Parametrized Surfaces
Just as it’s sometimes simpler to represent a curve in ℝ3 (𝑜𝑟 ℝ2 ) in terms of
parametric equations: 𝑥 = 𝑥(𝑡 ), 𝑦 = 𝑦(𝑡 ), 𝑧 = 𝑧(𝑡), instead of trying to
represent it as the intersection of 2 surfaces, 𝑧 = 𝑓 (𝑥, 𝑦) and 𝑧 = 𝑔(𝑥, 𝑦) (which
can’t always be done), it is sometimes simpler to represent a surface in ℝ3 in
parametric form: 𝑥 = 𝑥(𝑢, 𝑣), 𝑦 = 𝑦(𝑢, 𝑣), 𝑧 = 𝑧 (𝑢, 𝑣), instead of 𝑧 = 𝑓(𝑥, 𝑦)
(which can’t always be done).
Ex. Even a simple surface like the cylinder 𝑥 2 + 𝑧 2 = 4, can’t be represented as a
simple function 𝑧 = 𝑓(𝑥, 𝑦) (in this case we would have 2 functions 𝑧 = √4 − 𝑥 2
and 𝑧 = −√4 − 𝑥 2 ). However, parametrically we can represent this cylinder by:
𝑥 = 2𝑐𝑜𝑠𝑢, 𝑦 = 𝑣, 𝑧 = 2𝑠𝑖𝑛𝑢 ; 0 ≤ 𝑢 ≤ 2𝜋, 𝑣 ∈ ℝ (notice that 𝑥, 𝑦, and 𝑧
have to satisfy the original equation: 𝑥 2 + 𝑧 2 = 4).
In general, we can represent a surface in parametric form as:
𝑥 = 𝑥(𝑢, 𝑣), 𝑦 = 𝑦(𝑢, 𝑣), 𝑧 = 𝑧(𝑢, 𝑣), and in vector form by:
⃗ (𝑢, 𝑣) =< 𝑥(𝑢, 𝑣), 𝑦(𝑢, 𝑣), 𝑧(𝑢, 𝑣) >; where 𝛷
𝛷 ⃗ : 𝐷 ⊂ ℝ2 → ℝ3 .
⃗ , i.e. 𝛷
The surface, S, is the image of 𝛷 ⃗ (𝐷), and 𝛷
⃗ is called a parametrization of
S. For any surface there are an infinite number of parametrizations.
S is called a Differentiable Surface, (or a 𝑪𝟏 Surface) if 𝑥(𝑢, 𝑣), 𝑦(𝑢, 𝑣), 𝑧(𝑢, 𝑣),
are differentiable (or 𝐶 1)
⃗ (𝑢, 𝑣 ) =< 2𝑐𝑜𝑠𝑢, 𝑣, 2𝑠𝑖𝑛𝑢 >, 0 ≤ 𝑢 ≤ 2𝜋, 𝑣 ∈ ℝ is a
Ex. 𝛷
parametrization of the circular cylinder 𝑥 2 + 𝑧 2 = 4. 𝑧
2 2 2 2
(Notice: 𝑥 + 𝑧 =(2𝑐𝑜𝑠𝑢) + (2𝑠𝑖𝑛𝑢) = 4).
𝑥
, 2
Ex. Notice that any surface 𝑧 = 𝑓 (𝑥, 𝑦); e.g., 𝑧 = 𝑥 2 + 𝑦 2 , can be
parametrized by: 𝑥 = 𝑢, 𝑦 = 𝑣, 𝑧 = 𝑓(𝑢, 𝑣 ), ie,
⃗⃗⃗
𝛷(𝑢, 𝑣) =< 𝑢, 𝑣, 𝑓(𝑢, 𝑣) >.
In the case of
𝑧 = 𝑥 2 + 𝑦 2 ; 𝑥 = 𝑢, 𝑦 = 𝑣, 𝑧 = 𝑢2 + 𝑣 2 , i.e.,
⃗⃗⃗
𝛷(𝑢, 𝑣) =< 𝑢, 𝑣, 𝑢2 + 𝑣2 >.
𝑦
𝑥
Ex. (Important Example) Find a parametrization of the sphere of radius R,
𝑥 2 + 𝑦2 + 𝑧 2 = 𝑅2 .
One standard parametrization is to use spherical coordinates:
𝑥 = 𝑅 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 𝑧
𝑦
𝑦 = 𝑅𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙
𝑧 = 𝑅𝑐𝑜𝑠𝜙
𝑥
where 0 ≤ 𝜙 ≤ 𝜋, and 0 ≤ 𝜃 ≤ 2𝜋.
