Match the parameterizations (I)-(IV) with the surfaces...

Match the parameterizations (I)-(IV) with the surfaces (a)-(d). In all cases $0 \leq s \leq \pi / 2,0 \leq t \leq \pi / 2$. Note that only part of the surface may be described by the given parameterization. (a) Cylinder (b) Plane (c) Sphere (d) Cone I. $x=\cos s, \quad y=\sin t, \quad z=\cos s+\sin t$ II. $x=\cos s, y=\sin s, \quad z=\cos t$ III. $x=\sin s \cos t, y=\sin s \sin t, \quad z=\cos s$ IV. $x=\cos s, \quad y=\sin t, \quad z=\sqrt{\cos ^2 s+\sin ^2 t}$

Match the parameterizations (I)-(IV) with the surfaces (a)-(d). In all cases $0 \leq s \leq \pi / 2,0 \leq t \leq \pi / 2$. Note that only part of the surface may be described by the given parameterization.
(a) Cylinder
(b) Plane
(c) Sphere
(d) Cone
I. $x=\cos s, \quad y=\sin t, \quad z=\cos s+\sin t$
II. $x=\cos s, y=\sin s, \quad z=\cos t$
III. $x=\sin s \cos t, y=\sin s \sin t, \quad z=\cos s$
IV. $x=\cos s, \quad y=\sin t, \quad z=\sqrt{\cos ^2 s+\sin ^2 t}$

Key Concepts

Surface Parameterization

Surface parameterization is the process of representing a surface in space by expressing its coordinates as functions of one or more independent parameters. This approach facilitates the analysis of the surface’s geometric properties, such as curvature and area, and is a fundamental tool in multivariable calculus and differential geometry.

Cylinder

A cylinder is a surface generated by translating a closed curve (often a circle) along a direction parallel to itself. Its parameterization typically involves one parameter that defines the circular cross?section (with a constant radius) and another that represents the height or axial displacement. This structure allows easy identification of the circular cross-section regardless of the axial coordinate.

Plane

A plane is a flat, two-dimensional surface extending infinitely in all directions. Parameterizing a plane usually involves expressing the coordinates as linear functions of two independent parameters, reflecting its uniform, linear structure. This method often leads to an equation where one coordinate is a linear combination of the others, which is characteristic of planar surfaces.

Sphere

A sphere is the set of points that are all at a constant distance from a fixed center, forming a perfectly symmetric three-dimensional object. Its common parameterization uses angles (as in spherical coordinates) and a constant radius, allowing for the coverage of the entire surface of the sphere. This angular parameterization is particularly useful in addressing problems in physics and geometry.

Cone

A cone is a surface formed by connecting each point of a closed curve (the directrix) to a single point called the apex. The parameterization of a cone often reveals a relationship in which the distance from the apex is proportional to the radial distance in the base plane, typically resulting in an equation where the square of the vertical coordinate equals the sum of the squares of the horizontal coordinates. This relationship is key to identifying the conical structure.

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