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

# Bezier properties and order of continuity

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2072 Kartik, 8 marks
- Video: [Bezier Curve and Its Properties with Derivation of Bezier Polynomial Function | Computer Graphics |](https://www.youtube.com/watch?v=grYrC6_LDcY) — Vinay Mishra
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/bezier-properties-order-continuity
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

List out the properties of Bezier curve. What is order of continuity? Explain.

## Short answer

Properties include endpoint interpolation, degree one less than control points, end tangents along first/last polygon edges, convex hull containment, affine invariance, global control, symmetry, variation diminishing. Continuity order measures smoothness at joins: C0 means meeting, C1 equal tangent vectors, C2 equal second derivatives; geometric continuity G1 allows proportional tangents and G2 equal curvature.

## Answer

#### Properties of the Bezier curve
1. Passes through the first and last control points: P(0) = P0 and P(1) = Pn.
2. Degree = number of control points − 1.
3. Tangent at the ends is along the first and last control polygon edges: P'(0) = n(P1 − P0), P'(1) = n(Pn − Pn−1).
4. The curve lies inside the convex hull of the control polygon.
5. Blending (Bernstein) functions are non-negative and sum to 1.
6. Affine invariance: the curve can be transformed by transforming the control points.
7. Global control: moving one control point changes the whole curve.
8. Symmetry: the same curve is obtained when the point order is reversed.
9. Variation diminishing: the curve oscillates no more than the control polygon.

#### Order of continuity
When two curve sections are joined, the **order of continuity** tells how smoothly they meet. There are two kinds.

**Parametric continuity**
- **C⁰:** the sections meet: the end point of the first equals the start point of the second.
- **C¹:** the first derivatives (tangent vectors) are equal at the join.
- **C²:** the first and second derivatives (rate of change of tangent) are equal at the join.

**Geometric continuity**
- **G⁰:** same as C⁰.
- **G¹:** tangent directions are equal, but magnitudes may differ (proportional).
- **G²:** curvature is equal.

```text
 C0: joined      C1: joined, same slope     C2: also same curvature
  ___/\___          ___/---___
```

Example: for two Bezier sections, C⁰ needs P0' = Pn; C¹ needs P1' = Pn + (Pn − Pn−1).

## Check yourself

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

1. A Bezier curve defined by 5 control points has degree:
   A. 4
   B. 5
   C. 3
   D. 6
   Answer: A. Degree = number of control points - 1. (Key point: Degree)

2. Which control points does a Bezier curve pass through?
   A. All of them
   B. Only the first and last
   C. Only the middle ones
   D. None of them
   Answer: B. P(0)=P0 and P(1)=Pn; intermediate points only shape the curve. (Key point: Interpolated points)

3. What does C1 continuity at a join require?
   A. Equal first derivatives (tangent vectors) at the join
   B. Only that the end points coincide
   C. Equal curvature only
   D. Equal tangent directions with any magnitude
   Answer: A. C1 requires the tangent vectors to be equal; C0 requires only position match. (Key point: C1 continuity)

4. How does G1 differ from C1?
   A. G1 needs equal curvature
   B. G1 needs only equal tangent direction, magnitudes may differ
   C. G1 needs equal second derivatives
   D. G1 is the same as C0
   Answer: B. Geometric continuity relaxes parametric continuity; G1 tangents need only be proportional. (Key point: G1 vs C1)

## Related in Computer Graphics

- [Bezier curve and its properties](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-curve-properties) — Computer Graphics, chapter 5, asked 6×
- [C0 and C1 continuity of Bezier sections](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Ac0-c1-continuity-bezier-sections) — Computer Graphics, chapter 5
- [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 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

## 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
- [Short note: Setting out vertical curve](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Avertical-curve-setting-out-note) — Surveying II, chapter 6