---
title: "B-spline curve versus Bezier curve"
subject: "Computer Aided Design"
chapter: 3
key: "semester_4/computer_aided_design:bspline-vs-bezier"
source: https://www.bctengineer.com/q/semester_4/computer_aided_design/bspline-vs-bezier
site: BCT Engineering
---

# B-spline curve versus Bezier curve

Computer Aided Design · Chapter 3: Mathematical Representation of Curves

- Asked in: year not given, 5 marks
- Video: [S 2.44 Comparison Between Hermite Cubic Spline Curve, Bezier Curve and B-Spline Curve](https://www.youtube.com/watch?v=V_xDS-46Gao) — Dr. Khizar Ahmed Pathan
- 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-vs-bezier
- Whole chapter: https://www.bctengineer.com/md/semester_4/computer_aided_design/chapter-3

## Question

What is a B-spline curve? Differentiate between a B-spline curve and a Bezier curve. Explain the terms control points, knot vector, degree and order, and state the advantages of B-splines.

## Short answer

A B-spline curve is a piecewise polynomial defined by control points and B-spline basis functions over a knot vector. Unlike Bezier, degree is independent of point count, offers local control, and automatic C^{k-2} continuity. Control points attract the curve; knot vector defines parameter intervals; order = degree+1. Advantages: local control, low degree, high continuity, and NURBS extension for exact conics.

## Answer

A **B-spline** curve is a piecewise polynomial parametric curve defined by control points and B-spline basis functions. It is a generalisation of the Bezier curve and removes its two main weaknesses: global control and the link between the number of points and the degree.

#### Equation
$$
\mathbf P(u)=\sum_{i=0}^{n}\mathbf B_i\,N_{i,k}(u)
$$

$N_{i,k}$ are basis functions of order $k$ (degree $k-1$) given by the Cox-de Boor recursion over the knot vector $\{t_0,t_1,\dots\}$.

#### Terms
- **Control points** $\mathbf B_i$: vertices of the control polygon; the curve is attracted to them but usually does not pass through them.
- **Knot vector:** non-decreasing parameter values where the polynomial pieces join. Number of knots $=n+k+1$ for $n+1$ points. Uniform, open-uniform (end knots repeated $k$ times) and non-uniform vectors are used.
- **Degree:** polynomial degree of each segment (cubic is common); **order** $=$ degree $+1$.

#### Difference

| Point | Bezier | B-spline |
|---|---|---|
| Degree | $n$ = points $-1$ (fixed) | Chosen independently ($k-1$) |
| Control | Global | Local |
| Segments | One polynomial | Many joined polynomials |
| Continuity | Needs extra conditions to join | $C^{k-2}$ automatic |
| Passes through end points | Always | Only for open knot vector |
| Extra parameter | None | Knot vector |
| Computation | Simple | Higher |
| Special case | Bezier is a B-spline with $k=n+1$ and no interior knots | |

#### Advantages
1. Local control: moving one point changes only $k$ segments.
2. Degree stays low (cubic) for any number of points.
3. High continuity ($C^2$ for cubic) at joints.
4. Convex hull and variation diminishing properties.
5. NURBS (rational B-splines) can represent exact conics and free-form curves in one form.

## Check yourself

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

1. How does a B-spline differ from a Bezier regarding control?
   A. B-spline has global control
   B. Both are local
   C. Bezier is local
   D. B-spline has local control
   Answer: D. Moving one B-spline point changes only k segments. (Key point: Local control)

2. What is the order of a B-spline of degree 3?
   A. 4
   B. 3
   C. 2
   D. 5
   Answer: A. Order k = degree + 1 = 4. (Key point: Order)

3. What is a knot vector?
   A. List of control points
   B. Non-decreasing parameter values where polynomial pieces join
   C. The tangent list
   D. The weights
   Answer: B. Knots mark the joins of the polynomial pieces; count is n + k + 1. (Key point: Knot vector)

4. How is the degree of a B-spline chosen?
   A. Equal to points - 1
   B. Always 2
   C. Independently of the number of control points
   D. By the knot count only
   Answer: C. Unlike Bezier, B-spline degree is chosen independently, so it stays low for many points. (Key point: Degree)

## Related in Computer Aided Design

- [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
- [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
- [Hermite curve points and tangents](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Ahermite-curve-numerical) — Computer Aided Design, chapter 3
- [Cubic Bezier point and tangent calculation](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abezier-curve-numerical) — Computer Aided Design, chapter 3
- [Convert Hermite data to Bezier cubic](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Ahermite-to-bezier-conversion) — Computer Aided Design, chapter 3
- [B-spline surface and its advantages](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Abspline-surface-vs-bezier-surface) — Computer Aided Design, chapter 4

## Related in other subjects

- [Draw Bezier curve through 4 control points](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-four-points-p0-0-0) — Computer Graphics, chapter 5
- [Curve modeling using splines](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Acurve-modeling-using-splines) — Computer Graphics, chapter 5, asked 3×
- [Bezier curve through given points](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-two-properties-curve-through-points) — Computer Graphics, chapter 5
- [Bezier points at u=0.25,0.5,0.75: (10,15)...](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-points-control-10-15) — Computer Graphics, chapter 5, asked 4×