Just as it’s sometimes simpler to represent a curve in ℝ3 (𝑜𝑟 ℝ2 ) in terms of
parametric equations: 𝑥 = 𝑥(𝑡 ), 𝑦 = 𝑦(𝑡 ), 𝑧 = 𝑧(𝑡), instead of trying to
represent it as the intersection of 2 surfaces, 𝑧 = 𝑓 (𝑥, 𝑦) and 𝑧 = 𝑔(𝑥, 𝑦) (which
can’t always be done), it is sometimes simpler to represent a surface in ℝ3 in
parametric form: 𝑥 = 𝑥(𝑢, 𝑣), 𝑦 = 𝑦(𝑢, 𝑣), 𝑧 = 𝑧 (𝑢, 𝑣), instead of 𝑧 = 𝑓(𝑥, 𝑦)
(which can’t always be done).
Ex. Even a simple surface like the cylinder 𝑥 2 + 𝑧 2 = 4, can’t be represented as a
simple function 𝑧 = 𝑓(𝑥, 𝑦) (in this case we would have 2 functions 𝑧 = √4 − 𝑥 2
and 𝑧 = −√4 − 𝑥 2 ). However, parametrically we can represent this cylinder by:
𝑥 = 2𝑐𝑜𝑠𝑢, 𝑦 = 𝑣, 𝑧 = 2𝑠𝑖𝑛𝑢 ; 0 ≤ 𝑢 ≤ 2𝜋, 𝑣 ∈ ℝ (notice that 𝑥, 𝑦, and 𝑧
have to satisfy the original equation: 𝑥 2 + 𝑧 2 = 4).
In general, we can represent a surface in parametric form as:
𝑥 = 𝑥(𝑢, 𝑣), 𝑦 = 𝑦(𝑢, 𝑣), 𝑧 = 𝑧(𝑢, 𝑣), and in vector form by:
⃗ (𝑢, 𝑣) =< 𝑥(𝑢, 𝑣), 𝑦(𝑢, 𝑣), 𝑧(𝑢, 𝑣) >; where 𝛷
𝛷 ⃗ : 𝐷 ⊂ ℝ2 → ℝ3 .
⃗ , i.e. 𝛷
The surface, S, is the image of 𝛷 ⃗ (𝐷), and 𝛷
⃗ is called a parametrization of
S. For any surface there are an infinite number of parametrizations.
S is called a Differentiable Surface, (or a 𝑪𝟏 Surface) if 𝑥(𝑢, 𝑣), 𝑦(𝑢, 𝑣), 𝑧(𝑢, 𝑣),
are differentiable (or 𝐶 1)
⃗ (𝑢, 𝑣 ) =< 2𝑐𝑜𝑠𝑢, 𝑣, 2𝑠𝑖𝑛𝑢 >, 0 ≤ 𝑢 ≤ 2𝜋, 𝑣 ∈ ℝ is a
Ex. 𝛷
parametrization of the circular cylinder 𝑥 2 + 𝑧 2 = 4. 𝑧
2 2 2 2
(Notice: 𝑥 + 𝑧 =(2𝑐𝑜𝑠𝑢) + (2𝑠𝑖𝑛𝑢) = 4).
𝑥
, 2
Ex. Notice that any surface 𝑧 = 𝑓 (𝑥, 𝑦); e.g., 𝑧 = 𝑥 2 + 𝑦 2 , can be
parametrized by: 𝑥 = 𝑢, 𝑦 = 𝑣, 𝑧 = 𝑓(𝑢, 𝑣 ), ie,
⃗⃗⃗
𝛷(𝑢, 𝑣) =< 𝑢, 𝑣, 𝑓(𝑢, 𝑣) >.
In the case of
𝑧 = 𝑥 2 + 𝑦 2 ; 𝑥 = 𝑢, 𝑦 = 𝑣, 𝑧 = 𝑢2 + 𝑣 2 , i.e.,
⃗⃗⃗
𝛷(𝑢, 𝑣) =< 𝑢, 𝑣, 𝑢2 + 𝑣2 >.
𝑦
𝑥
Ex. (Important Example) Find a parametrization of the sphere of radius R,
𝑥 2 + 𝑦2 + 𝑧 2 = 𝑅2 .
One standard parametrization is to use spherical coordinates:
𝑥 = 𝑅 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 𝑧
𝑦
𝑦 = 𝑅𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙
𝑧 = 𝑅𝑐𝑜𝑠𝜙
𝑥
where 0 ≤ 𝜙 ≤ 𝜋, and 0 ≤ 𝜃 ≤ 2𝜋.
