11.7
Cylinders in Three-Dimensional Space
In three-dimensional space, a cylindrical surface forms when a two-dimensional curve extends along a straight line. This curve is called a profile curve and represents the cross-section of a shape. This process creates a surface made up of parallel lines, called rulings, that follow a fixed direction.
In Cartesian coordinates, a cylindrical surface is often identified when an equation does not include one of the three variables. For example, in y = x2, the variable z does not appear.
This equation gives a parabolic curve in the xy-plane, and the absence of z shows that the curve extends infinitely along the z-axis. The surface formed in this way is called a parabolic cylinder.
A similar idea applies to circular profile curves. For example, a circle in the xy-plane can be extended along the z-axis to form a circular cylinder, like a pipe with the same cross-section at every height.
In architecture and engineering, 3D models of structures such as tunnels and arches can be created in parametric modeling software by defining a cross-sectional curve mathematically. This curve is then extended in a chosen direction to form a cylindrical surface.
A cylindrical surface is generated when a two-dimensional profile curve is translated along a straight line in three-dimensional space. The translated copies of the curve form a surface composed of parallel rulings, each oriented in the same fixed direction. This construction allows many three-dimensional forms to be described using relatively simple planar equations.
In Cartesian coordinates, a cylindrical surface is often recognized by an equation that omits one of the three variables. For example,
\begin{equation*}y = x^2\end{equation*}
does not include the variable z. This equation describes a parabola in the xy-plane. Because z is unrestricted, the parabolic curve extends infinitely in the z-direction, producing a parabolic cylinder. Every plane parallel to the xy-plane intersects the surface in the same parabolic profile.
The same principle applies to circular profile curves. A circle in the xy-plane, such as
\begin{equation*}x^2 + y^2 = r^2\end{equation*}
can be extended along the z-axis to form a circular cylinder. The resulting surface has the same circular cross-section at every height, similar to a pipe or column. Other profile curves, including ellipses or hyperbolas, can generate corresponding cylindrical surfaces when translated along a fixed direction.
In three-dimensional space, a cylindrical surface forms when a two-dimensional curve extends along a straight line. This curve is called a profile curve and represents the cross-section of a shape. This process creates a surface made up of parallel lines, called rulings, that follow a fixed direction.
In Cartesian coordinates, a cylindrical surface is often identified when an equation does not include one of the three variables. For example, in y = x2, the variable z does not appear.
This equation gives a parabolic curve in the xy-plane, and the absence of z shows that the curve extends infinitely along the z-axis. The surface formed in this way is called a parabolic cylinder.
A similar idea applies to circular profile curves. For example, a circle in the xy-plane can be extended along the z-axis to form a circular cylinder, like a pipe with the same cross-section at every height.
In architecture and engineering, 3D models of structures such as tunnels and arches can be created in parametric modeling software by defining a cross-sectional curve mathematically. This curve is then extended in a chosen direction to form a cylindrical surface.
From Chapter 11:
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11.7 : Cylinders in Three-Dimensional Space
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11.1 : Introduction to Vectors
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11.2 : Vectors in Engineering Applications
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11.3 : Dot Product of Two Vectors
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11.4 : Cross Product and Its Geometry
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11.5 : Lines in Space
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11.6 : Planes in Space
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11.8 : Quadric Surfaces
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11.9 : Vectors in 2D: Problem Solving
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11.10 : Vectors in Space: Problem Solving
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