Understanding Line Parametrization - Explanation and Examples

[ranger1716]

could someone please explain...

first of all hi!, I'm new here, even though I've lurked these forums before.

Anyways, my question is:

could someone define what exactly a parametrization of a line means?

my calc teacher just kind of jumped headlong into it, and I'm having trouble grasping the concept.

thanks

 

[quantumdude]

I think the best way to introduce it is by a physical example.

Consider a race car blazing down a straight drag strip for a quarter mile (1320 feet), and let's say he covers the distance in 10 seconds. Let the origin of a coordinate system lie at the Start line and let the Finish line lie to the right of the Start line on the x-axis. The equation of the line is y=0.

Now consider a beat up Chevy Nova taking on the same drag strip, but it runs the quarter mile in 30 seconds. What's the equation of the line it follows? Again, it's y=0.

Both cars followed the same trajectory, but clearly both trips weren't identical. One car was much faster than the other. This difference can be expressed quantitatively by introducing a parameterization[/color] of the line y=0. In this case, the parameter is time.

The drag racer covered 1320 feet in 10 seconds, which means it had a velocity of 132 ft/s. The parameterization of his run is then:

$$x=0$$
$$y=132t$$
$$0 \leq t \leq 10$$.

Similarly, the parameterization of the Nova's run is:

$$x=0$$
$$y=44t$$
$$0 \leq t \leq 30$$.

If you want to see a physical example of parameterization of a parabola, you have only to look in your Physics I book in the chapter on projectile motion.

Does that help?

 

[stunner5000pt]

you can write a line in space relating vectors (if you know what a vector is this will come easily)
imagine you have a line taht points in some direction. That line is a vector so it can be moved anywhere in space as long as its orientation (angle it makes with x,y,z axes) is kept the same. Ok now since you can move this anywhere, you can make it go through some artirary point. Now you have a line that is unique because it passes through one point and heads in only two directions. You can make this line longer or shorter depending on how you want it, by simply multiplying the original vector (directional vector) by some parameter

for a line which passes through a point Xo,Yo,Zo with directional vector d1,d2,d3, a point on thsi line is defined by

$$(x,y,z) = (x_{0},y_{0},z_{0}) + t (d_{1},d_{2},d_{3})$$
t can be any real number

 

[ranger1716]

I think that does help to clear up what we've been doing.

so essentially, a parametrization is basically just a description of the line or curve?

 

[stunner5000pt]

I think that does help to clear up what we've been doing.

so essentially, a parametrization is basically just a description of the line or curve?

it could apply for both a line or a curve or a surface (plane,cone,cylinder, and other fun looking shapes)