal vector.

4. How are the unit tangent and unit normal vectors useful in mathematics?

The unit tangent and unit normal vectors are important for understanding the behavior of curves in mathematics. They can be used to calculate the curvature of a curve, which is a measure of how much the curve deviates from a straight line at a given point. They are also used in differential geometry, which is a branch of mathematics that studies the properties of curves and surfaces.

5. Are there any real-world applications of the unit tangent and unit normal vectors?

Yes, the unit tangent and unit normal vectors have various applications in fields such as physics, engineering, and computer graphics. For example, in physics, they are used to describe the motion of objects along curved paths, and in computer graphics, they are used to create smooth and realistic animations of objects moving along curves.

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