Name: modeling cylindrical surfaces in 3d
Uploaded: 2026-06-08T09:38:50.491Z
Duration: 1 min 28 s
Channel: JoVE

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.

cylinders in three-dimensional space

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.

Cylinders in Three-Dimensional Space

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Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Vectors in Space

المتجهات في الفضاء

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الرئيسية

الأسطوانات في الفضاء ثلاثي الأبعاد

modeling cylindrical surfaces in 3d

Cylindrical surfaces in three-dimensional space come from moving a two-dimensional curve in a straight line. As the curve slides, it leaves behind translated copies that build a surface. These copies form parallel rulings, which are straight lines in the same fixed direction.

This idea helps describe some 3D shapes with simple equations in Cartesian coordinates. A cylindrical surface is often shown by an equation that leaves out one of the three variables. For example, y = x^2 does not include z. The equation gives a parabola in the xy-plane, and because z is unrestricted, the curve extends forever in the z-direction to make a parabolic cylinder.

Cross-sections show the same shape at every matching slice. Any plane parallel to the xy-plane cuts the parabolic cylinder in the same parabolic profile. The same pattern also works for circles. A circle such as x^2 + y^2 = r^2 in the xy-plane can be extended along the z-axis to form a circular cylinder, like a pipe or column, with the same circular cross-section at every height.

Other profile curves can make their own cylindrical surfaces too. Ellipses and hyperbolas are two examples. When one of these curves is translated along a fixed direction, it creates the matching cylindrical form.

Modeling Cylindrical Surfaces in 3D

introduction to vectors

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vectors in engineering applications

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dot product of two vectors

Vector Projection and the Dot Product

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cross product and its geometry

Cross Product, Area, and Torque

The cross product connects vector direction, area, and rotation in three-dimensional space. When two non-zero vectors are not parallel, they define a unique plane and form a parallelogram. Their cross product gives a third vector that is perpendicular to that plane. The direction of this new vector is found with the Right-Hand Rule. Point the fingers of your right hand along the first vector and curl them toward the second vector. Your thumb shows the direction of the result. This convention...

lines in space

Vector Form for a Line in Space

A line in three-dimensional space can be described with a vector equation. This method uses a known point on the line and a direction vector. It is a useful way to model straight paths in space. The transcript uses a surveying laser beam as an example. A laser beam from a surveying instrument can be treated as a line across a construction site. The beam starts at a known point, called P₀, whose position vector is r₀. The beam also has a direction vector v, which is parallel to its path. Any...

planes in space

Plane Equations from a Point and Normal

A plane in three-dimensional space can be described by one point on the plane and a normal vector. The normal vector is perpendicular to the surface, and it sets the plane’s orientation. In geometry, this gives a clear way to model a flat surface in space. The transcript uses a sloped glass panel on a building façade as a real-world example. Let r₀ be the position vector of a known point on the panel, such as the base mounting point. Let n be the normal vector to the panel. Any other point r...

Cylindrical surfaces in three-dimensional space come from moving a two-dimensional curve in a straight line. As the curve slides, it leaves behind translated copies that build a surface. These copies form parallel rulings, which are straight lines in the same fixed direction. This idea helps describe some 3D shapes with simple equations in Cartesian coordinates. A cylindrical surface is often shown by an equation that leaves out one of the three variables. For example, y = x^2 does not include...

quadric surfaces

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vectors in 2d: problem solving

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vectors in space: problem solving

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key terms and concepts

Learn how translating a curve creates cylindrical surfaces in 3D, including parabolic and circular cylinders, cross-sections, and equations that omit one variable.

Modeling Cylindrical Surfaces in 3D | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Modeling Cylindrical Surfaces in 3D

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