Section 17.1 Parameterized Curves

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

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

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?

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

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