Geom Notes

7/27/2019 Geom Notes

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Differential Geometry Notes

Differential Geometry is the study of the local structure of geometric objects (such ascurves and surfaces).

We normally assume that these objects are continuous and differentiable (to whatever

degree necessary).Local properties are characterized by the derivatives (in different combinations) of the

object.Much of DG relies on expressing quantities in terms of a local coordinate system that

depends on the shape of the object.Often, DG is concerned with the characterization of invariants.An invariant is a measure (scalar, vector, matrix, tensor) which does not change under

certain kinds of transformations (e.g. translation-rotation).Invariants are interesting because we would like to make measurements about objects

that do not depend on their position/orientation wrt the camera.

1 Representing shapes (e.g. curves)

Parametric:

Sampled:

Implicit:

Plane Curves

A space curve is a 1D manifold (number of independent variables) embedded in a 3D space.Therefore there are two dimensions that are orthongonal to the manifold.

A surface is a 2D manifold embedded in a 3D space, and therefore it has 1 dimensionthat is orthogonal to the manifold.

For this reason, it is useful to study a 1D manifold in a 2D space, which also has 1orthogonal dimension. Such an object is called a plane curve.

C = {x(s) : a x b}

The function x(s) is a vector function of a scaler argument (s).That is

x : S IR2 where S IR

The tangent unit vector is

t(s) =dx(s)/ds

||dx(s)/ds||

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The normal unit vector is

n(s) =dt(s)/ds

||dt(s)/ds||

Draw picture of coordinate system

The speed of a parameteric curve is defined as

(s) = ||dx(s)/ds||

The curvatureof a parametric curve is

(s) =n(s) dt(s)/ds

(s)

Given an initial location and orientation, only (s) and (s) are necessary to completelyreconstruct a planar curve.

Note: one way to describe a planer curve is with the scaler function (s). This descriptionis invariant to rotation/translation. Therefore, we could compare curvature signatures ofobjects to compare.

Space Curves

A curve in 3D.

C = {x(s) : a x b}

The function x(s) is a vector function of a scaler argument (s).

That isx : S IR3 where S IR

Only smooth curves are considered (continuous first and second derivatives).The tangent unit vector is

t(s) =dx(s)/ds

||dx(s)/ds||

The normal unit vector is

n(s) =dt(s)/ds

||dt(s)/ds||

The binormal unit vector is

b(s) = t(s) n(s)

Draw picture of coordinate system (from page 67 of Besl)

The speed of a parameteric curve is defined as

(s) = ||dx(s)/ds||

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Any curve without singularities, i.e. dx(s)/ds = 0 s S, can be reparameterized interms of arc length. The curve has the same shape, but has a new parameter u, such thatdx(u)/du = 1 u U.

Such a curve is said to be parameterized by arc length.The curvatureof a parametric curve is

(s) =n(s) dt(s)/ds

(s)

The torsion of a parameter curve C is defined as

(s) = n(s) db(s)/ds

(s)

Space curves without torsion must lie in a plane in 3D, and thus a plane curve can berepresented as a space curve without torsion.

These definitions provide a complete differential description of a space curve. That is,the curve can be reconstructed from these functions.

Considering orthongal group transformations, the directions of the vectors rotation withthe rotation of the coordinate system, but the values of , , and are invariant to or-thogonal group transformations.

These properties are intrinsic to the curve. They dont depend on arbitrary aspects likethe parameterization or the coordinate system in which we express the curve.

Surfaces in 3D

Curvature, torsion, and speed, uniquely determine the shape of a space curve. These charac-teristics are useful because they are are invariant to orthongonal coordinate transformations

and they have a one-to-one relationship with the shape of the curve.Similar quantities exist for surfaces.A surface S is defined as

S =

x(s) =

xy

z

:

xy

z

=

x(u, v)y(u, v)

z(u, v)

where (u, v) D IR2

That isx : D IR3 where D IR2

The invariants that describe surface shape are not scalers, they are vectors/tensors. Thereare two. They are called the first and second fundimental forms.

FFF is denoted I. It is

I(u,v,du,dv) = dx dx = Edu2 + 2Fdudv + Gdv2

= (du dv)

g11 g12g21 g22

dudv

= duT (g) du

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g11 = E = xu xu

g22 = G = xv xv

g12 = g21 = F = xu xv

where the subscripts denote the partial derivatives

xu(u, v) =x

u=

xuy

uzu

xv(u, v) =x

v=

xvy

vzv

Where xu and xv are called the x-tangent vector and u-tangent vector respectively.They may or may not be orthongal, but they both lie in the tangent plane to the surface.

Draw picture of coordinate system in tangent plane

The FFF is the generalization of the speed associated with a space curve. Because thereare two directions you can go, you need a 2x2 matrix.

The FFF tells how much movement (and in what direction) you get on the surface dxfor a little bit of movement (du, dv).

Notice the FFF is invariant to orthogonal group transformations.The Second Fundimental Form (or II) is:

II(u,v,du,dv) = dx dn = edu2

+ 2fdudv + gdv2

= (du dv)

b11 b12b21 b22

dudv

= duT (b) du

where the matrix elements are defined as

b11 = e = xuu n

b22 = f = xvv n

b21 = b12 = g = xuv n

Where the subscripts denote the partial derivatives

xuu(u, v) =2x

u2=

2xu2

2y

u2

2zu2

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The surface normal is defined by

n(u, v) =xu xv

||xu xv||= Unit Normal Vector

The SFF characterizes the relationship between small movements in parameters du and

changes in surface normal dn.Consider a special case where the parameterization is orthormal. Then

b11 = E = xu xu = 1

b22 = G = xv xv = 1

b12 = g21 = F = xu xv = 0

And the first fudamental form is the identity matrix. The SFF is the same as the shapeoperator.

