Calc 2 Lecture Notes Section 9.1 Page 1 of 11

Section 9.1: Plane Curves and Parametric EquationsBig idea:. Points on a graph can be represented by two separate equations for the x and y coordinates in terms of a third variable (usually called t), instead of a single equation. The variable t represents time in many applications, which makes parametric equations useful for describing the location of objects as a function of time.

Big skill: You should be able to graph parametric equations by hand and by using a calculator or Winplot. You should also be able to recognize and easily graph the parametric representation of some common curves like lines, parabolas, circles, and ellipses.

Parametric equations are obtained by setting coordinates equal to functions of a third variable:

Both functions have the same domain D for the independent variable t.

When the variables are interpreted as being the coordinates of a point, then the collection of all such points is called the graph of the parametric equations.

The independent variable t is often thought of as representing time.

Calc 2 Lecture Notes Section 9.1 Page 2 of 11

Example: Graph the following parametric equations by hand and on your calculator and on Winplot for the domain -2 t 4. Notice how a graph (which is not the graph of a function) is generated without using cumbersome y = … notation.

t x y

Calc 2 Lecture Notes Section 9.1 Page 3 of 11

Example: Graph the path of a projectile given by the parametric equations:

When does the projectile hit the ground? What is the shape of the trajectory?

Example: Create two sets of parametric equations to graph the equation x2 + y3 – 2y = 3 by setting y = t.

Notice that even though two sets of parametric equations are required, they at least provide a way to graph this equation without solving for y, as required by a graphing calculator (Winplot can graph implicit equations, of course).

Note: must have t 1.89329 for real values of x. Note: this is an elliptic curve (see page 10)…

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Example: Parametric equations for lines.

If , then ,

OR if , then .

Practice:

1. Write parametric equations for the line .

2. Write a Cartesian equation for the line .

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Example: Parametric equations for circles.

If , then (and vice-versa) for .

Or for

Practice:3. Write parametric equations for the circle .

4. Write a Cartesian equation for the circle .

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Example: Parametric equations for ellipses.

If , then (and vice-versa) for .

Or for

Practice:5. Write parametric equations for the ellipse .

6. Write a Cartesian equation for the ellipse .

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Example: Parametric equations for hyperbolas.

If , then (and vice-versa) for .

Or for Or for

Practice:7. Write parametric equations for the hyperbola .

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Example: Lissajous curves are the family of curves described by the parametric equations

. They were first studied by Bowditch, and then later and independently by

Lissajous. If is rational, then the curve closes in on itself.

Citation: Weisstein, Eric W. "Lissajous Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LissajousCurve.html

Practice:

8. Graph the Lissajous curve .

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Example: The trisectrix of MacLaurin is given by the Cartesian equation , and was first studied by Colin Maclaurin in 1742 to help solve the

problem of trisecting an angle (hence the name “trisectrix”).

Citation: Weisstein, Eric W. "Maclaurin Trisectrix." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MaclaurinTrisectrix.html

A trivial parameterization of the MacLaurin trisectrix is:

A better parameterization is: . Verify and graph this parameterization.

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More unique equations:Elliptic curve: y2 = Ax3 + Bx + CElliptic curves are important in number theory, and constitute a major area of current research; for example, they were used in the proof, by Andrew Wiles (assisted by Richard Taylor), of Fermat's Last Theorem. They also find applications in cryptography and integer factorization. Interesting note: elliptic curves can not be parameterized using rational expressions.Citation:http://en.wikipedia.org/wiki/Elliptic_curve

Calc 2 Lecture Notes Section 9.1 Page 11 of 11

Cissoid of Diocles: x3 + xy2 – 2ay2 = 0

or

The cissoid of Diocles is a cubic curve invented by Diocles in about 180 BC in connection with his attempt to duplicate the cube by geometrical methods.Citation:Weisstein, Eric W. "Cissoid of Diocles." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CissoidofDiocles.html

More specially named curves:Bipartite Cubic: y2 = x(x – a)(x – b)Semi-Cubical Parabola: ay2 = x3 (a special case of an elliptic curve)The Witch of Agnesi: x2y + 4a2y – 8a3 = 0The Strophoid: x3 + xy2 + ax2 – ay2 = 0The Serpentine:The Folium:

The Lemniscate:

The Top:

The Bifolium: