---
title: "Bezier points at u=0.25,0.5,0.75: (2,10)..."
subject: "Computer Graphics"
chapter: 5
key: "semester_5/computer_graphics:bezier-points-control-2-10"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/bezier-points-control-2-10
site: BCT Engineering
---

# Bezier points at u=0.25,0.5,0.75: (2,10)...

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2074 Chaitra, 6+2 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-2-10
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

Find the coordinates at u = 0.25, u = 0.5, and u = 0.75 with respect to the control points (2, 10), (6, 20), (12, 5) and (16, 15) using Bezier function. And plot the Bezier curve with your calculated coordinates.

## Short answer

Using the cubic Bezier formula with control points (2,10),(6,20),(12,5),(16,15): P(0.25)=(5.31,13.59), P(0.5)=(9,12.5), P(0.75)=(12.69,11.41). The curve starts at (2,10), bends toward P1 and P2, ends at (16,15), staying inside the convex hull.

## Answer

Control points: P0(2,10), P1(6,20), P2(12,5), P3(16,15). Cubic Bezier:

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

**Blending functions**

| u | B0 = (1−u)³ | B1 = 3u(1−u)² | B2 = 3u²(1−u) | B3 = u³ |
|---|---|---|---|---|
| 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 = 0.4219·2 + 0.4219·6 + 0.1406·12 + 0.0156·16 = 0.844 + 2.531 + 1.688 + 0.250 = 5.3125
y = 0.4219·10 + 0.4219·20 + 0.1406·5 + 0.0156·15 = 4.219 + 8.438 + 0.703 + 0.234 = 13.594

**u = 0.5**
x = 0.125·2 + 0.375·6 + 0.375·12 + 0.125·16 = 0.25 + 2.25 + 4.5 + 2 = 9
y = 0.125·10 + 0.375·20 + 0.375·5 + 0.125·15 = 1.25 + 7.5 + 1.875 + 1.875 = 12.5

**u = 0.75**
x = 0.0156·2 + 0.1406·6 + 0.4219·12 + 0.4219·16 = 0.031 + 0.844 + 5.063 + 6.750 = 12.6875
y = 0.0156·10 + 0.1406·20 + 0.4219·5 + 0.4219·15 = 0.156 + 2.813 + 2.109 + 6.328 = 11.406

**Answer**

| u | x | y |
|---|---|---|
| 0 | 2 | 10 |
| 0.25 | 5.31 | 13.59 |
| 0.5 | 9.00 | 12.50 |
| 0.75 | 12.69 | 11.41 |
| 1 | 16 | 15 |

**Plot:** mark the control points P0 to P3 (joined by the dashed control polygon), then the calculated points, and draw a smooth curve from P0 through (5.31, 13.59), (9, 12.5) and (12.69, 11.41) to P3.

```text
20 |            o P1
   |
15 |                                o P3
   |           *      *      *
10 |    o P0
   |
 5 |                        o P2
   +---------------------------------- x
        2       6     9     12      16

 o = control points, * = calculated points at u = 0.25, 0.5, 0.75
```

The curve starts at P0, bends towards P1, dips towards P2 and ends at P3, staying inside the convex hull of the four control points.

## Check yourself

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

1. What are the cubic blending values at u = 0.5?
   A. 0.25, 0.25, 0.25, 0.25
   B. 0.125, 0.375, 0.375, 0.125
   C. 0.5, 0.25, 0.125, 0.125
   D. 0.125, 0.125, 0.375, 0.375
   Answer: B. (0.5)^3 = 0.125 and 3(0.5)(0.5)^2 = 0.375. (Key point: Blending values at u=0.5)

2. What is P(0.5) for P0(2,10), P1(6,20), P2(12,5), P3(16,15)?
   A. (9, 15)
   B. (9, 12.5)
   C. (8, 12.5)
   D. (10, 12.5)
   Answer: B. x = 0.25+2.25+4.5+2 = 9 and y = 1.25+7.5+1.875+1.875 = 12.5. (Key point: Point at u=0.5)

3. At u = 0.75 compared with u = 0.25, the blending weights are:
   A. Identical
   B. Reversed in order
   C. All equal
   D. Zero for P1 and P2
   Answer: B. B0..B3 at 0.75 are 0.0156, 0.1406, 0.4219, 0.4219, the reverse of those at 0.25. (Key point: Symmetry of weights)

4. Which points does the drawn curve actually touch?
   A. All four control points
   B. P0 and P3 only (plus the computed points)
   C. P1 and P2 only
   D. The control polygon edges
   Answer: B. Only the end points lie on the curve; P1 and P2 pull it. (Key point: Plotting)

## 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×

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