1. Section 17.1 Parameterized Curves

2. Parameterized curves in 2-space
Consider the graph of y = f(x), a ≤ x ≤ b
The equation for this graph can be expressed using the following parametric equations:
In general let C be the curve given by
Let’s look at a couple examples in maple

3. Parameterized curves in polar coordinates
In cylindrical coordinates we have r = f(θ), α ≤ θ ≤ β
As parametric equations we let C be the curve
Let’s take a look in maple

4. Parametric curves in 3-space
In rectangular coordinates (x,y,z) we have
Let’s take a look in maple to see how we can plot a circle that is perpendicular to the xy-plane
We have seen how to make a helix, now let’s make the parametric equations for a helix with a radius of 10 and a height of 10 that has 2 windings
Notice that z is a linear function of t
Can you find the equations in cylindrical coordinates?
What if we want our helix to have a conical shape?
What about a spherical shape?

5. Parameterized curves in spherical coordinates
In spherical coordinates we have (ρ, , θ)
As parametric equations we let C be the curve
Let’s take a look in maple

6. Relationship between parametric equations and vectors
Given parametric equations
we have a line that goes through the point (a,b,c) with slopes of m, n and p in the x, y and z directions, respectively
Thus the slopes are giving us the direction and give us the make up of our vector
The corresponding vector is