---
title: "Bezier points at u=0.25,0.5,0.75: (10,15)..."
subject: "Computer Graphics"
chapter: 5
key: "semester_5/computer_graphics:bezier-points-control-10-15"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/bezier-points-control-10-15
site: BCT Engineering
---

# Bezier points at u=0.25,0.5,0.75: (10,15)...

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2079 Baisakh, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/bezier-points-control-10-15
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

Find the coordinates of Bezier curve at u = 0.25, 0.5 and 0.75 with respect to the control points (10, 15), (15,20), (20, 35), (25, 10) using Bezier function.

## Short answer

With control points (10,15),(15,20),(20,35),(25,10), the cubic Bezier yields P(0.25)=(13.75,19.84), P(0.5)=(17.5,23.75), P(0.75)=(21.25,22.03). These are weighted sums using the Bernstein blending functions, and the curve passes through the first and last control points.

## Answer

Control points P0(10,15), P1(15,20), P2(20,35), P3(25,10). Cubic Bezier:

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

| u | B0 | B1 | B2 | B3 |
|---|---|---|---|---|
| 0.25 | 0.4219 | 0.4219 | 0.1406 | 0.0156 |
| 0.50 | 0.1250 | 0.3750 | 0.3750 | 0.1250 |
| 0.75 | 0.0156 | 0.1406 | 0.4219 | 0.4219 |

**u = 0.25**
x = 0.4219·10 + 0.4219·15 + 0.1406·20 + 0.0156·25 = 4.219 + 6.328 + 2.813 + 0.391 = 13.75
y = 0.4219·15 + 0.4219·20 + 0.1406·35 + 0.0156·10 = 6.328 + 8.438 + 4.922 + 0.156 = 19.84

**u = 0.5**
x = 0.125·10 + 0.375·15 + 0.375·20 + 0.125·25 = 1.25 + 5.625 + 7.5 + 3.125 = 17.5
y = 0.125·15 + 0.375·20 + 0.375·35 + 0.125·10 = 1.875 + 7.5 + 13.125 + 1.25 = 23.75

**u = 0.75**
x = 0.0156·10 + 0.1406·15 + 0.4219·20 + 0.4219·25 = 0.156 + 2.109 + 8.438 + 10.547 = 21.25
y = 0.0156·15 + 0.1406·20 + 0.4219·35 + 0.4219·10 = 0.234 + 2.813 + 14.766 + 4.219 = 22.03

**Answer**

| u | Point |
|---|---|
| 0.25 | (13.75, 19.84) |
| 0.50 | (17.50, 23.75) |
| 0.75 | (21.25, 22.03) |

## Check yourself

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

1. Which formula is used for four control points?
   A. Quadratic Bezier with weights (1-u)^2, 2u(1-u), u^2
   B. Linear interpolation (1-u)P0 + uP3
   C. Cubic Bezier with weights (1-u)^3, 3u(1-u)^2, 3u^2(1-u), u^3
   D. Hermite blending functions
   Answer: C. Four points mean n=3. (Key point: Formula)

2. What is x at u = 0.5 for x-values 10, 15, 20, 25?
   A. 15
   B. 20
   C. 17.5
   D. 18.75
   Answer: C. 0.125(10)+0.375(15)+0.375(20)+0.125(25) = 17.5. (Key point: Midpoint x)

3. What is y at u = 0.5 for y-values 15, 20, 35, 10?
   A. 20
   B. 23.75
   C. 27.5
   D. 19.84
   Answer: B. 1.875+7.5+13.125+1.25 = 23.75. (Key point: Midpoint y)

4. Which is P(0.75)?
   A. (13.75, 19.84)
   B. (21.25, 22.03)
   C. (17.5, 23.75)
   D. (25, 10)
   Answer: B. Weights 0.0156, 0.1406, 0.4219, 0.4219 give x = 21.25 and y = 22.03. (Key point: Point at 0.75)

## Close variants set in other exams

- [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×
- [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×

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