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

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

Computer Graphics · Chapter 5: Curve Modeling

- Asked in: 2075 Chaitra, 8 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-10-10
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-5

## Question

Find the coordinates at U=0.25, 0.5, and 0.75 with respect to the control points (10,10), (15,25), (20,30), and (25,5) using Bezier function. Draw your curve with given control points.

## Short answer

With control points (10,10),(15,25),(20,30),(25,5), the cubic Bezier gives P(0.25)=(13.75,19.06), P(0.5)=(17.5,22.5), P(0.75)=(21.25,18.44). The curve starts at (10,10), rises through these points, and ends at (25,5), inside the convex hull.

## Answer

Control points P0(10,10), P1(15,25), P2(20,30), P3(25,5). 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 = 0.4219·10 + 0.4219·25 + 0.1406·30 + 0.0156·5 = 4.219 + 10.547 + 4.219 + 0.078 = 19.06

**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 = 0.125·10 + 0.375·25 + 0.375·30 + 0.125·5 = 1.25 + 9.375 + 11.25 + 0.625 = 22.5

**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 = 0.0156·10 + 0.1406·25 + 0.4219·30 + 0.4219·5 = 0.156 + 3.516 + 12.656 + 2.109 = 18.44

**Answer**

| u | x | y |
|---|---|---|
| 0 | 10 | 10 |
| 0.25 | 13.75 | 19.06 |
| 0.50 | 17.50 | 22.50 |
| 0.75 | 21.25 | 18.44 |
| 1 | 25 | 5 |

**Drawing:** plot P0 to P3 and join them with straight lines (the control polygon). Then plot the points in the table and join them smoothly. The curve starts at (10,10) heading towards P1, rises to about y = 22.5 in the middle, and falls to (25,5), staying inside the convex hull of the four control points.

```text
30 |                    o P2
   |
25 |          o P1
   |               *
20 |        *
   |                       *
15 |
   |
10 |o P0
   |
 5 |                              o P3
   +-------------------------------- x
    10        15        20        25

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

## Check yourself

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

1. How are points on the Bezier curve found at given u values?
   A. Average the end points only
   B. Multiply each control point by its blending value and add
   C. Solve the Hermite matrix
   D. Join the control points with straight lines
   Answer: B. P(u) is the sum of control points weighted by Bernstein blending functions. (Key point: Method)

2. What is y at u = 0.25 for y-values 10, 25, 30, 5?
   A. 22.5
   B. 19.06
   C. 18.44
   D. 13.75
   Answer: B. 4.219+10.547+4.219+0.078 = 19.06. (Key point: u = 0.25 y)

3. What is P(0.5)?
   A. (17.5, 18.75)
   B. (15, 22.5)
   C. (20, 22.5)
   D. (17.5, 22.5)
   Answer: D. x = 17.5 and y = 1.25+9.375+11.25+0.625 = 22.5. (Key point: u = 0.5)

4. How should the final curve be drawn?
   A. Join only the control points with straight lines
   B. Draw a circle through P0 and P3
   C. Plot the control polygon, plot computed points, and join them smoothly in order of u
   D. Pass the curve through all four control points
   Answer: C. The curve starts at P0, ends at P3 and stays in the hull. (Key point: Drawing)

## 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

- [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×
- [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
- [Bresenham: line (25,20) to (15,10)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_graphics%3Adigitize-line-25-20-15-10) — Computer Graphics, chapter 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

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