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MATH 251, Section 516 : REVIEW
Parametric equations of some ”classical” surfaces
A surface can be parametrized with only 2 parameters, it means that every point (x, y, z) of a surface
S can be described as (x(u, v), y(u, v), z(u, v)) where (u, v) ∈ D where D is the domain of parameters and
depends on S.
S is traces out by the position vector r(u, v) as (u, v) moves throughout the domain D.
Cone :
– with x-axis :
symmetric equation : y 2 + z 2 = x2 , parametric equation : ~r(u, v) = hu, u cos v, u sin vi, (u, v) ∈ D
– with y-axis :
symmetric equation : y 2 = x2 + z 2 , parametric equation : ~r(u, v) = hu cos v, u, u sin vi, (u, v) ∈ D
– with z-axis :
symmetric equation : z 2 = x2 + y 2 , parametric equation : ~r(u, v) = hu cos v, u sin v, ui, (u, v) ∈ D
Sphere :
symmetric equation of a sphere with center (0, 0, 0) and radius a : x2 + y 2 + z 2 = a2 .
parametric equation (think to use spherical coordinates where r is fixed equal to a :
~r(θ, φ) = hcos θ sin φ, sin θ sin φ, cos φi.
Cylinder :
– with x-axis : y 2 + z 2 = 1, parametric equation : ~r(x, t) = hx, cos t, sin ti
– with y-axis : x2 + z 2 = 1, parametric equation : ~r(y, t) = hcos t, y, sin ti
– with z-axis : x2 + y 2 = 1, parametric equation : ~r(z, t) = hcos t, sin t, zi
When the surface is given by the explicit graph of a function f of several variables :
– z = f (x, y)
– y = f (x, z)
– x = f (y, z)
~r(x, y) = hx, y, f (x, y)i
~r(x, z) = hx, z, f (x, z)i
~r(y, z) = hf (y, z), y, zi.
Examples :
1. Cylinder : x2 + y 2 = 9, 1 ≤ z ≤ 5.
A parametric equation : ~r(t, z) = h3 cos t, 3 sin t, zi, (t, z) ∈ D = {(t, z) | 0 ≤ t ≤ 2π, 1 ≤ z ≤ 5}.
p
2. upper half sphere : z = 100 − x2 − y 2 .
The radius is 10 since we have x2 + y 2 + z 2 = 102 = 100.
A parametric equation is :
~r(θ, φ) = h10 cos θ sin φ, 10 sin θ sin φ, 10 cos φi,
where (θ, φ) verifies 0 ≤ θ ≤ 2π and 0 ≤ φ ≤
π
2
(upper half sphere).
3. Elliptic paraboloid : x = 3y 2 + z 2 + 1, where 1 ≤ x ≤ 2.
We have x = f (y, z), so ~r(y, z) = h3y 2 + z 2 + 1, y, zi, where 1 ≤ x ≤ 2 implies 1 ≤ 3y 2 + z 2 + 1 ≤ 2,
2
so D = {(y, z) | 0 ≤ 3y 2 + z 2 ≤ 1}, so in polar coordinates (y =
and 0 ≤ r ≤ 1.
√1 r cos t, z
3
= r sin t) with 0 ≤ t ≤ 2π
4. Elliptic paraboloid : y = x2 + 4z 2 , 0 ≤ y ≤ 1. A parametric equation is ~r(x, z) = hx, x2 + 4z 2 , zi, with
D = {(x, z) | 0 ≤ x2 + 4z 2 ≤ 1}. So, in polar coordinates, we can also obtain x = r cos θ, z = 21 r sin θ
for 0 ≤ r ≤ 1 and 0 ≤ θ ≤ 2π.
We can also have the following parametric equation : ~r(r, θ) = hr cos θ, r2 , 21 r sin θi.
remark= we can do the same with the previous example.