Unit 1: Paths and Curves

Sarah Harrelson, Dec 20, 2022

Unit 1 of a high school Multivariable Calculus Class for LASA High School: [url]https://www.lasahighschool.org/[/url]

Table of Contents

  1. 1.1 Plane Curves
    1. 1.1.1 Introduction to parameterized curves
    2. 1.1.2 A subtlety
    3. 1.1.3 Properties of paths
    4. 1.1.4 Directed Lines and Line Segments
    5. 1.1.5 Transformations of paths (part 1)
    6. 1.1.6 Transforming the Parameter (Circles)
    7. 1.1.7 Transforming the parameter (directed lines)
    8. 1.1.8 Circles and Ellipses
  2. 1.2 Calculus of Plane Curves
    1. 1.2.1 The Cycloid Problem
    2. 1.2.2 The Velocity Vector
    3. 1.2.3 Tangent Line to a Plane Curve
    4. 1.2.4 A few product and chain rules
  3. 1.3 Speed and Arclength
    1. 1.3.1 Speed
    2. 1.3.2 Estimating length via line segments
    3. 1.3.3 Creating an integral
    4. 1.3.4 Arc length
    5. 1.3.5 Arc length parameterization
  4. 1.4 Curvature
    1. 1.4.1 The Unit Tangent Vector
    2. 1.4.2 Curvature (round 1)
    3. 1.4.3 Curvature (Round 2)
    4. 1.4.4 The Unit Normal
    5. 1.4.5 The Osculating Circle
  5. 1.5 Space Curves
    1. 1.5.1 Intro to Space Curves
    2. 1.5.2 Directed Lines
    3. 1.5.3 Circles
    4. 1.5.4 Projections onto Coordinate Planes
  6. 1.6 The Frenet Frame
    1. 1.6.1 The Osculating Circle for Space Curves
    2. 1.6.2 Acceleration
    3. 1.6.3 The Binormal Vector
    4. 1.6.4 Torsion

1.

1.1 Plane Curves

  • 1. 1.1.1 Introduction to parameterized curves
  • 2. 1.1.2 A subtlety
  • 3. 1.1.3 Properties of paths
  • 4. 1.1.4 Directed Lines and Line Segments
  • 5. 1.1.5 Transformations of paths (part 1)
  • 6. 1.1.6 Transforming the Parameter (Circles)
  • 7. 1.1.7 Transforming the parameter (directed lines)
  • 8. 1.1.8 Circles and Ellipses

1.1.

1.1.1 Introduction to parameterized curves

A [b][color=#ff0000]parameterized curve[/color][/b] is a function from a subset of [math]\mathbb{R}[/math] into Euclidean space ([math]\mathbb{R}^n[/math]). For the time being we are going to stick to [b][color=#ff0000]plane[/color][/b] curves, meaning our codomain is [math]\mathbb{R}^2[/math].[br][br]Notation:[br][br][math]\vec{c}:\left[a,b\right]\to\mathbb{R}^2[/math] where [math]\vec{c}\left(t\right)=\left(x\left(t\right),y\left(t\right)\right)[/math].[br][br]The input variable [math]t[/math] is often called the [b][color=#ff0000]parameter[/color][/b]. The functions [math]x\left(t\right)[/math] and [math]y\left(t\right)[/math] are called the [b][color=#ff0000]component functions[/color][/b] of [math]\vec{c}[/math]. The function itself is called a [b][color=#ff0000]path[/color] in [math]\mathbb{R}^2[/math] [/b]and the image of the function is called the [color=#ff0000][b]image curve[/b] [/color]of the path. The points [math]\vec{c}\left(a\right)[/math] and [math]\vec{c}\left(b\right)[/math] are the [b][color=#ff0000]endpoints[/color][/b] of the curve.[br][br]The first thing I want you to do is just play. In the GeoGebra applet below you can type in two component functions and a domain of definition to see the resulting parameterized curve. Take some time to experiment. Try fixing a couple component functions and changing the domain to see how the resulting curve changes. Then hold the domain steady and edit the component functions creating different curves. Can you parameterize a line segment? A circle? A spiral? Can you create a curve whose endpoints are the same? How about a curve with a loop-de-loop?
– Sarah Harrelson

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2.

1.2 Calculus of Plane Curves

  • 1. 1.2.1 The Cycloid Problem
  • 2. 1.2.2 The Velocity Vector
  • 3. 1.2.3 Tangent Line to a Plane Curve
  • 4. 1.2.4 A few product and chain rules

2.1.

1.2.1 The Cycloid Problem

The classic cycloid curve is generated by tracing the path of fixed point on the rim of a circle as it rolls on a flat surface. When the circle has radius one, the curve is parameterized by the path [math]\vec{c}\left(t\right)=\left(t-\sin t,1-\cos t\right),t\in\mathbb{R}[/math]
Which of the following describes [math]\vec{c}\left(t\right)[/math]?
Define a vector as follows:[br][math]\vec{v}_h\left(t\right)=\left(x'\left(t\right),0\right)[/math].[br]Describe this vector: [br][list][*]What does this vector look like? [/*][*]Is it defined at all points along the path? [/*][*]When is it long? [/*][*]How long does it get? [/*][*]When does it vanish? [/*][*]In which direction does it point at each value of [math]t[/math]?[/*][/list]
Repeat the previous exercise for the vector [math]\vec{v}_v\left(t\right)=\left(0,y'\left(t\right)\right)[/math].
– Sarah Harrelson

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3.

