Skip to main content

Ruled surface between two lines: points

Short answer

A ruled surface joins corresponding points of two curves with straight lines: P(u,w)=(1-w)P1+wP2. Here P(u,w) = (6u, w(4+2u), w(2+3u)). Points: (0.5,0.5) -> (3,2.5,1.75); (0.25,0.5) -> (1.5,2.25,1.375); (0.5,0.75) -> (3,3.75,2.625).

Define a ruled surface. A ruled surface is generated between two straight-line boundary curves, P1(u) from (0, 0, 0) to (6, 0, 0), and P2(u) from (0, 4, 2) to (6, 6, 5). Write its equation and find the surface points at (u, w) = (0.5, 0.5), (0.25, 0.5) and (0.5, 0.75).

A ruled surface is generated by moving a straight line (a ruling) so that its end points slide along two boundary curves. Corresponding points of the two curves (same uu) are joined by straight lines:

P(u,w)=(1−w) P1(u)+w P2(u),0≤u,w≤1\mathbf P(u,w)=(1-w)\,\mathbf P_1(u)+w\,\mathbf P_2(u),\qquad 0\le u,w\le1

The surface is linear in ww; it is a plane if the two curves are coplanar and straight, otherwise a twisted (hyperbolic-paraboloid type) patch.

Boundary curves

P1(u)=(0,0,0)+u(6,0,0)=(6u, 0, 0)P2(u)=(0,4,2)+u(6,2,3)=(6u, 4+2u, 2+3u)\begin{aligned} \mathbf P_1(u)&=(0,0,0)+u(6,0,0)=(6u,\ 0,\ 0) \\ \mathbf P_2(u)&=(0,4,2)+u(6,2,3)=(6u,\ 4+2u,\ 2+3u) \end{aligned}

Surface equation

P(u,w)=(6u, w(4+2u), w(2+3u))\mathbf P(u,w)=\big(6u,\ w(4+2u),\ w(2+3u)\big)

Points

(u,w)(u,w)P1(u)\mathbf P_1(u)P2(u)\mathbf P_2(u)P(u,w)\mathbf P(u,w)
(0.5, 0.5)(3, 0, 0)(3, 5, 3.5)(3, 2.5, 1.75)
(0.25, 0.5)(1.5, 0, 0)(1.5, 4.5, 2.75)(1.5, 2.25, 1.375)
(0.5, 0.75)(3, 0, 0)(3, 5, 3.5)(3, 3.75, 2.625)

For example, at (0.5,0.75)(0.5,0.75): y=0.75×5=3.75y=0.75\times5=3.75, z=0.75×3.5=2.625z=0.75\times3.5=2.625.

Answer: P(u,w) = (6u, w(4+2u), w(2+3u)); points (3, 2.5, 1.75), (1.5, 2.25, 1.375), (3, 3.75, 2.625).

✓ Answer checked by Claude Opus 5.5 (high effort)

Related questions

All Computer Aided Design chapter 4 questions →