Section 17.5 Parameterized Surfaces

Recall Parametric curves in 3-space In rectangular coordinates (x,y,z) we have We used these to make a helix

Parameterized curves in spherical coordinates In spherical coordinates we have (ρ, , θ) As parametric equations we let C be the curve We used these to make a spherical helix

Now we are going to move on to parameterized surfaces What would we get from the following set of parametric equations? Let’s take a look with Maple

In the first two cases we have a parameterized curve in 3 space In the third case we have a parameterized surface as our parametric equations were in terms of two variables In rectangular coordinates, parameterized surfaces are given by where a ≤ u ≤ b and c ≤ v ≤ d Both u and v are object parameters Can do in cylindrical and spherical also

Let’s see how we can parameterize a sphere In rectangular coordinates In spherical coordinates How about a Torus One way of thinking about a Torus is a circle rotated around the z-axis Let’s take a look in maple

Generalized Torus Say we want to create a Torus that is not circular Essentially we want to rotate some region in a plane around the third axis (5,1) (12,6) (8,14) x z

Generalized Torus We need to make parametric equations for the triangular region This will be easier to do in cylindrical coordinates (5,1) (12,6) (8,14) x z

Parameterizing Planes The plane through the point with position vector and containing the two nonparallel vectors and is given by So if The parametric equations are

Parameterizing Surfaces of Revolution We can create surfaces that have an axis rotational symmetry and circular cross sections to that axis For example, how about a cone that has a base that is a circle of radius 3 in the xy-plane and a height of 10. The following structure can be used to revolve a curve around the z axis This can be modified to revolve around other axes as well

Parameter Curves Parameter curves are obtained by setting one of the parameters to a constant and letting the other vary Take the following parametric equations What do they give us? What would we get if z is held constant? What would we get if t is held constant? These parameter curves are cross sections of our parameterized surface