Review and continue geodesics on the cylinder

cone and cylinder coverings in Maple

Applications of unwrapping: surface area of a cylinder

parametrizing the cylinder via coordinate systems:

Clicker questions on the hw readings 1-4

180 degree cone and variable cone

isoperimetric inequality proof and applications Mention other results from the global differential geometry of curves.

Glossary on Surfaces.

Clicker question

Define manifolds, orbifolds, surfaces, and geodesics. helix on cylinder and cone

The generalized helix on the sphere is called loxodrome or rhumb line. Its tangent lines have constant angle to the direction connecting the two poles

Visual Intelligence Continue with the cylinder. Use covering arguments to answer questions about the geodesics.

Clicker 1: Should the Frenet Frame be named after Frenet?

maple

A second argument that implies constant positive curvature in a plane is a part of a circle to motivate the fundamental theorem of curves for the plane and R^3. The embeddings make a difference as we'll see when we examine curves on other kinds of surfaces. Torsion is a spacecurve construct. Replaced with other curvatures more generally.

study guide

Given a fixed piece of string, what figure bounds the largest area? motivation, begin isoperimetric inequality proof and applications

Clicker #1 and #2

Constant positive curvature in a plane is a part of a circle. TNB slides.

Clicker #3 and #4

Discuss a parametrization of the strake and the annulus to motivate surfaces.

Curvature/torsion ratio is a constant then helix.

Discuss the fundamental theorem of curves for the plane and R^3.

Review Curve applications: Strake and more and connections to 3-D printing (once we have surfaces)

curve clicker questions including formulas and results from last week. TNB slides

Prove that curvature 0 iff a line. Prove that torsion 0 iff planar.

radius and curvature comic

Discuss that non-zero curvature constant for a plane curve means part of a circle.

Clicker questions on Rudy Rucker's How Flies Fly: Kappatau Space Curves

It is not true that a third coordinate nonzero means torsion is nonzero, via examples.

B=TxN. Since a nonplanar curve cannot be contained in a single plane, the osculating plane changes, which means that the normal vector to the osculating plane B changes. Since B' is not the 0 vector and B' = -tau N, then tau can't be 0.

Desmos. +add image ballmer_peak.png. put in function on next line.

Wolfram Demonstrations Project

Review TNB slides

T moves towards N and B moves away from N. How about N'?

Derive N' in the Frenet frame equations in two different ways.

The geometry of helices and applications. Maple commands:

with(VectorCalculus): with(plots):

helix:=<r*cos(t), r*sin(t), h*t> ;

TNBFrame(helix,t);

simplify(Curvature(helix,t));

simplify(Torsion(helix,t),trig);

spacecurve({[5*cos(t), 5*sin(t), 3*t, t = 0 .. 7]});

simplify(Curvature(helix, t)) assuming 0<h, 0<r;

simplify(Torsion(helix,t),trig) assuming 0<h, 0<r;

Twisted shirt

Curve applications: Strake and more

Torsion/curvature constant condition.

lolcatenary and Johann Bernoulli [1691]

Discuss a curve from #1 (or #3).

Warehouse 13's Mathematical Artifact (32:11-33:41) and the Lemniscate of Bernoulli.

with(plots): with(VectorCalculus):

plot([(t+t^3)/(1+t^4), (t-t^3)/(1+t^4), t = -10 .. 10]);

ArcLength(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4)>, t = -10 .. 10);

simplify(Curvature(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4)>),t);

Torsion(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4),0>,t);

TNBFrame(<(t+t^3)/(1+t^4), (t-t^3)/(1+t^4),0>);

then add assuming t::real to the TNBFrame command (look at last coordinate of B).

Examine TNBapplet.mw from -10 to -.005 and from .005 to 10.

Clicker questions on hw2

Review TNB slides

Mention that T, k and N work in higher dimensions, but the osculating plane is not defined by a normal, nor does cross product make sense - that is replaced by tensors and forms.

Continue deriving the Frenet equations. osculate comic Show that B'=-tau N. B' has no tangential component via a cross product argument, and B' has no B component via a dot product argument.

Clicker questions on derivatives with respect to arc length

Calculate T and T' for a circle of arbitrary frequency.

Why the curvature vector is perpendicular to T(s) (and that the derivative of a unit vector is perpendicular to itself).

TNB slides

geom

Animated torus knot

Normal

Discuss the curvature of a circle or radius r (1/r) and the osculating circle. B and the torsion

Clickers on curves article

MacTutor's Famous Curve Index

National Curve Bank pretzel as a curve

Wolfram's Astroid

Note that in 1.1, v^1 versus v_1--book getting you ready for Einstein summation notation. Lots of examples that we'll be exploring.

Clicker question on arc length

1.2 on arc length including proof of why regular curves can be reparamatrized by arc length to have unit speed.

Tractrix arc length challenge by hand and using the Maple Applet spacecurve.mw that calculates the Velocity, Acceleration, Jerk, Speed, ArcLength, Curvature, and Torsion

Jerk and higher time derivatives.

Begin 1.3 on Frenet frames. Visualization using Frenet Frame, and your hand, TNB slides, T, N, curvature vector and the magnitude as a scalar. Connect to earlier proof to explain why T(s) is a unit vector, and how chain rul comes in to computing the curvature vector from T(t)

Hand out glossary review: ideas from ideas from calc 3 and linear algebra that will be helpful here. Fill in as we go along, including within relevant hw.

Clicker questions

Talk about the hw 1 problems that the class struggles with. Solutions on ASULearn.

shortest distance comic

arc length shirt

e-book 9781614446088

Grading Policies

Tractrix. Discuss why arc length is defined as it is, and discuss local to global issues that relate.

Parametrized curves comic.

Examples of paramatrized differentiable curves in space and Maple Applet TNBapplet.mw

Prove that alpha is a curve that is a (constant speed) straight line iff the acceleration is 0.

Why is a line the shortest distance path between 2 points? Our intuition might be that a curve is inefficient since it starts off pointing away from the endpoint. However this intuition is false on a sphere.

Prove that a line in R