EMAT 4680/6680 Explorations 10:

Parametric Curves

Last modified on

A parametric curve in the plane is a pair of functionsx = f(t)

y = g(t)

where the two continuous functions define ordered pairs

(x,y). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range oftand your work with parametric equations should pay close attention the range oft. In many applications, we think ofxandy"varying with timet" or the angle of rotation that some line makes from an initial location.

Various graphing technology, such as the TI-81, TI-82, TI-83, TI-84,TI-85, TI-86, TI-89, TI- 92, and Graphing Calculator 3.5 or 4.0 can be readily used with parametric equations. TryGraphing Calculator 3.5or GC 4.0 for what is probably the friendliest software. Geogebra and Desmos can also deal with parametric curves. Graphing Calculator 4.0 Lite DOES NOT handle parametric equations.Note: Graphing technologies compute values of (x,y) for increments of

tand then construct a line segment connecting them. When the increment oftis small these are very short segments and the curve is simulated. The TI instruments include a 'step' setting for the increments of 't' and it is possible for consecutive (x,y) to be rather far apart. This can produce interesting drawings but misrepresent the parametric curve given by the set of points. In other words, the 'step' setting is a way of drawing segments between regularly spaced but not adjacent points on the parametric curve.

1. Graph

How would you change the equations to explore other graphs?

2. For various

aandb, investigate

3. For various

aandb, investigate

4. Graph

Interpret. What would you change to explore and understand the graphs?

5. Graph several sets of curves for

for selected values of

a,b, andkin an appropriate range fort. (Setaandband then overlay graphs for several values ofk. Repeat for new values ofaandb.)

6. Graph

for some appropriate range for

t.

Interpret. Is there anything to vary to help understand the graph?

7. Write parametric equations of a line segment through (7, 5) with slope of 3. Graph the line segment using your equations. As a line segment, it will have end points. Explore how you would choose endpoints of such that the two distances from (7, 5) are 2 units and 3 units.

8. Investigate

for different values of

aandb. What is the curve whena<b?a=b?a>b?

Describe fully. What is changed if the equations are

where

his any real number?

Investigate with graphs for smallh(e.g. -3 <h< 3).

NOTE:These latter parametric equations describe the locus of the vertex (x,y) of a triangle with altitude h whose other two vertices are moved, one along the x-axis and the other along the y-axis.

9. Derive the parametric equations for the locus of a point (x, y) on a line segment that is moved so that one end is on the x-axis and the other end is on the y-axis.

10. Investigate each of the following for

Describe each when

a=b,a<b, anda>b.

11. Consider

The equation is easy to graph with Graphing Calculator 3.5 or 4.0 but would be difficult to graph with most Function Grapher applications. Putting it in parametric form makes it possible to graph with many other applications (including the TI-81 or TI-82).

Lety = txcross the curve at(0,0)and at(x,y). Then, substituting,

and so we have the parametric equations

Graph the curve using the parametric equations for a suitable range of

t.

Graph

using Graphing Calculator 3.5 or 4.0 for comparison.

12. Investigation. Consider the parametric equations

Graph these for

Describe fully. You may have to increase the range of

tfor the larger fractions. This class of parametric curves are called theLissajouscurves. Compare with

x = sin((a)t)

y = sin((b)t)

13. A

cycloidis the locus of a point on a circle that rolls along a line. Write parametric equations for the cycloid and graph it. Consider also a GSP construction of the cycloid. ClickHEREto see a GSP animation showing the cycloid.

Click

HEREfor a derivation of parametric equations for the cycloid.

Return to