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Tabulated cylinder surface

Short answer

A tabulated cylinder translates a plane curve along a vector: P(u,w) = C(u) + wL. With C(u)=(10cos u, 10sin u, 0) and L=(0,0,25), at u=30°, w=0.4 the point is (8.660, 5.000, 10.0).

What is a tabulated cylinder? Write its parametric equation. A planar curve is the quarter circle x = 10 cos(u), y = 10 sin(u), z = 0, 0 <= u <= 90 deg. It is translated along the vector L = (0, 0, 25). Find the surface point at u = 30 deg, w = 0.4.

A tabulated cylinder is a surface generated by translating a plane curve (the directrix) along a straight line of fixed direction and finite length (the generator). All generators are parallel. It is the surface produced by an extrude feature in solid modelling; if the directrix is a circle and the direction is perpendicular to its plane, a right circular cylinder results.

Equation

P(u,w)=C(u)+w L,0≤u≤1 (curve parameter), 0≤w≤1\mathbf P(u,w)=\mathbf C(u)+w\,\mathbf L,\qquad 0\le u\le1\ (\text{curve parameter}),\ 0\le w\le1

where C(u)\mathbf C(u) is the directrix and L\mathbf L is the vector giving the direction and length of translation.

Numerical

P(u,w)=(10cos⁡u, 10sin⁡u, 25w)\mathbf P(u,w)=\big(10\cos u,\ 10\sin u,\ 25w\big)

At u=30∘u=30^\circ and w=0.4w=0.4:

x=10cos⁡30∘=8.660y=10sin⁡30∘=5.000z=25(0.4)=10.0\begin{aligned} x&=10\cos30^\circ=8.660 \\ y&=10\sin30^\circ=5.000 \\ z&=25(0.4)=10.0 \end{aligned}

The surface is a quarter of a cylinder of radius 10 and height 25, with its curved area =14(2π×10×25)=392.7=\tfrac14(2\pi\times10\times25)=392.7 square units.

Answer: P(u,w) = C(u) + wL; the point is (8.660, 5.000, 10.0).

✓ Answer checked by Claude Opus 5.5 (high effort)

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