BBC Micro Elite

# Drawing circles

## The routines that draw planets and the hyperspace and docking tunnels

References: CIRCLE, CIRCLE2, TT128

You never forget your first journey in Elite, and a lot of that is down to the circle routine. The launch tunnel rushing past as you punch your way out of the station - that's the circle routine. The planet Lave, hanging in space in front of you in all its rotating glory - that's the circle routine. The nearby systems you can choose to visit on the Short-range Chart are all those inside a circle drawn by the circle routine. And the hyperspace tunnel? You guessed it. It's the circle routine again.

Circles are drawn by the CIRCLE2 routine. This routine draws a circle by starting at the bottom of the circle - or at 6 o'clock if you think of it as a clock face - and moving anti-clockwise in steps defined by the size of the step size in STP. The whole circle is divided into 64 steps and the step number is stored in CNT, so if STP were 2, CNT would be 0, 2, 4 and so on up to and including 64. So we work our way around the circle like this:

• CNT = 0-16 is the bottom-right quadrant (6 o'clock to 3 o'clock)
• CNT = 16-32 is the top-right quadrant (3 o'clock to 12 o'clock)
• CNT = 32-48 is the top-left quadrant (12 o'clock to 9 o'clock)
• CNT = 48-64 is the bottom-left quadrant (9 o'clock to 6 o'clock)

If we can work out the coordinates of the point on the circle at step CNT, then we can draw the circle by simply drawing lines between each point, with each line being a segment of the circle. We can draw smooth circles by having smaller segments, as with the circles on the charts, or we can draw more polygonal circles by having large segments, as with the launch tunnel. The circle's "step size" determines how many of the 64 points make up each segment, so smaller step sizes give smoother circles (the step size is typically 2, 4 or 8 points).

So let's consider the step where CNT is around 5, say, so that's around 5 o'clock. The sine table at SNE contains 32 values that cover half a circle, so we can think of CNT as the angle that we have travelled through as we work our way round the circle, with 64 covering the whole thing. So 5 o'clock looks like this (I've put a "c" for the angle CNT as it's a bit of a tight squeeze):

```        _ - _
=         =                     |`.                          |`.
=             =                   |c `.                        |c `.  K
=               =                  |    `. K         K * cos(c) |    `.
=       |`.     =        __-->     |      `.                    |      `.
=      |  `.  =  ___.--´          |        `.                  +--------`
=     |    `.                    |     __--´    ----->        K * sin(c)
`--__|__--´                     |__--´
```

So if the centre of the circle (the top of the triangle above) is at the origin (0, 0), then using basic trigonometry, we can see that at step number CNT, the point on the circle is at these coordinates:

```  x = K * sin(CNT)
y = K * cos(CNT)
```

The SNE table only gives us positive results, so for other quadrants of the circle, we'll need to set the signs of x and y according to the particular quadrant we're in, but the magnitude of the coordinates will be as above. Specifically, we need to do the following (as screen y-coordinates are positive down the screen and screen x-coordinates are positive to the right):

• CNT = 0-16 is the bottom-right quadrant so x is +ve and y is +ve
• CNT = 16-32 is the top-right quadrant so x is +ve and y is -ve
• CNT = 32-48 is the top-left quadrant so x is -ve and y is -ve
• CNT = 48-64 is the bottom-left quadrant so x is -ve and y is +ve

To get the final screen coordinates of the point at count CNT, we have to add the results from above to the coordinates of the centre of the circle, as the origin of the screen is at the top-left, not in the centre of the circle. We do this with the following:

```  x = K * sin(CNT) + K3(1 0)
y = K * cos(CNT) + K4(1 0)
```

And that's how we draw the planet and chart circles in a step-wise fashion using the sine table.