Lecture 2: a Short Atlas of Curves

14 Lecture 2:

12 A Short Atlas of Curves

8 a·sin(X) Y= X 6

a·sin(θ) 2 r = θ θ

-14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14

-2 Chapter 1 – Functions and Equations

Chapter 1: Functions and Equations

LECTURE TOPIC 0 GEOMETRY EXPRESSIONS™ WARM-UP 1 EXPLICIT, IMPLICIT AND PARAMETRIC EQUATIONS 2 A SHORT ATLAS OF CURVES 3 SYSTEMS OF EQUATIONS 4 INVERTIBILITY, UNIQUENESS AND CLOSURE

Lecture 2 – A Short Atlas of Functions 2 Learning Calculus with Geometry Expressions™

Calculus Inspiration Carolyn Shoemaker

Chapter 1 – Functions and Equations

Cocheloid and Sinc Function Lecture02-Cocheloid.gx 

8 a·sin(X) Y= X 6

a·sin(θ) 2 r = θ θ

-14 -12 -10 -8 -6 -4 -2 2 4 6 8 10 12 14

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Lecture 2 – A Short Atlas of Functions 4 Learning Calculus with Geometry Expressions™

CARTESIAN VS. POLAR COORDINATE SYSTEMS

Up to now we have worked in a Cartesian Coordinate System, a regular grid, in the x and y directions. We can also work in a complementary coordinate system called a Polar Coordinate System or just Polar Coordinates for short. A point in Polar Coordinates is not located by its (x, y) point pair, but rather by its radial distance r out, and its counter clockwise rotation,  from a horizontal axis. In Cartesian Coordinates the units of the point pair are often the same for x and y, feet, meters, etc. In Polar Coordinates, the radius r has units of distance, and  has angular units of measure. In Calculus, radians are always used instead of degrees so that the relationship between arc length s, and angular measure , are preserved as in:

Chapter 1 – Functions and Equations

CARTESIAN VS. POLAR COORDINATE SYSTEMS

(r,θ)

2 4 6 8 10 12 14 16 18

Plotting the same function in Polar Coordinates looks much different because instead of going over and up, in x and y-2, we are going out a distance r, and rotating by an angle .

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Lecture 2 – A Short Atlas of Functions 6

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-10 Learning Calculus with Geometry Expressions™

FileOpenLecture02-Bifolium.gx Bifolium Function 4.5

4.0

Exercises 3.5 1) Open Geometry Expressions™.

2) Open this example. 3.0 3) Vary parameter a in Variables Dialog. 4) Find an implicit expression for this curve. 2.5

2.0

1.5 B

1.0 a·sin(θ)·cos(θ)2

0.5

A θ

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-0.5

Chapter 1 – Functions and Equations

2.0 Cardiod Function Lecture02-Cardioid.gx

1.5

Explicit Polar Form: B ra (1  cos( )) 1.0 Implicit Cartesian Form: 0.5 (x2 y 2 ax) 2 a(x 2 2 y) 2

θ -2.5 -2.0 -1.5 -1.0 -0.5 A 0.5 1.0 1.5 2.0

-0.5 a·(1+cos(θ))

-1.0

-1.5

Lecture 2 – A Short Atlas of Functions 8 Learning Calculus with Geometry Expressions™

Cissoid of Diocles Function Lecture02-Cissoid.gx

4 Explicit Polar Form: B

ra sin( )  tan( ) 2 Implicit Cartesian Form: θ y23 ( a x ) x -10 -8 -6 -4 -2 A 2 4 6 8 10 Exercise:

• Vertical Asymptote at x = a -2 • Cusp at origin. a·sin(θ)·tan(θ)

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Chapter 1 – Functions and Equations

Sine and Cosine Functions Lecture02-SinCos.gx 6 Explicit Polar Form B

r a sin() 4

r a cos( ) a·sin(θ) C Explicit Cartesian Form 2

ysin ( ) 4 θ y cos( ) 6 -10 -8 -6 -4 -2 A 2 4 6 8 10 Exercise a·cos(θ) Drag Points P and B -2

P (x,-4+sin(x)) -4

-6 (x,-6+cos(x))

Lecture 2 – A Short Atlas of Functions 10 Learning Calculus with Geometry Expressions™

