---
title: "Lissajous"
id: 54989
type: "computer_media"
slug: "lissajous"
url: "http://localhost/computer_media/lissajous/"
markdown_url: "http://localhost/computer_media/lissajous.md"
published_at: "2024-06-10T01:44:45+00:00"
modified_at: "2026-03-30T21:43:29+00:00"
author: "David Anderson"
featured_image:
  url: "http://localhost/wp-content/uploads/2019/02/20230809-041332.jpg"
excerpt: "Watch continuously morphing Lissajous curves drawn in real time using parametric sine and cosine equations with an ever-shifting frequency ratio."
category:
  - name: "Archived Media"
    slug: "archived-media"
    taxonomy: "category"
    url: "http://localhost/category/archived-media/"
post_tag:
  - name: "Downloadable"
    slug: "downloadable"
    taxonomy: "post_tag"
    url: "http://localhost/tag/downloadable/"
  - name: "TS 2068"
    slug: "ts2068"
    taxonomy: "post_tag"
    url: "http://localhost/tag/ts2068/"
model:
  - name: "Timex/Sinclair 2068"
    slug: "ts-2068"
    taxonomy: "model"
    url: "http://localhost/model/ts-2068/"
genre:
  - name: "Art"
    slug: "art"
    taxonomy: "genre"
    url: "http://localhost/type/art/"
  - name: "Demo"
    slug: "demo"
    taxonomy: "genre"
    url: "http://localhost/type/demo/"
media_contents:
  - id: 54964
    title: "ISTUG Public Domain Library 6"
    type: "computer_media"
    url: "http://localhost/computer_media/istug-public-domain-library-6/"
media_type: "Program"
mediadate: "198x"
images:
  - url: "http://localhost/wp-content/uploads/2024/06/SCR-20260329-nuhq.png"
media_type_tags: "Art, Demo"
---

# Lissajous

This program draws Lissajous figures on screen using parametric equations driven by a loop variable. The X coordinate is computed as 127·sin((m+1)·a)+127 and the Y coordinate as 87·cos((n+1)·a)+87, where m and n both equal 1/w and w increments each iteration, causing the frequency ratio to shift continuously. A border rectangle is drawn first using DRAW commands, and then each frame plots a line segment between consecutive points on the parametric curve. The program saves itself with an auto-run line using SAVE “LISSAJOUS” LINE 5.

***

## Program Analysis

### Program Structure

The program is organized into a short initialization block followed by a single drawing loop. Line `5` sets colors and clears the screen. Lines `10`–`20` initialize the frequency ratio variables. Lines `120`–`140` draw a border rectangle and set up the angle accumulator. Lines `210`–`350` form the core drawing loop, and line `360` jumps back to line `200`, which does not exist — this causes the interpreter to resume at the next available line, which is `210`, effectively looping continuously.

### Parametric Curve Mathematics

The program implements Lissajous figures using the standard parametric form:

- `x1 = 127·SIN((m+1)·a) + 127`
- `y1 = 87·COS((n+1)·a) + 87`

Here `m` and `n` are initialized to `1/w` (with `w=1`, so initially 1.0), and `w` increments by 1 each iteration of the outer loop. This causes the ratio between the sine and cosine frequency arguments to change continuously, producing an evolving series of Lissajous patterns rather than a static figure.

### Coordinate Mapping

The offsets of 127 and 87 center the curves in the screen’s pixel space. The screen is 256 pixels wide and 176 pixels tall, so the amplitude factors (127 and 87) nearly fill the display area while the additions center the origin. The border is drawn at lines `120`–`120` using four chained `DRAW` commands from pixel origin (0,0).

### Drawing Technique

Rather than plotting individual pixels, the program computes two successive points on the curve — `(x1,y1)` using `(m+1)·a` and `(x2,y2)` using `m·a` — then draws a line segment between them using `PLOT x1,y1: DRAW x2-x1,y2-y1`. This produces a connected curve rather than a dotted scatter plot, at the cost of two full parametric evaluations per step.

### Notable Techniques and Anomalies

- The `GO TO 200` on line `360` targets a non-existent line, causing execution to fall through to line `210` — a well-known BASIC technique to skip re-initialization on each loop iteration.
- `LET m=1/w: LET n=m` on line `20` ensures `m` and `n` are always equal, meaning the X and Y frequency arguments differ only by the constant offset of 1 in `(m+1)` vs. `m`.
- The angle step of `0.1` radians provides reasonable curve resolution; finer steps would slow drawing noticeably given the floating-point cost of two `SIN`/`COS` evaluations per iteration.
- Line `140` executes `PLOT 134,92` (approximately the screen center) before the loop, which has no visible effect since subsequent `PLOT` calls in the loop override the graphics cursor immediately.

### Variable Summary

| Variable | Role |
| --- | --- |
| `w` | Incrementing counter controlling frequency ratio |
| `m`, `n` | Frequency ratio (always equal to `1/w`) |
| `a` | Parametric angle accumulator, incremented by 0.1 each step |
| `x1`, `y1` | First point on curve (using `(m+1)·a`) |
| `x2`, `y2` | Second point on curve (using `m·a`) |

## Source Code

```
    5 INK 0: PAPER 7: BORDER 7: CLS 
   10 LET w=1
   20 LET m=1/w: LET n=m
  120 PLOT 0,0: DRAW 255,0: DRAW 0,175: DRAW -255,0: DRAW 0,-175
  130 LET a=0
  140 PLOT 134,92
  210 LET x1=127*SIN ((m+1)*a)+127
  220 LET y1=87*COS ((n+1)*a)+87
  310 LET x2=127*SIN (m*a)+127
  320 LET y2=87*COS (n*a)+87
  330 PLOT x1,y1: DRAW x2-x1,y2-y1
  340 LET a=a+.1
  350 LET w=w+1
  360 GO TO 200
 9998 SAVE "LISSAJOUS" LINE 5
```