---
title: "HiResHat"
id: 58394
type: "computer_media"
slug: "hireshat"
url: "http://localhost/computer_media/hireshat/"
markdown_url: "http://localhost/computer_media/hireshat.md"
published_at: "2024-11-17T08:48:55+00:00"
modified_at: "2026-03-30T21:42:58+00:00"
author: "David Anderson"
featured_image:
  url: "http://localhost/wp-content/uploads/2024/11/HRH.png"
excerpt: "A nested-loop parametric surface renderer plots a 3D hat shape using a sum-of-sines height function and an oblique isometric projection — entirely in BASIC."
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: "Demo"
    slug: "demo"
    taxonomy: "genre"
    url: "http://localhost/type/demo/"
media_type: "Program"
download_url: "https://archive.org/download/timex-sinclair-software-archive/HiResHat-2068%20%28198x%29%28UNK%29%28TS2068%29%28US%29%28Program%29.zip"
mediadate: "198x"
images:
  - url: "http://localhost/wp-content/uploads/2024/11/HRH.png"
media_type_tags: "Demo"
---

This program draws a high-resolution 3D hat-shaped surface using the PLOT command. Lines 20 draws a full-screen border rectangle, then the main nested loop (lines 60–160) iterates over a circular domain in the k and t variables to compute a parametric surface where height is defined by a sum-of-sines formula: SIN(xt) + 0.4·SIN(3·xt). The x and y screen coordinates are derived via an oblique isometric-style projection using scaling factors xs=0.78 and ys=0.85, with centre point p=128, q=95. No machine code is used; the program relies entirely on BASIC floating-point maths, making it computationally intensive but straightforward to follow.

***

## Program Structure

The program is divided into three logical phases:

1. **Border drawing** (line 20): uses `PLOT`/`DRAW` to trace a rectangle around the full 256×176 pixel screen area.
2. **Parameter initialisation** (lines 35–50): pre-computes all scaling and frequency factors to avoid repeated arithmetic inside the loops.
3. **Surface rendering** (lines 60–160): a double nested loop iterates over the circular domain and plots each surface point individually.

## Mathematical Surface Definition

The surface is a classic “sombrero” or hat function. For each column slice `k` (ranging from `-zp` to `zp`, i.e. −64 to 64), the half-width of the circular cross-section is computed at line 80:

`xl = INT(0.5 + SQR(xp*xp - zt*zt))`

where `zt = k * xq` scales `k` into the same coordinate space as the x-axis radius `xp=144`. This enforces a circular boundary on the (t, k) domain. The height at each point is then:

`yy = (SIN(xt) + 0.4 * SIN(3*xt)) * yf`

where `xt = SQR(t*t + zt*zt) * xf` is the radial distance scaled by frequency factor `xf = xr/xp` (with `xr = 1.5*PI`). The second harmonic term `0.4*SIN(3*xt)` adds a secondary ripple ring around the central peak.

## Projection and Scaling

The 3D-to-2D projection is an oblique parallel (cabinet-style) projection rather than true isometric. Screen coordinates are computed as:

| Variable | Formula | Purpose |
| --- | --- | --- |
| `x` | `xs*(t+k) + p` | Horizontal screen position; `t+k` gives the oblique axis blend |
| `y` | `ys*(yy-k) + q` | Vertical screen position; `yy` is height, `-k` recedes into the screen |

The factors `xs=0.78` and `ys=0.85` compress the axes to keep the image within the bordered area. The centre offsets `p=128` and `q=95` place the origin near the middle of the screen.

## Pre-computed Constants

All derived constants are computed once in lines 35–50 rather than inside the loops, which is good practice for BASIC performance:

- `xf = xr/xp` — radial-to-angle frequency scale
- `yf = yp/yr` — height amplitude scale (evaluates to 56 since `yr=1`)
- `zf = xr/zp` — defined but never used (see anomalies below)
- `xq = xp/zp` — ratio used to scale `k` relative to `xp`

## Notable Techniques

- Using `INT(0.5 + SQR(...))` at line 80 is a standard rounding idiom, avoiding a dedicated `ROUND` function.
- The circular loop limit (`FOR t=-xl TO xl`) ensures only points inside the circular base are plotted, keeping the rendered surface geometrically clean.

## Source Code

```
    5 SAVE "HI RES HAT"
   10 REM HI RES HAT
   20 PLOT 0,0: DRAW 0,175: DRAW 255,0: DRAW 0,-175: DRAW -255,0:
   30 REM Lines 40 to 130 calcu X and Y for Plotting
   35 LET xs=.78: LET ys=.85
   40 LET p=128: LET q=95: LET xp=144: LET xr=1.5*PI: LET yp=56: LET yr=1
   50 LET zp=64: LET xf=xr/xp: LET yf=yp/yr: LET zf=xr/zp: LET xq=xp/zp
   60 FOR k=-zp TO zp
   70 LET zt=k*xq
   80 LET xl=INT (.5+SQR (xp*xp-zt*zt))
   90 FOR t=-xl TO xl
  100 LET xt=SQR (t*t+zt*zt)*xf
  110 LET yy=(SIN (xt)+.4*SIN (3*xt))*yf
  120 LET x=(xs*(t+k))+p
  130 LET y=(ys*(yy-k))+q
  140 PLOT x,y
  150 NEXT t
  160 NEXT k
  170 STOP
```