Normal curvature
1
2
Theorema Egregium
1
Theorema Egregium 1
Curvature and torsion of torus knots
With the definitions
torus[a, b, c][u, v] := {(a + b*Cos[v])*Cos[u], (a + b*Cos[v])*Sin[u], c*Sin[v]}
and
torusknot[a, b, c][p, q][t] :=
{(a + b*Cos[q*t])*Cos[p*t], (a + b*Cos[q*t])*Sin[p*t], c*Sin[q*t]}
this graphic shows the torusknot[8,3,5][1,2][t]
on the torus torus[8,3,5], shown in a yellow wireframe view, along with the curve
alone. Also shown are the graphs of the
curvature (red) and torsion (green) functions for this curve, graphed over the
t-axis.
Clicking on the graphic runs an animation which is determined by letting
t increase from 0 to 2 pi in increments of 2Pi/20, for the torusknot curve, its curvature
and its torsion functions.
So in the animation the 'leading edge' of the torusknot curve has curvature and
torsion values given by the (y-coordinates of the) leading edge of the graphs
of those corresponding
functions.
This graphic also
illustrates 'which' points on the curvature and
torsion graphs correspond to which points on the torusknot curve, for the given
parameterization.
Click on the picture to see the animated view.
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Curvature and torsion of torus knots 2
With the definitions torus[a, b, c][u, v] := {(a + b*Cos[v])*Cos[u], (a + b*Cos[v])*Sin[u], c*Sin[v]} and torusknot[a, b, c][p, q][t] := {(a + b*Cos[q*t])*Cos[p*t], (a + b*Cos[q*t])*Sin[p*t], c*Sin[q*t]} this graphic shows the curvature and torsion for the curve torusknot[8,3,8][1,2][t] on the torus[8,3,8], shown in a yellow wireframe view,along with the curve alone. Clicking on the graphic runs an animation which shows how the torus, the torus knot, and its curvature and torsion change when we graph torusknot[8,3,c][1,2][t] on the torus[8,3,c], and let c vary. We vary c from 5 to 8, then from 8 to 3, then back to 5. This animation is meant to illustrate how curvature and torsion change when the curve (the torusknot in this case) changes in a controlled way. Click on the picture to see the animated view.
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Curves with prescribed curvature and torsion
In the right graphic, the red line is the graph of Curvature[ t ] = t, the green line is the graph of Torsion[ t ; n ] = n t, with n = 0, with t in [0,10]. In the left graphic is the numerically-generated unit-speed space curve alpha[ t ; n ] with curvature = Curvature[ t ] and torsion = Torsion[ t ; n ] for t in [0,10]. The initial conditions for alpha[ t ; n ] are alpha[ 0 ; n ] = {0,0,0}, t-derivatives alpha ' [ 0 ; n ] = {1,0,0} and alpha '' [0;n] is normalized to {0,1,0}. We then vary n from 0 to 1 in increments of .1 The changes in the curve alpha[ t ; n ] and the graph of Torsion[ t ; n ] are shown in the animated view. This animation gives a 'dynamic' view of what the torsion of a space curve controls (how much the curve is 'stretched out'). The alpha[ t ; n ] curve is obtained by solving the Frenet frame equations numerically with a program Gray wrote using the Mathematica function NDSolve. Click on the picture to see the animated view.
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In the right graphic,
the red line is the graph of
Curvature [ t ; n ] = n t, with n = 0 the
green line
is the graph of
Torsion[ t ] = t, with t in [0,10]. In
the left graphic is the numerically-generated unit-speed space curve alpha[ t ; n ] with
curvature = Curvature[ t ; n ] and 'torsion' = Torsion[ t ] for t in [0,10] (see below
for some remarks about this). The initial
conditions for alpha[ t ; n ] are
alpha[ 0 ; n ] = {0,0,0}, t-derivatives
alpha ' [ 0 ; n ] = {1,0,0} and alpha '' [0;n] is normalized to {0,1,0}.
We then vary n from 0 to 1 in increments of .1. The changes in the curve alpha[ t ; n ] and the
graph of Curvature[ t ; n ]
are shown in the animated view. This animation gives a 'dynamic' view of what
the curvature of a space curve controls (how much the curve
'winds around'). The alpha[ t ; n ] curve is obtained by
solving the Frenet frame equations numerically with a program Gray
wrote using the Mathematica function
NDSolve.