Cycloid Function Lecture02-Cycloid.gx

6 Parametric Cartesian Form: Flipping sign flips curve x a ( sin( )) 5 upside down. y a (1 cos( )) Features: 4 This Rolling Wheel Curve Flipped upside down solves the Brachistochrone Problem 3 a Exercises: 1) Animate Variables 2 P 2) Dragging P φ 3) Flip Sign

1 2 3 4 5 6 7 8 9 (a·(φ-sin(φ)),a·(1-cos(φ)))

Chapter 1 – Functions and Equations

Hyperbolic Functions Lecture02-SinhCoshTanh.gx 10 Explicit Form:

8 eexx sinh( x ) 2 6 eexx (x,cosh(x)) cosh( x ) 2 xx ee 4 tanh( x ) x,ex eexx

1 2 Exercise: x, Drag Key Colored Points. ex

-10 -8 -6 -4 -2 2 4 6 8

(x,tanh(x)) -2 (x,sinh(x))

Lecture 2 – A Short Atlas of Functions 12 Learning Calculus with Geometry Expressions™

8 Epicycloid: Wheel on Wheel Lecture02-Epicycloid.gx

6 Explicit Cartesian Form:

ab x ( a b )cos( ) b cos 4 b P ab b y ( a b )sin( ) b sin 2 b a

Exercise: θ -12 Animate-10 Parameters-8 a and-6 b. -4 -2 2 4 6 8

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-4 θ·(a+b) θ·(a+b) (a+b)·cos(θ)-b·cos ,(a+b)·sin(θ)-b·sin b b

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Chapter 1 – Functions and Equations

The Ellipse Lecture02-Ellipse.gx 7 Parametric Cartesian Form: Eccentricity : x a cos( ) 6 The quantity: y b sin( ) b2 5  1 a2 Exercise: Drag P, a, and b. 4 is the eccentricity of the ellipse. 2·a-t P 3

2 P' t (a·cos(θ),b·sin(θ))

θ Focus

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 b -1 b2 a· 1- a2 a -2

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Lecture 2 – A Short Atlas of Functions 14 Learning Calculus with Geometry Expressions™

Involute of A Circle Lecture02-InvoluteOfCircle.gx

12 B AB is a string unwrapped from a spool. The involute is the locus, the path of B. 10 Parametric Form: 8 x r cos( ) r  sin(  ) r·θ

y r sin( ) r  cos(  ) 6 Exercise: Animate the parameters r and . 4

r 2 A θ

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Chapter 1 – Functions and Equations 0.4 Evolute of an Ellipse Lecture02-EvoluteOfEllipse.gx

(a,b)

0.2

(a·sin(θ),b·cos(θ))

Evolutes and Involutes are -0.6 -0.4 -0.2 0.2 inverses0.4 operations.0.6 0.8

Ellipse Parametric Form: x a sin( ) -0.2 y b cos( ) Exercise: a2-b2 ·sin(θ)3 -a2+b2 ·cos(θ)3 1) Drag the point (a,b). , a·sin(θ)2+a·cos(θ)2 b·sin(θ)2+b·cos(θ)2 2) What is the evolute -0.4 of a circle?

Lecture 2 – A Short Atlas of Functions 16 (x,exp(x))

Learning Calculus with Geometry Expressions™

5 Natural Logarithm Spiral Lecture02-LogSpiral.gx

4 Explicit Polar Form:

3 r loge ( )

Exercise: 2 Animate . 1 θ

-8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5

log(θ) -1 P

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Chapter 1 – Functions and Equations

Lecture02-OffsetCurves.gx Offset Curves: A Powerful Tool 3.0

Steps to Create: 2.5 1) Define a curve with Point P.

2) Create tangent line to curve at P 2.0 3) Create perpendicular to tangent. 4) Create Point P’ on perpendicular. 1.5 5) Constrain Offset distance from P to P’. 6) Create the locus of P’. 7) Vary Offset to see various curves. 1.0 Offset

0.5 Original P' Curve

-4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0 2.5 3.0

-0.5 P

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Exercises: -1.5 1) Animate  and Offset . 2) Make an offset curve of your own. -2.0

Lecture 2 – A Short Atlas of Functions 18 Learning Calculus with Geometry Expressions™

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