---
title: "B-spline surface and its advantages"
subject: "Computer Aided Design"
chapter: 4
key: "semester_4/computer_aided_design:bspline-surface-vs-bezier-surface"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/bspline-surface-vs-bezier-surface
site: BCT Engineering
---

# B-spline surface and its advantages

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

- Asked in: year not given, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/computer_aided_design/bspline-surface-vs-bezier-surface
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-4

## Question

Explain the B-spline surface with its equation. Compare it with a Bezier surface and state its advantages in free-form design.

## Short answer

B-spline surface is a tensor-product surface from B-spline bases, control net and two knot vectors: P(u,w)=sum B_ij N_i,k(u) M_j,l(w). Unlike Bezier, degree is independent of net size, control is local, and multiple patches join with automatic continuity. It enables low-degree one-piece complex surfaces, local editing, smoothness, convex hull, and NURBS exact conic representation.

## Answer

A **B-spline surface** is a tensor-product surface built from B-spline basis functions in two directions. It is defined by a control net and two knot vectors.

#### Equation
For an $(m+1)\times(n+1)$ net $\mathbf B_{ij}$, with orders $k$ in $u$ and $l$ in $w$:

$$
\mathbf P(u,w)=\sum_{i=0}^{m}\sum_{j=0}^{n}\mathbf B_{ij}\,N_{i,k}(u)\,M_{j,l}(w)
$$

$N_{i,k}$ and $M_{j,l}$ are B-spline basis functions defined by the knot vectors $U$ and $W$ (Cox-de Boor recursion). A bicubic B-spline surface uses $k=l=4$.

#### Comparison

| Point | Bezier surface | B-spline surface |
|---|---|---|
| Degree | Fixed by net size ($m$, $n$) | Chosen freely ($k-1$, $l-1$) |
| Shape control | Global | Local: a point affects only $k\times l$ patches |
| Size of surface | One patch | Many patches joined smoothly |
| Continuity at joins | Must be arranged | $C^{k-2}$ automatically |
| Corner interpolation | Always | Only with open knot vectors |
| Knot vectors | None | Two (U and W) |
| Computation | Simpler | More |

#### Advantages
1. A large, complex surface (car roof, wing) is one entity with a low degree.
2. Local modification does not disturb the rest of the surface.
3. Smooth ($C^2$ for bicubic) without extra constraints.
4. Convex hull property and affine invariance hold.
5. The rational version (NURBS) also represents exact cylinders, cones, spheres and tori, and is the standard in CAD and IGES/STEP exchange.

## Check yourself

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

1. How is a B-spline surface defined?
   A. By four boundary curves blended together
   B. By a control net and a single Bernstein basis in each direction, with no knots
   C. By a tensor product of B-spline basis functions in u and w, using a control net and two knot vectors
   D. By a rolling ball of fixed radius between two surfaces
   Answer: C. P(u,w) = sum B_ij N_i,k(u) M_j,l(w), where the bases come from knot vectors U and W. (Key point: B-spline surface definition)

2. What is the main difference in shape control between B-spline and Bezier surfaces?
   A. B-spline control is global, Bezier is local
   B. B-spline control is local (a point affects only k x l patches), Bezier is global
   C. Both are local
   D. Neither can be modified after creation
   Answer: B. Moving one B-spline net point changes only nearby patches, whereas moving a Bezier net point changes the whole patch. (Key point: Local control)

3. Which statement about degree and continuity is correct?
   A. B-spline degree is fixed by net size and joins need manual arrangement
   B. B-spline degree is chosen freely and joins have C^(k-2) continuity automatically
   C. Bezier surfaces give C^(k-2) at joins automatically
   D. B-spline surfaces always lose the convex hull property
   Answer: B. B-spline order is independent of net size and gives automatic C^(k-2) continuity (C^2 for bicubic). (Key point: Degree and continuity)

## Related in Computer Aided Design

- [Bezier surface equation and properties](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-surface-properties) — Computer Aided Design, chapter 4
- [B-spline curve versus Bezier curve](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abspline-vs-bezier) — Computer Aided Design, chapter 3
- [Biquadratic Bezier surface point calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-surface-numerical) — Computer Aided Design, chapter 4
- [Bezier curve: equation and properties](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-curve-properties) — Computer Aided Design, chapter 3
- [Surface of revolution: equation, points, area](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Asurface-of-revolution-numerical) — Computer Aided Design, chapter 4
- [Uniform cubic B-spline segment points](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Auniform-cubic-bspline-numerical) — Computer Aided Design, chapter 3

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