For n = 0, the curve
alpha[ t ; n ] is a plane curve, even though its 'torsion' is nonzero. This is
because we chose the curvature to be identically
zero. In this case the function we have called the torsion function does not
behave like the usual notion of torsion, which is defined when the curvature is nonzero.
One can pick 'any' two (say smooth) functions, K[ t ] and T[ t ] to play the roles of the curvature and
torsion functions appearing in the Frenet frame equations, and a
solution curve to those equations is obtained, C [ t ; K ; T ]
(given initial conditions).
The relation between the 'usual' notions of curvature
and torsion for C [ t ; K ; T ] and K[ t ] and T[ t ], especially when K[ t ] takes zero or
negative values, is explored in exercises in the course.
Click on the picture to see the animated view.
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In the right graphic, the red line is the graph of Curvature[ t ; n] = n t, the green line is the graph of Torsion[ t ; n ] = n t, both with n = 0 , with t in [0,10]. In the left graphic is the numerically-generated unit-speed space curve alpha[ t ; n ] with curvature = Curvature[ t ; n ] and torsion = Torsion[ t ; n] for t in [0,10]. The initial conditions for alpha[ t ; n ] are alpha[ 0 ; n ] = {0,0,0}, t-derivatives alpha ' [ 0 ; n ] = {1,0,0} and alpha '' [0;n] is normalized to {0,1,0}. We then vary n from 0 to 1 in increments of .1 The changes in the curve alpha[ t ; n ] and the graph of Curvature[ t ; n ] and Torsion[ t ; n ] are shown in the animated view. This animation gives one possible view of what happens to a space curve when both its curvature and torsion are varied (a sort of competition between streching out and winding around!). Many other choices of curvature and torsion functions can be examined by minor modifications to the program Gray wrote to generate the space curves. The alpha[ t ; n ] curve is obtained by solving the Frenet frame equations numerically with a program Gray wrote using the Mathematica function NDSolve. Click on the picture to see the animated view.
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First fundamental form animations
This animation
illustrates how certain basic surface parameterizations distort lengths of curves.
This idea leads to a discussion of intrinsic versus non-intrinsic properties
of surfaces.
In the
left graphic, the green
paraboloid[1,2][u,v]= {u , v, u^2 + 2 v^2} surface
(in 'wireframe' view) is shown over the x-y parameter
plane. A copy of the u-v plane is shown the plane z = -1.
In this plane are red line segments
of length 1 starting at the point (0,0). The images of these line
segment are shown are thick black curves on the surface.
In the right graphic the lengths of the curves on the surface are
shown in black as a function of the 'polar angle' in the plane.
Also shown in red are the lengths of the original
line segments (all of which = 1). Click on the picture to see the animated
view.
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This animation
illustrates how certain basic surface parameterizations distort lengths of tangent
vectors.
This idea leads to a discussion of induced metrics on surfaces and on the domains of the
parameterizing
coordinate patches. Later, this leads to a discussion of
a metric on a plane domain which may or may not
arise from an embedding
into three-space.
In the upper left graphic, the
blue tangent vectors with yellow tips in the
u-v plane are based at the point (0,1). These
have length 1 and their endpoints trace out the
red (Euclidean) circle
of radius 1. These vectors are mapped to the surface
z = x^2 + y^2 by the tangent map to the parameterization
paraboloid[1,1][u,v]= {u , v, u^2 + v^2}. In the
graphic on the lower left, the image vectors on the
paraboloid[1,1] at the point {0,1,1} are shown in
blue, and the image of
the circle is shown in red.
The lengths of these image vectors (labeled G-length
) are shown in the blue
curve in the lower right graphic, as a function of the
'polar angle' in the u-v plane. In the upper right graphic is
shown the original blue vectors,
now measured with the 'induced metric', that is, with the lengths
given by the lengths of the images tangent to the surface.Click on
the picture to see the animated view.
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Normal curvature animations
The surface z = y^2 - x^2 is shown in red. The green 'upward' normal is located at (0,0,0). The normal curvature is computed in the direction of the blue tangent vector at (0,0,0). The blue and green vectors span the yellow slicing plane. The normal curvature is (up to sign) the ordinary space curve curvature of the curve given by the intersection of the slicing plane and the surface. The normal curvature is positive when the slice curve 'bends up towards' the chosen normal, and negative when it 'bends down away'. The intersection curves are shaded black. It is an exercise in the course to prove that these curves are the images of straight lines through {0,0} in the u-v parameter plane under the parameterization { u , v , -u^2 + v^2 }. In the right-hand frame the normal curvature is graphed as a function of the 'polar angle' of the blue tangent vector. The 'polar angle' is defined using the parameterization for a basis of the tangent space, and applying the Gram-Schmidt process to that basis. Explaining this was also part of an exercise. Click on the picture to see the animated view.
