Signed Curvature Graphs

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Andrew Iskauskas 6 years, 5 months ago

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Andrew Iskauskas 6 years, 5 months ago

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Andrew Iskauskas 6 years, 5 months ago

Created this as a copy of Signed Curvature Graphing.

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Question statement

The Fundamental Theorem of Planar Curves states that every curve in $\mathbb{R}^{2}$ with non-zero curvature is completely determined by its signed curvature, $\kappa(t)$, and some initial condition for a point on the curve.

Below are six curves $\gamma(t)$, all of which satisfy $\gamma(0)=(0,0)$ and $\gamma^{\prime}(0)=(0,1)$. The corresponding signed curvatures are as follows:

A: $\simplify{1/{quadratic[0]}*(t+{quadratic[1]})(t+{quadratic[2]})}$

B: $\simplify{{sin[0]}*t*sin({sin[1]}*t)}$

C: $\simplify{exp({exp}*t)}$

D: $\simplify{1/{cubic[0]}*(t+{cubic[1]})(t+{cubic[2]})(t+{cubic[3]})}$

E: $\simplify{({quotient[0]}*t)/({quotient[1]}+{quotient[2]}*t^2)}$

F: $\simplify{{cos[0]}*t*cos({cos[1]}*t)}$

Give any introductory information the student needs.

			Name	Type	Generated Value

quadratic	list	[ 3, 1, -3 ]
cubic	list	[ 4, 2, 4, -4 ]
quotient	list	[ 3, 2, 2 ]
cos	list	[ 2, 2 ]
sin	list	[ -2, 2 ]
exp	integer	-1

			Name	Type	Generated Value

quadstr	string	(Math.pow(#,3)/3+-2##/2+-3*#
cubstr	string	(Math.pow(#,4)/4+2*Math.pow(#,
quostr	string	Math.log(2+2##)3/(22)
cosstr	string	2(Math.cos(2#)/(4)+#*Math.si
sinstr	string	-2(Math.sin(2#)/(4)+#*Math.c
expstr	string	Math.exp(-1*#)

			Name	Type	Generated Value

collected	list	Nested 6×2 list
markingmatrix1	matrix	Matrix of size 6×3
markingmatrix2	matrix	Matrix of size 6×3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
[random(2..4),random(-3..3 except 0),random(-3..3 except 0)]

JME function reference

Description

Generates the list of constants {a,b,c} for a quadratic (t+b)(t+c)/a.

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `list`

[ 3, 1, -3 ]

This variable doesn't seem to be used anywhere.

Match choices with answers

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{graph({collected[0][0]})}
{graph({collected[1][0]})}
{graph({collected[2][0]})}

For each combination of answer and choice, specify the number of marks to add or subtract when the student picks it.

	{graph({collected[0][0]})}	{graph({collected[1][0]})}	{graph({collected[2][0]})}
A
B
C
D
E
F

Advice

In principle, given a signed curvature function $\kappa(t)$ and an intitial condition $\gamma(0)$, $\gamma^{\prime}(0)$, we can find the equation $\gamma:\,\mathbb{R}\to\mathbb{R}^{2}$ of the curve using

$\theta(t)=\int\kappa(t)\mathrm{d}t$

$\gamma=(\int \cos(\theta(t)-\theta_{0})\mathrm{d}t,\,\int \sin(\theta(t)-\theta_{0})\mathrm{d}t),$

where $\theta_{0}$ is determined by the initial conditions. However, this is often not feasible to work out explicitly: for example, if $\kappa(t)=t$ and $\gamma(0)=(0,0)$ and $\gamma^{\prime}(0)=(0,1)$, then we must find

$\int \cos(\frac{t^{2}}{2})\mathrm{d}t,$

which has no nice form.

Instead, we can relate properties of the signed curvature to properties of the curve:

Symmetry: if $\kappa(t)$ is an even function, then $\gamma$ has reflection symmetry; if $\kappa(t)$ is odd, $\gamma$ has rotational symmetry.
Turning points: if $\kappa(t)$ changes sign at a point, then the curve $\gamma$ will change from anticlockwise to clockwise rotation (and vice versa). Note that $\kappa(t)$ must change sign: it is not enough for $\kappa(t)$ to equal $0$ at some point.
Asymptotics: As $t\to\pm\infty$, we can determine the behaviour of $\kappa(t)$: if it tends to $0$ as $t\to\infty$, then $\gamma$ curves less as we travel rightwards along it; if $\kappa(t)\to\infty$ as $t\to\infty$ then $\gamma$ curves more and more sharply; if $\kappa(t)$ has no limit as $t\to\infty$ then we expect oscillations.

Consider, as an example,

$\kappa(t)=\sin\left(\frac{t^{2}}{2}\right).$

This is an even function, as the argument is $t^{2}$ so $\kappa(-t)=\kappa(t)$; hence $\gamma$ has reflection symmetry. $\sin(x)$ changes sign an infinite number of times, so we expect infinitely many changes from clockwise to anticlockwise. Finally, $\sin(x)$ has no limit as we take $x\to\infty$, corresponding to $t\to\pm\infty$, so we would expect the curve to oscillate. Then we can see that this $\kappa(t)$ corresponds to the following graph:

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Andrew Iskauskas 6 years, 5 months ago

Andrew Iskauskas 6 years, 5 months ago

Andrew Iskauskas 6 years, 5 months ago

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Signed Curvature Graphs

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

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Comment

Checkpoint description

Andrew Iskauskas 6 years, 5 months ago

Andrew Iskauskas 6 years, 5 months ago

Andrew Iskauskas 6 years, 5 months ago

Error in variable testing condition

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Generated value: list

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Match choices with answers

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Generated value: `list`