# Analysis of Planar Curves

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## Lecture Notes

#### Parametric Equations

PDF
Section 1, Page 1 to page 2

Representing Cartesian coordinate curves using explicit and implicit forms. Representing curves using parametric equations which define x and y in terms of a third variable. Includes examples of parametric equations for a circle, ellipse, and projectile fired at an angle.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Implicitization

PDF
Section 2, Page 2 to page 3

Finding implicit forms for parameterized curves. Uses examples from the previous section of the notes.

Instructor: Prof. Jason Starr
Prior Knowledge: Parametric equations (section 1 of this lecture)

#### Polar Coordinate Curves

PDF
Section 3, Page 5 to page 6

Definition, with examples of circles and a horizontal line defined in polar coordinates.

Instructor: Prof. Jason Starr
Prior Knowledge: Parametric equations (section 1 of lecture 21)

## Online Textbook Chapter

#### Parametric Representation of a Curve

Document

Using parametric equations to define a curve in two or three dimensions and properties of parametric equations.

Instructor: Prof. Daniel J. Kleitman
Prior Knowledge: None

## Exam Questions

#### Parametric Equations

PDF - 2.2 MB#
Problem 4E-1 (page 31) to problem 4E-9 (page 31)

Nine questions which involve finding equations in rectangular coordinates for those given in parametric form, or putting a rectangular equation in parametric form.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Polar Coordinate Graphs

PDF- 2.2 MB#
Problem 4H-1 (page 32) to problem 4H-3 (page 33)

Three multi-part questions which involve converting rectangular coordinates to polar coordinates, converting polar equations to rectangular equations, and graphing curves given in polar coordinates.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Polar Curves

PDF
Problem 4 (page 5)

Sketching a curve in polar coordinates, and labeling the quadrants, endpoints, tangent slopes, and angles for the curve.

Instructor: Prof. Jason Starr
Prior Knowledge: None

#### Area Inside a Polar Curve

PDF
Problem 5 (page 1)

Sketching a curve defined in polar coordinates and finding the area inside it.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Polar Coordinates

PDF
Problem 9 (page 1)

Sketching a curve given in polar coordinates and finding points of intersection between that and other curves.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Polar Coordinates

PDF
Problem 5 (page 1)

Representing a circle using both rectangular and polar coordinates.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Integrals in Polar Coordinates: Lunar Eclipse

PDF
Problem 6 (page 1)

Setting up and evaluating an integral to represent the uncovered area of the two moons involved in a lunar eclipse on another planet.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Polar Coordinates: Spiral

PDF
Problem 5 (page 1)

Sketching a spiral defined in polar coordinates, counting the times it crosses the x-axis, and finding the area of specific regions of the spiral.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Lines in Polar Coordinates

PDF
Problem 6 (page 1)

Finding an equation in polar coordinates and the appropriate range of theta for a line given in rectangular coordinates.

Instructor: Prof. David Jerison
Prior Knowledge: None

#### Polar Curves

PDF
Problem 14 (page 2)

Sketching a curve given in polar coordinates and finding the area swept by a line segment as one of the endpoints moves along this curve.

Instructor: Prof. David Jerison
Prior Knowledge: None

## Java Applets

#### Polar Plotter

Java Applet
Requires Java Virtual Machine

Applet for showing the graph of a function defined in polar coordinates.

Instructor: Prof. Daniel J. Kleitman

#### Curves in Two Dimensions

Java Applet
Requires Java Virtual Machine

Applet for plotting curves defined in rectangular or parametric form.

Instructor: Prof. Daniel J. Kleitman