Drawing Circles with Different Aspect Ratios Using a Programming Language

Introduction

When dealing with graphical representation, understanding the aspect ratio is crucial. The aspect ratio of a circle (or an ellipse) can be manipulated to draw a perfect circle, an ellipse, or a flattened circle, depending on the aspect ratio's value. This article will guide you through how to write a program in a programming language of your choice to draw a circle with an aspect ratio that is equal to 1, less than 1, or greater than 1.

Motivation

The challenge presented in this article comes from your homework. The circle is a special case of the ellipse, and one can compensate for different aspect ratios by applying the inverse aspect ratio to the ellipse. When the aspect ratio is 1, the circle remains a circle regardless of the underlying pixel aspect ratio. This article will explore this concept and provide a practical example.

First Steps: Choosing a Programming Language

The phrase “In a language of your choice” sets the stage for this exploration. For the purposes of this article, we will use MSX BASIC as a basis for illustration, although the principles can be applied to any programming language with graphical support.

MSSX BASIC Example

Let's start with a simple example in MSX BASIC, a classic language from the 1980s. The following code snippet demonstrates how to draw a circle in the center of the screen with an aspect ratio of 0.7:

10 SCREEN 220 CIRCLE 128, 96, 40, 130.730 GOTO 30

Line 10 switches to a graphics mode, specifically mode 2, which is a popular choice for MSX BASIC. Line 20 uses the CIRCLE command to draw a circle with a center at (128, 96) and a radius of 40 pixels in color 13 (which is purple). The aspect ratio of the circle is 0.7, as the angle 130.7 is provided to account for the screen’s aspect ratio of 0.7.

Line 30 simply loops the program to keep the circle on the screen.

Underlying Principles: Drawing Circles vs. Ellipses

A circle is a special case of an ellipse, where the radii are equal. However, if these radii are different, you get an ellipse. To draw a circle with a different aspect ratio, you can treat the circle as a special case of an ellipse by applying a coefficient adjustment:

Mathematical Understanding

Suppose you center the circle at coordinates (Xc, Yc) and want to draw a circle with a radius R. If the aspect ratio is not 1, you will need to adjust the horizontal and vertical radii. Let's denote the horizontal radius as Rx and the vertical radius as Ry. The relationship can be expressed as:

When the aspect ratio is 1, Rx Ry R. When the aspect ratio is less than 1, Rx R / aspect ratio, Ry R. When the aspect ratio is greater than 1, Rx R, Ry R * aspect ratio.

To implement this in code, you can use trigonometric functions like sine and cosine to calculate the coordinates on the circle's perimeter and then scale them according to the aspect ratio.

Example Code: Drawing a Circle with a Different Aspect Ratio

Here's an example in pseudocode to demonstrate the concept. This example uses a loop to iterate over the angle range to draw the circle:

CLS ' Clear the screenFOR angle  0 TO 360    Xc  128 ' Center X    Yc  96  ' Center Y    Rx  40  ' Radius    AspectRatio  0.7 ' Aspect ratio    ' Adjusting the aspect ratio    IF AspectRatio  1 THEN        Ry  Rx / AspectRatio    ELSE        Ry  Rx * AspectRatio    END IF    X  Xc   Rx * COS(angle * 3.14159265 / 180) ' X coordinate    Y  Yc - Ry * SIN(angle * 3.14159265 / 180) ' Y coordinate, note the minus sign for up-down alignment    ' Draw the point    PLOT X, YNEXT angle

This code iterates over the angles from 0 to 360, calculating the X and Y coordinates of the circle's perimeter and adjusting the radius based on the aspect ratio. The sine and cosine functions are used to calculate the coordinates, and the radius is scaled accordingly.

Conclusion

In conclusion, the aspect ratio plays a crucial role in determining the shape of a circle or an ellipse. By adjusting the radii based on the aspect ratio, you can draw a perfect circle, an ellipse, or a flattened circle, depending on the required aspect ratio. This knowledge is essential in many areas, including game development, graphic design, and data visualization.