---
title: "Ruled surface between two lines: points"
subject: "Computer Aided Design"
chapter: 4
key: "semester_4/computer_aided_design:ruled-surface-numerical"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/ruled-surface-numerical
site: BCT Engineering
---

# Ruled surface between two lines: points

Computer Aided Design · Chapter 4: Representation of Surfaces and Solids

- Asked in: year not given, 6 marks
- Video: [Ruled surfaces: worlds from lines](https://www.youtube.com/watch?v=q6k9YH-8nFw) — Zenzicubic
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/ruled-surface-numerical
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-4

## Question

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).

## 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).

## Answer

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 $u$) are joined by straight lines:

$$
\mathbf P(u,w)=(1-w)\,\mathbf P_1(u)+w\,\mathbf P_2(u),\qquad 0\le u,w\le1
$$

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

#### Boundary curves
$$
\begin{aligned}
\mathbf P_1(u)&=(0,0,0)+u(6,0,0)=(6u,\ 0,\ 0) \\
\mathbf P_2(u)&=(0,4,2)+u(6,2,3)=(6u,\ 4+2u,\ 2+3u)
\end{aligned}
$$

#### Surface equation
$$
\mathbf P(u,w)=\big(6u,\ w(4+2u),\ w(2+3u)\big)
$$

#### Points
| $(u,w)$ | $\mathbf P_1(u)$ | $\mathbf P_2(u)$ | $\mathbf P(u,w)$ |
|---|---|---|---|
| (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 $(0.5,0.75)$: $y=0.75\times5=3.75$, $z=0.75\times3.5=2.625$.

**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).**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the ruled surface equation?
   A. (1-u)P1 + uP2
   B. (1-w)P1(u) + wP2(u)
   C. P1 + P2
   D. P1(u) x P2(u)
   Answer: B. Same-u points are joined linearly in w. (Key point: Equation)

2. What is P2(u)?
   A. (6u, 4, 2)
   B. (6, 4+2u, 2+3u)
   C. (6u, 4+2u, 2+3u)
   D. (u, 4u, 2u)
   Answer: C. From (0,4,2) with direction (6,2,3). (Key point: P2(u))

3. What is the point at (0.5, 0.5)?
   A. (3, 5, 3.5)
   B. (1.5, 2.5, 1.75)
   C. (3, 2.5, 3.5)
   D. (3, 2.5, 1.75)
   Answer: D. Half of P2(0.5) = (3,5,3.5) plus half of (3,0,0). (Key point: Point)

4. When is the ruled surface twisted rather than a plane?
   A. When the boundary curves are not coplanar
   B. Always flat
   C. When w = 0
   D. When u = 1
   Answer: A. A plane results only if the curves are coplanar and straight. (Key point: Shape)

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