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A cylinder (from Ancient Greek (klindros) 'roller, tumbler')[1] has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.

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A solid bounded by a cylindrical surface and two parallel planes is called a (solid) cylinder. The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder. All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of the parallel planes is called a base of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a right cylinder, otherwise it is called an oblique cylinder. If the bases are disks (regions whose boundary is a circle) the cylinder is called a circular cylinder. In some elementary treatments, a cylinder always means a circular cylinder.[2]

The cylinder obtained by rotating a line segment about a fixed line that it is parallel to is a cylinder of revolution. A cylinder of revolution is a right circular cylinder. The height of a cylinder of revolution is the length of the generating line segment. The line that the segment is revolved about is called the axis of the cylinder and it passes through the centers of the two bases.

The bare term cylinder often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an open cylinder. The formulae for the surface area and the volume of a right circular cylinder have been known from early antiquity.

A right circular cylinder can also be thought of as the solid of revolution generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution.[3]

A cylindric section is the intersection of a cylinder's surface with a plane. They are, in general, curves and are special types of plane sections. The cylindric section by a plane that contains two elements of a cylinder is a parallelogram.[4] Such a cylindric section of a right cylinder is a rectangle.[4]

A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section.[5] If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively.

For a right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element. The right sections are circles and all other planes intersect the cylindrical surface in an ellipse.[6] If a plane intersects a base of the cylinder in exactly two points then the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle.

In the case of a right circular cylinder with a cylindric section that is an ellipse, the eccentricity e of the cylindric section and semi-major axis a of the cylindric section depend on the radius of the cylinder r and the angle tag_hash_129 between the secant plane and cylinder axis, in the following way:

For a given volume, the right circular cylinder with the smallest surface area has h = 2r. Equivalently, for a given surface area, the right circular cylinder with the largest volume has h = 2r, that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle).[8]

A right circular hollow cylinder (or cylindrical shell) is a three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis, as in the diagram.

In some areas of geometry and topology the term cylinder refers to what has been called a cylindrical surface. A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line.[12] Such cylinders have, at times, been referred to as generalized cylinders. Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder.[13] Thus, this definition may be rephrased to say that a cylinder is any ruled surface spanned by a one-parameter family of parallel lines.

A cylinder having a right section that is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder and hyperbolic cylinder, respectively. These are degenerate quadric surfaces.[14]

If AB > 0 this is the equation of an elliptic cylinder.[15] Further simplification can be obtained by translation of axes and scalar multiplication. If {\displaystyle \rho } has the same sign as the coefficients A and B, then the equation of an elliptic cylinder may be rewritten in Cartesian coordinates as:

This equation of an elliptic cylinder is a generalization of the equation of the ordinary, circular cylinder (a = b). Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plcker conoid.

In projective geometry, a cylinder is simply a cone whose apex (vertex) lies on the plane at infinity. If the cone is a quadratic cone, the plane at infinity (which passes through the vertex) can intersect the cone at two real lines, a single real line (actually a coincident pair of lines), or only at the vertex. These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively.[17]

A solid circular cylinder can be seen as the limiting case of a n-gonal prism where n approaches infinity. The connection is very strong and many older texts treat prisms and cylinders simultaneously. Formulas for surface area and volume are derived from the corresponding formulas for prisms by using inscribed and circumscribed prisms and then letting the number of sides of the prism increase without bound.[18] One reason for the early emphasis (and sometimes exclusive treatment) on circular cylinders is that a circular base is the only type of geometric figure for which this technique works with the use of only elementary considerations (no appeal to calculus or more advanced mathematics). Terminology about prisms and cylinders is identical. Thus, for example, since a truncated prism is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes would be called a truncated cylinder.

Then I select the outside and inside of the horizontal cylinder, Reverse Faces, Select the inside and outside of both cylinders and Intersect Faces With Selection. That lets me delete the faces I need to to get to the tee shape. I have to reverse the faces of the new cylinder again to clean it up:

Tee E607572 37.5 KB

Next Create a surface at the same height as the outline curves with the same dimensions as the full labyrinth.

Next flow the curves from the flat surface over the cylinder.

Next split the cylinder with the flowed curves

Next delete all parts that are not part of the labyrint

Next jlin the remaining surface (if more than one)

Next Offset the cylinder using OffsetSrf Command

I found a workaround by creating a new circle curve at the desired diameter, centering it at the end of the cylinder, then using Scale2-D to transform the cylinder and having it snap to the new curve.

In general, when applying the boolean tools, please always make sure to have a clear intersection instead of just overlapping faces like on the cylinder on the right.

After you have the first tube, you can array it along the perimeter by replacing the pivot point of the Move/Rotate tool. At least for the remaining few weeks until we release the Circular Pattern feature!

The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder (a.k.a. cylindrical surface) and two parallel planes (Kern and Bland 1948, p. 32; Harris and Stocker 1998, p. 102). A cylinder of this sort having a polygonal base is therefore a prism (Zwillinger 1995, p. 308). Harris and Stocker (1998, p. 103) use the term "general cylinder" to refer to the solid bounded a closed generalized cylinder.

Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself (Zwillinger 1995, p. 311). To make matters worse, according to topologists, a cylindrical surface is not even a true surface, but rather a so-called surface with boundary (Henle 1994, pp. 110 and 129).

As if this were not confusing enough, the term "cylinder" when used without qualification commonly refers to the particular case of a solid of circular cross section in which the centers of the circles all lie on a single line (i.e., a circular cylinder). A cylinder is called a right cylinder if it is "straight" in the sense that its cross sections lie directly on top of each other; otherwise, the cylinder is said to be oblique. The unqualified term "cylinder" is also commonly used to refer to a right circular cylinder (Zwillinger 1995, p. 312), and this is the usage followed in this work. 17dc91bb1f