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

# Bezier curve through given points

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2076 Asoj, 2+6 marks
- Video: [Bezier Curve Example | CG | Computer Graphics | Lec-44 | Bhanu Priya](https://www.youtube.com/watch?v=Un1gToUjYso) — 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-two-properties-curve-through-points
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

Mention two important properties of Bezier Curve and find the Bezier Curve which passes through (0,0,0) and (-2,1,1) and is controlled by (7,5,2) and (2,0,1).

## Short answer

Two properties: endpoint interpolation and convex hull containment. With P0=(0,0,0), P1=(7,5,2), P2=(2,0,1), P3=(-2,1,1), the cubic Bezier is x=13u^3-36u^2+21u, y=16u^3-30u^2+15u, z=4u^3-9u^2+6u.

## Answer

#### Two properties
1. The curve passes through the first and last control points (P(0) = P0, P(1) = Pn) and is tangent to the first and last edges of the control polygon there.
2. The curve lies inside the convex hull of its control points (the blending functions are non-negative and sum to 1).

#### Curve
The curve passes through the end points, and the other two points are control points:

P0 = (0,0,0), P1 = (7,5,2), P2 = (2,0,1), P3 = (−2,1,1)

Cubic Bezier: P(u) = (1−u)³P0 + 3u(1−u)²P1 + 3u²(1−u)P2 + u³P3

In power form: P(u) = a u³ + b u² + c u + P0, with

- a = −P0 + 3P1 − 3P2 + P3 = (21 − 6 − 2, 15 − 0 + 1, 6 − 3 + 1) = (13, 16, 4)
- b = 3P0 − 6P1 + 3P2 = (−42 + 6, −30 + 0, −12 + 3) = (−36, −30, −9)
- c = −3P0 + 3P1 = (21, 15, 6)

**x(u) = 13u³ − 36u² + 21u**
**y(u) = 16u³ − 30u² + 15u**
**z(u) = 4u³ − 9u² + 6u**

Check at u = 1: x = 13 − 36 + 21 = −2, y = 16 − 30 + 15 = 1, z = 4 − 9 + 6 = 1, which is P3. Sample point at u = 0.5: (3.125, 2, 1.25).

**Answer:** the Bezier curve above, with start (0,0,0) and end (−2,1,1).

## Check yourself

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

1. Which is a property of a Bezier curve?
   A. It passes through every control point
   B. It lies inside the convex hull of its control points
   C. It changes only locally when a point moves
   D. Its degree equals the number of control points
   Answer: B. Blending functions are non-negative and sum to 1, which gives the convex hull property. (Key point: Property)

2. In the problem, which points are the end points P0 and P3?
   A. (7,5,2) and (2,0,1)
   B. (0,0,0) and (-2,1,1), the points the curve passes through
   C. (0,0,0) and (7,5,2)
   D. (2,0,1) and (-2,1,1)
   Answer: B. A Bezier curve passes through its first and last points; the other two act as control points. (Key point: Assigning points)

3. What is a = -P0 + 3P1 - 3P2 + P3 for the x-coordinate?
   A. 13
   B. 21
   C. -13
   D. 9
   Answer: A. x: 0 + 21 - 6 - 2 = 13. (Key point: Coefficient a)

4. What does the check at u = 1 confirm?
   A. The curve ends at P3 = (-2,1,1)
   B. The curve ends at P2
   C. The curve passes through P1
   D. The curve is closed
   Answer: A. x: 13-36+21 = -2, y: 16-30+15 = 1, z: 4-9+6 = 1, matching P3. (Key point: Check)

## Related in Computer Graphics

- [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 curve and its properties](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-curve-properties) — Computer Graphics, chapter 5, asked 6×
- [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×
- [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×
- [Bezier points at u=0.25,0.5,0.75: (2,10)...](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-points-control-2-10) — Computer Graphics, chapter 5, asked 2×
- [Bezier points at u=0.25,0.5,0.75: (10,10)...](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-points-control-10-10) — Computer Graphics, chapter 5, asked 2×

## 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
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- [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
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