---
title: "Bezier properties; point at u=0.2"
subject: "Computer Graphics"
chapter: 5
key: "semester_5/computer_graphics:bezier-properties-point-u-0-2"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/bezier-properties-point-u-0-2
site: BCT Engineering
---

# Bezier properties; point at u=0.2

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2074 Asoj, 8 marks
- Video: [Bezier Curve Numerical Problem | Computer Graphics](https://www.youtube.com/watch?v=wZbazXBkGv4) — 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-point-u-0-2
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

Explain properties of Bezier curve. Find the coordinate at u = 0.2 with respect to the control points (1,1), (4,6), (8,-3) and (12,2) using Bezier function.

## Short answer

For the four control points, cubic Bezier at u=0.2 with weights 0.512, 0.384, 0.096, 0.008 gives P(0.2)=(2.912, 2.544). Key properties are endpoint interpolation, end tangents along first/last edges, convex hull containment, affine invariance, global control, and symmetry.

## Answer

#### Properties of Bezier curve
1. Passes through the first and last control points.
2. Degree is one less than the number of control points.
3. End tangents are along the first and last edges of the control polygon.
4. The curve lies inside the convex hull of the control points.
5. Blending functions are positive and sum to 1.
6. Affine invariance: transformations can be applied to control points only.
7. Global control: moving any control point changes the whole curve.
8. Symmetric: reversing the order of points gives the same curve.

#### Point at u = 0.2
Control points P0(1,1), P1(4,6), P2(8,−3), P3(12,2): cubic (n = 3).

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

| Function | Formula at u = 0.2 | Value |
|---|---|---|
| B0 | (0.8)³ | 0.512 |
| B1 | 3(0.2)(0.8)² | 0.384 |
| B2 | 3(0.2)²(0.8) | 0.096 |
| B3 | (0.2)³ | 0.008 |

(Sum = 1.000)

x = 0.512·1 + 0.384·4 + 0.096·8 + 0.008·12 = 0.512 + 1.536 + 0.768 + 0.096 = 2.912

y = 0.512·1 + 0.384·6 + 0.096·(−3) + 0.008·2 = 0.512 + 2.304 − 0.288 + 0.016 = 2.544

**Answer: P(0.2) = (2.912, 2.544)**

## Check yourself

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

1. Four control points give which type of Bezier curve?
   A. Quadratic (n = 2)
   B. Quartic (n = 4)
   C. Linear (n = 1)
   D. Cubic (n = 3)
   Answer: D. Degree = points - 1 = 3. (Key point: Degree from points)

2. What is B1 = 3u(1-u)^2 at u = 0.2?
   A. 0.512
   B. 0.096
   C. 0.384
   D. 0.192
   Answer: C. 3(0.2)(0.8)^2 = 0.6 x 0.64 = 0.384. (Key point: Blending value)

3. The four blending values at u = 0.2 (0.512, 0.384, 0.096, 0.008) sum to:
   A. 0.2
   B. 1
   C. 0.8
   D. 1.2
   Answer: B. Bezier blending functions always sum to 1, a check on the calculation. (Key point: Blending sum)

4. What is the x-coordinate of P(0.2)?
   A. 2.544
   B. 3.2
   C. 4.0
   D. 2.912
   Answer: D. x = 0.512(1)+0.384(4)+0.096(8)+0.008(12) = 2.912. (Key point: Result)

## Close variants set in other exams

- [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×

## 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 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 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: (10,10)...](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Abezier-points-control-10-10) — Computer Graphics, chapter 5, asked 2×
- [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
- [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×

## 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
- [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
- [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
- [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