---
title: "Bezier curve and its properties"
subject: "Computer Graphics"
chapter: 5
key: "semester_5/computer_graphics:bezier-curve-properties"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/bezier-curve-properties
site: BCT Engineering
---

# Bezier curve and its properties

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2073 Shrawan, 6 marks; 2072 Chaitra, 2+4 marks; 2073 Chaitra, 3+4 marks; 2068 Chaitra, 3+2 marks; 2067 Mangsir (old course), 5 marks; 2079 Baisakh, 1 marks (asked 6 times)
- Video: [Bezier Curve in Computer Graphics | Properties | CG | Lec-42 | Bhanu Priya](https://www.youtube.com/watch?v=TMub68iLqLs) — Education 4u
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/bezier-curve-properties
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

What is a Bezier curve? Explain its properties (with examples).

## Short answer

A Bezier curve is a parametric curve defined by control points P0…Pn, starting at P0 and ending at Pn, shaped by the others. Properties: lies within convex hull of control points, tangents at ends are along P0P1 and Pn-1Pn, blending functions sum to one, affine invariance, global control. Example: quadratic with (0,0),(2,4),(4,0) stays inside triangle.

## Answer

A **Bezier curve** is a parametric curve defined by a set of control points P0 … Pn. It starts at P0, ends at Pn, and is pulled towards the other points, which it generally does not pass through.

P(u) = Σ (i = 0 to n) P_i · B_{i,n}(u), 0 ≤ u ≤ 1

where B_{i,n}(u) = C(n,i) · u^i · (1 − u)^(n−i) are the Bernstein blending functions.

**Properties**
1. The curve passes through the first and last control points: P(0) = P0, P(1) = Pn.
2. The degree is one less than the number of control points (4 points give a cubic).
3. Tangent at the start is along P0P1 and at the end along Pn−1Pn: P'(0) = n(P1 − P0), P'(1) = n(Pn − Pn−1).
4. The curve lies inside the convex hull of its control points.
5. Blending functions are non-negative and sum to 1 (partition of unity).
6. Affine invariance: transforming the control points transforms the curve in the same way.
7. Global control: moving one control point changes the whole curve (no local control).
8. Symmetry: reversing the control point order gives the same curve.
9. Variation diminishing: the curve does not cross a line more often than the control polygon does.

**Example (quadratic):** control points P0(0,0), P1(2,4), P2(4,0). At u = 0.5: P = 0.25(0,0) + 0.5(2,4) + 0.25(4,0) = (2, 2). The curve starts at (0,0), ends at (4,0) and does not reach P1(2,4); it stays inside the triangle.

```text
        P1
       /  \
      / .-.\
     / /   \\
   P0        P2
```

## Check yourself

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

1. A Bezier curve passes through:
   A. The first and last control points
   B. All control points
   C. Only the middle point
   D. None of them
   Answer: A. P(0) = P0 and P(1) = Pn. (Key point: End points)

2. The curve lies:
   A. Outside the hull
   B. Inside the convex hull of control points
   C. On the polygon
   D. Anywhere
   Answer: B. Blending functions are non-negative and sum to 1. (Key point: Convex hull)

3. Moving one control point:
   A. Changes only a local part
   B. Changes nothing
   C. Changes the whole curve (no local control)
   D. Changes only the ends
   Answer: C. Bezier has global control. (Key point: Control)

4. For P0(0,0), P1(2,4), P2(4,0), P at u = 0.5 is:
   A. (2, 4)
   B. (1, 2)
   C. (2, 0)
   D. (2, 2)
   Answer: D. 0.25(0,0) + 0.5(2,4) + 0.25(4,0) = (2,2). (Key point: Example)

## Related in Computer Graphics

- [Bezier properties; point at u=0.2](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-properties-point-u-0-2) — Computer Graphics, chapter 5, asked 2×
- [Derive cubic Bezier equation and properties](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-cubic-equation-properties) — Computer Graphics, chapter 5
- [Bezier properties and order of continuity](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-properties-order-continuity) — Computer Graphics, chapter 5
- [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
- [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
- [Bezier equation and point at u=0.8](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-equation-point-u-08) — Computer Graphics, chapter 5

## Related in other subjects

- [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
- [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
- [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