1.3 Speed and Arclength

  • 1. 1.3.1 Speed
  • 2. 1.3.2 Estimating length via line segments
  • 3. 1.3.3 Creating an integral
  • 4. 1.3.4 Arc length
  • 5. 1.3.5 Arc length parameterization

3.1.

1.3.1 Speed

You may have noticed that when I animate a path by tracing the movement of a point, the dots along the path are not evenly spaced apart. Unsurprisingly this is related to the way the vector [math]\vec{c}\left(t\right)[/math] is changing. In the GeoGebra applet below, a path is animated on the left screen while the length of the velocity vector [math]\vec{c}'\left(t\right)[/math] is tracked on the right screen.
The length of the velocity vector is called the [b][color=#ff0000]speed[/color][/b] of the path [math]\vec{c}\left(t\right)[/math]. That is:[br]The [b][color=#ff0000]speed[/color][/b] of the path [math]\vec{c}\left(t\right)[/math] is the scalar quantity [math]\left|\left|\vec{c}'\left(t\right)\right|\right|[/math].[br][br]Why do you think we chose the word "speed" to describe this scalar quantity?
– Sarah Harrelson

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4.

1.4 Curvature

  • 1. 1.4.1 The Unit Tangent Vector
  • 2. 1.4.2 Curvature (round 1)
  • 3. 1.4.3 Curvature (Round 2)
  • 4. 1.4.4 The Unit Normal
  • 5. 1.4.5 The Osculating Circle

4.1.

1.4.1 The Unit Tangent Vector

The velocity vector gives two pieces of information about a path - the direction of movement (the direction in which the velocity vector points) and the speed (the length of the velocity vector). If we normalize the velocity vector (remember normalize means create a parallel vector of length 1) then we create a vector that solely tells us about direction of movement (and how rapidly that direction is changing). This vector is called the [b][color=#ff0000]unit tangent[/color][/b] vector to the path [math]\vec{c}[/math]. In other words:[br][br][math]\vec{T}\left(t\right)=\frac{\vec{c}\,'\left(t\right)}{\left|\left|\vec{c}\,'\left(t\right)\right|\right|}[/math][br][br]In the GeoGebra applet below you can see a path traced out by a moving point with the unit tangent anchored on the moving point. In the upper window, the unit tangent is drawn in standard position (i.e. anchored at the origin). Spend some time experimenting. Consider the following questions:[br][list][*]For every differentiable path [math]\vec{c}[/math], the unit tangent (when anchored at the origin) traces a portion of the same curve. Explain.[/*][*]What is happening in the path when the unit tangent is swiveling slowly? What about when it is swiveling rapidly?[/*][*]Can you design a path whose unit tangent (when drawn in standard position) traces out an injective path?[/*][*]Can you design a path whose unit tangent (when drawn in standard position) traces out an entire circle?[/*][*]Can you design a path whose unit tangent (when drawn in standard position) does not move at all?[/*][/list]
– Sarah Harrelson

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5.

1.5 Space Curves

  • 1. 1.5.1 Intro to Space Curves
  • 2. 1.5.2 Directed Lines
  • 3. 1.5.3 Circles
  • 4. 1.5.4 Projections onto Coordinate Planes

5.1.

1.5.1 Intro to Space Curves

So far we have been studying plane curves - that is the image curves of paths whose codomain is [math]\mathbb{R}^2[/math]. Today we extend our study to space curves - paths whose image curves live in [math]\mathbb{R}^3[/math]. The GeoGebra file below illustrates several interesting space curves.[br]Tip - for this GeoGebra applet I enabled "right clicking". When you right-click on the 3d graphics screen you will be offered a menu that includes the option "Zoom to Fit". This is a handy quick way to get good screen dimensions for viewing a curve.
– Sarah Harrelson

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6.

1.6 The Frenet Frame

  • 1. 1.6.1 The Osculating Circle for Space Curves
  • 2. 1.6.2 Acceleration
  • 3. 1.6.3 The Binormal Vector
  • 4. 1.6.4 Torsion

6.1.

1.6.1 The Osculating Circle for Space Curves

Recall we defined the osculating circle to a plane curve to be a circle of best fit to the curve. We did this by making the curvature of the osculating circle equal to the curvature of the curve at the point of tangency. Nearly everything we did for plane curves can be transferred nicely to space curves - curvature has the same definition (length of [math]\vec{T}'\left(s\right)[/math] where [math]\vec{c}\left(s\right)[/math] is understood to be an arclength parameterization). The same formulas for curvature that were introduced earlier still apply.[br][br]However when we consider the osculating circle to a space curve we are confronted with a choice. At each point along the curve there are infinitely many planes containing that point in which we could draw the osculating circle - which one should we choose?[br][br]The GeoGebra applet below illustrates the choice of plane - the osculating circle is drawn in the plane defined by [math]\vec{T}\left(t\right)[/math] and [math]\vec{N}\left(t\right)[/math] (recall the unit normal is the unit tangent to the unit tangent). As you've seen already, these two vectors are always orthogonal. This plane is called the [b][color=#ff0000]osculating plane[/color][/b].
Write a parameterization for the osculating circle to a space curve in terms of the curvature [math]\kappa[/math], unit tangent [math]\vec{T}[/math], and unit normal [math]\vec{N}[/math] vectors.
– Sarah Harrelson

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