---
title: "Second region condition, ellipse midpoint"
subject: "Computer Graphics"
chapter: 2
key: "semester_5/computer_graphics:ellipse-second-region-condition"
source: https://www.bctengineer.com/q/semester_5/computer_graphics/ellipse-second-region-condition
site: BCT Engineering
---

# Second region condition, ellipse midpoint

Computer Graphics · Chapter 2: Scan-Conversion

- Asked in: 2072 Kartik, 2 marks
- Video: [Midpoint Ellipse Drawing Algorithm](https://www.youtube.com/watch?v=a6XFLAxqqQI) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/computer_graphics/ellipse-second-region-condition
- Whole chapter: https://www.bctengineer.com/md/semester_5/computer_graphics/chapter-2

## Question

State the condition to identify that you are in the second region of the ellipse using the midpoint algorithm.

## Short answer

The second region is identified when 2·ry²·x ≥ 2·rx²·y, meaning the slope magnitude is at least 1. Region 1 continues while 2ry²x < 2rx²y; otherwise switch to region 2 where y is stepped.

## Answer

In the midpoint ellipse algorithm, the slope of the ellipse at (x, y) is dy/dx = −(ry²x)/(rx²y). We are in the **second region** when the slope magnitude becomes 1 or more, i.e. when

**2·ry²·x ≥ 2·rx²·y**

The loop for region 1 runs while 2ry²x < 2rx²y; once this is false, the algorithm switches to region 2 (y is then the unit-step direction).

## Check yourself

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

1. The midpoint ellipse algorithm is in region 2 when:
   A. 2ry^2 x ≥ 2rx^2 y
   B. 2ry^2 x < 2rx^2 y
   C. rx x ≥ ry y
   D. x ≥ rx
   Answer: A. Slope magnitude ≥ 1 corresponds to this inequality. (Key point: Condition)

2. The slope of the ellipse at (x,y) is:
   A. (ry^2 x)/(rx^2 y)
   B. −(ry^2 x)/(rx^2 y)
   C. −(rx^2 y)/(ry^2 x)
   D. −(ry x)/(rx y)
   Answer: B. Differentiating ry^2x^2 + rx^2y^2 = rx^2ry^2 gives this. (Key point: Slope)

3. In region 2, which direction is the unit step?
   A. x
   B. Both
   C. y (decreasing by 1)
   D. Neither
   Answer: C. Once the slope magnitude is ≥ 1, y becomes the unit step. (Key point: Unit step)

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