1.1 Surface Curvature

We could think of this as a local second-order model of the surface. Fit a parabola. It hasa long axis, short axis, and orientation. This this the info contained in SFF.

We can express the derivatives of the normal (in terms of the parameterization) as

= (b)(g)1 =1

EG F2

f F eG eF f EgF f G f F gE

This is called the Weingarten Mapping matrix or the shape operator. It is the generalizationof the curvature of plane curves (which has a term that accounts for speed). It expresses thecurvature independent of the parameterization.

The eigenvectors of are called the principle axes and their associated eigenvalues arecalled the principle curvatures as one moves a vector around in the tangent plane, thecurvature increases, reaches a maximum, decreases reaches a minimum, and then reaches amax again at 180 degrees.

Notice that each du has a corresponding dxThere are several aspects of curvature that are important:

Principle curvatures: These are the eigenvalues of .

k =

1

2(11 + 22) 1

4(11 + 22)2

(1122 1221)

Gaussian Curvature: This is the product of the eigenvalues (or the determinant of ).

k = det

=

eg f2

EG F2

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Mean Curvature:

H =1

2(k1 + k2) =

1

2(a11 + a22) =

1

2

eG 2f F + gE

EG F2

Deviation from Flatness (total curvature):

D2 = |||| = 211

+ 2212

+ 222

This is the total magnitude of the curvature. It can also be computed

D2 = k21

+ k22

= H2 K

Using these measures we can classify points on surfaces according to type.k1 < 0 k1 = 0 k1 > 0

k2 < 0 convex/peak parabolic/ridge hyperbolic/saddle

k2 = 0 parabolic/ridge planar/flat parabolic/valleyk2 > 0 hyperbolic/saddle parabolic/valley concave/pit

Describe how these properties can be determinded from K and H

Dubin indicatrix.

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Differential Geometry ExamplesScientific Visualization, CS6630

2 Space Curve: Helix with Radius R and length H

X(s) =

R cos sR sin s

Hs

(1)

Xs =

R sin sR cos s

H

(2)

= ||Xs|| = (Xs Xs)1

2 =

H2 + R2 12 (3)

t =1

(H2 + R2)1

2

R sin sR cos s

H

(4)

t

s=

1

(H2 + R2)1

2

R cos sR sin s

0

(5)

k = 1ts

= RH2 + R2 (6)

n =tsts

=

cos s sin s

0

(7)

b =t n

||t n||=

1

(H2 + R2)1

2

Hsin sHcos s

R

(8)

b

s==

1

(H2 + R2)1

2

Hcos sHsin s

0

(9)

= 1b

s n =

1

H2 + R2

Hcos sHsin s

R

cos s sin s

0

= H

H2 + R2(10)

Questions:

1. Describe the direction of normal in terms of the geometry of the helix.

2. What happens to the curvature as H and why?

3. What happens to the binormal as H 0.

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Surface: Quadratic

This is kind of a complicated example to compute by hand, but the result gives some insightinto how this stuff works and insight into the shapes of quadratic surfaces.

X(u, v) = uv

Au2 + Bv2

(11)

Xu =

10

2Au

(12)

Xv =

01

2Bv

(13)

n = Xu Xv = 1S

2Au2Bv1

(14)where

S =

1 + 4A2u2 + 4B2v2 12 (15)

I =

1 + 4A2u2 4AuBv

4AuBv 1 + 4B2v2

(16)

det I = 1 + 4A2u2 + 4B2v2 + 16A2u2B2v2 16A2u2B2v2 = 1 + 4A2u2 + 4B2v2 = S2 (17)

I1 = S2 1 + 4B2v2 4AuBv

4AuBv 1 + 4A2

u2 (18)

Xuu =

00

2A

(19)

Xvv =

00

2B

(20)

Xuv =

00

0

(21)

II = S1

2A 00 2B

(22)

= II(I1) = S3

2A + 8AB2v2 8AuB2v8A2uBv 2B + 8A2u2Bv

(23)

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The eigenvectors of this matrix are not easy to compute (for me perhaps one of you allcan see the answer). However, we can compute the important invariants.

H =1

2Tr = S3

A + B + 4AB(Au2 + Bv2)

(24)

K = det = S

3

4AB + 16A3u2B + 16A3u2B

(25)

Questions:

At the origin u = v = 0, the analysis is quite simple. What are the principle directionsand curvatures? How are they consistent with our understanding that shape matrixgives a quadratic approximation to the local surface structure?

Describe how the special cases AB > 0, AB = 0, and AB < 0, give the expectedresults for H and K everywhere on the surface. Describe the shape of the surfaceunder each of these circumstances.

What happens to the overall curvature of the surface as u and v become very large?How is this consistent with our understanding of the curvature of a parabola (drawpicture to explain).

3 Graph Surface curvature from Derivatives

A height surface or range map is defined as

z = f(x, y)

When descretley sampled, this can be thought of as an image that contains the depth of

a scene/object/shape at every point.This is also called a 2.5D representation.

Why is it not a truly 3D representation.

The parameterization for this surface is therefore:

x(u, v) =

uv

f(u, v)

The derivatives are as follows:

xu =

10

fu

(26)

xv =

01

fv

(27)

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xuu =

00

fuu

(28)

xuv = xvu =

00

fuv

(29)

xvv =

00

fvv

(30)

n =1

1 + f2u + f2v

fufv

1

(31)

(32)

Now we can get the frist fundamental form:

E = g11 = 1 + f2

u and F = g21 = fufv and E = g22 = 1 + f2

v

and the SFF is

L = b11 =fuu

1 + f2u + f2v

M = b12 =fuv

1 + f2u + f2v

N = b22 =fvv

1 + f2u + f2v

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