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The surface z = y^2 - x^2 is shown in red. The green 'upward' normal is located at (0,1,1). The normal curvature is computed in the direction of the blue tangent vector based at (0,1,1). The blue and green vectors span the yellow slicing plane. The normal curvature is (up to sign) the ordinary space curve curvature of the curve given by the intersection of the slicing plane and the surface. The normal curvature is positive when the slice curve 'bends up towards' the chosen normal, and negative when it 'bends down away'. In the right-hand frame the normal curvature is graphed as a function of the 'polar angle' of the blue tangent vector. The 'polar angle' is defined using the parameterization for a basis of the tangent space, and applying the Gram-Schmidt process to that basis. Explaining this was part of an exercise in the course.Click on the picture to see the animated view.
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Gauss map animations
The above graphic
is one frame of an animated view of the Gauss map of a catenoid. On the left side,
the catenoid, the surface of revolution with profile curve
the catenary {v, cosh[v]}, is shown with a
red spiral curve on it.
This curve obtained by mapping a spiral
centered at the origin of the u-v plane to
the catenoid by the parameterization
catenoid[u,v] = {cos[u]*cosh[v],sin[u]cosh[v],v}.
The blue 'outward'
unit normal with a yellow tip is shown at a
point of this curve.
On the right side of the graphic the blue
unit normal with yellow
tip is shown translated to the origin,
with its yellow
tip on a gray
sphere (shown in wireframe view).The
yellow curve on the sphere is the image of
the red curve under the
Gauss map of the catenoid. We can think of the
unit normal as 'painting' that portion of the
sphere corresponding to the image of the red spiral
under the Gauss map. If we imagine the outermost part of the spiral on the
catenoid as enclosing a two-dimensional region D, then it appears from
the graphic that the image under the
Gauss map of D will be a two-dimensional region
on the sphere. The area of this image is controlled by the Gauss curvature
of the catenoid. If p is any point on the catenoid, and
D any two-dimensional region with p in D,
then the ratio
Area(G(D))/Area(D), where G is the Gauss map, is a certain number.
The limiting
value of such numbers (with an appropriate sign) as D 'shrinks down to p' is the
Gauss curvature of
the surface at p, K(p).
This is how Gauss introduced the idea of what
we now call Gauss curvature (Gauss called it the "measure of curvature").
For the catenoid, the Gauss curvature is strictly negative at
every point, taking a minimum value of -1
at the "center" of the red spiral (the negativity of
the curvature is reflected in the opposing motions of
the normals on the catenoid as they move along the red spiral
and the sense in which
the image of the spiral is traced out on the
sphere). That K = -1 at the center of the spiral
can be interpretted to mean that near the center of the red
spiral,the area of a small disc will have approximately the same area
as its image under the Gauss map, and have an opposite
orientation.
Contrast this animation with the
next one, the Gauss map for a cylinder. There the Gauss curvature
is identically 0 and any such disc will be mapped to a curve.
Click on
the picture to see the animated view.
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The above graphic
is one frame of an animated view of the Gauss map of a cylinder.
On the left side,
the cylinder surface x^2 + y^2 =1 is shown with a
red spiral on it.
This curve obtained by mapping the sprial
centered at the origin of the u-v plane to
the cylinder by the parameterization
cyl[u,v] = {cos[u],sin[u],v}.
The blue 'outward'
unit normal with a yellow tip is shown at a
point of this curve.
In the right side the blue
unit normal with yellow
tip is shown translated to the origin,
with its yellow
tip on a gray
sphere (shown in wireframe view).The
yellow curve on the sphere is the image of
the red curve under the
Gauss map of the cylinder. We can think of the
unit normal as 'painting' that portion of the
sphere corresponding to the image of the red spiral
under the Gauss map.
If we imagine the outermost part of the spiral on the
cylinder as enclosing a two-dimensional region D then the image under the
Gauss map of D will be not be a two-dimensional region
on the sphere, but a one-dimensional region (i.e. a curve).
Hence Area(G(D))= 0, since "area" means "two-dimensional
measure", and then Area(G(D))/Area(D)= 0.
As discussed in the
previous example , this type of ratio of areas, Area(G(D))/Area(D),
is related to the Gauss curvature of the surface. The reason
one gets
Area(G(D))/Area(D)= 0 for the cylinder is that
the Gauss curvature of the cylinder
is equal to zero at every point. This contrasts with the
previous example of the Gauss map of the catenoid. There,
the curvature was nonzero (with value near -1 at the spiral 'center')
and a corresponding two-dimensional region D on the
catenoid mapped to a two-dimensional
region under
the Gauss map.
Click on
the picture to see the animated view.
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Frenet frame animations
This graphic shows part of a [2,5]-torus knot on an '[8,3,5]'-torus. The '[8,3,5]'-torus means the surface of revolution obtained from an ellipse in the x-y plane centered at (8,0) with major axis 2 x 5 in the y-direction and minor axis 2 x 3 in the x-direction, rotated about the y-axis. The [2,5]-torus knot means a curve which wraps 'around' the torus twice as it 'spirals' five times over and under the torus. The Frenet frame for this knot is shown with moving along the curve with red tangent, blue normal,and green binormal. For a reference, a second copy of the Frenet frame is shown with its 'center' at the origin of 3-space. At points on the curve where the curvature is small and the torsion is large in absolute value, there is dramatic 'twisting' in the plane spanned by the normal and binormal vectors. This topic is discussed further in Gray's book and also in the next graphic. Click on the picture to see the animated view.
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This graphic again shows part of a [2,5]-torus knot on an '[8,3,5]'-torus, along with its Frenet frame moving along the curve with red tangent,blue normal, and green binormal. A second copy of this Frenet frame is shown with its 'center' at the origin of 3-space. Additionally, a third Frenet frame is shown at the origin, which has longer and thinner tangent, normal and binormal vectors. For this longer thinner frame we take the initial conditions to be the same as those for torus knot Frenet frame, at a point on the torus knot where curvature is small and torsion is large (right after the first 'hilltop' is crossed by the torus knot Frenet frame). But for the longer thinner Frenet frame, we set the curvature function equal to 0 and solve the resulting Frenet frame equations with torsion equal to that of the torus knot. This new frame keeps the tangent vector fixed and has the normal and binormal vectors 'spinning' in the orthogonal plane. When the torsion is large in absolute value, the 'spinnning' is very fast. The torus knot Frenet frame behaves 'like' this longer thinner frame when the torus knot has small curvature and large torsion (and matching initial conditions). This phenomenon can be observed by advancing the animation slowly using the 'arrow' buttons on most .mov viewers. Mathematically, these ideas can be made quite precise by comparing solutions to systems of differential equations when those systems are 'close' using Gronwall-type inequalities. These topics were explored in exercises in the course.Click on the picture to see the animated view.
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Theorema Egregium
This graphic shows the first frame of an animated view of the Theorema Egregium of Gauss, that
the (Gauss) curvature K of a surface is invariant under isometry. The
green/yellow surface shown is a portion of the round sphere of
radius 1. It is the starting surface in a family of isometric surfaces all of which
share the features of being surfaces of revolution with Gauss curvature K = 1 at every point.
To the right of the spherical surface is a purple graph of its Gauss curvature
as a function of the sphere parameterization
variables { u , v } (the u-curves are mapped to the parallels, the v-curves to meridians).
Below these are graphs of the principal
curvatures, k1 (red) and k2 (blue). When the animation runs
the sphere portion varies through
the family of isometric surfaces. The corresponding
principal curvature functions also vary, but the Gauss curvature, which is
given
by the product,
k1*k2 = K, remains constant. This exhibits the Theorema
Egregium in action.
Click on the picture to see the animated view.
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There
are lots of other Differential Geometry topics explored with
Mathematica and
Gray's programs in the Spring 2000, Fall 2002 and Fall 2003 courses.
The Mathematica notebooks are written
for an audience which I assume has NOT used Mathematica before.
The notebooks
were written so that students can
modify the input easily in order to examine
geometric attributes of different curves or
surfaces, or different points on given
curves and surfaces. These modifications are
included as exercises. I will be adding
further graphics and topics to those given above in the future.
Please write me if you would like copies of these Mathematica files emailed to you.
I would be happy to received
comments, suggestions, corrections, ideas....
You may modify the files, use them however you want, and let me know how to improve them!
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