Curves

Introduction

Dr Logo has no primitive that draws a circle. It does not need one.

Go forward a little, turn a little, and do it enough times, and you are back where you started having drawn something no one would call a polygon.

A circle

repeat 36 [fd 5 rt 10]

Thirty-six turns of ten degrees is one full revolution. Each side is five steps long, so the turtle walks 180 steps in all - and a loop 180 round has a diameter of about 57, which is what you get.

Two numbers control it and they do different jobs:

• The step sets the size. Thirty-six steps of 10 give a circle 57 across; thirty-six steps of 15 give one nearly three times that, which is too big for the screen.
• How many, and how far each turn must multiply to 360, or you get an arc instead of a circle. That is not a fault. That is the next idea.

Why not repeat 360 [fd 1 rt 1]?

Because you will be waiting. Dr Logo draws about eleven turtle steps a second, so a 360-sided circle takes the best part of half a minute, and on a 256-pixel screen it looks exactly like the 36-sided one that took three seconds.

Ten degrees a step is smooth enough for anything the Einstein can display. Save the fine steps for when you can see the difference.

An arc is a circle you stopped early

to arc :size :steps
repeat :steps [fd :size rt 10]
end

arc 5 36 is the whole circle. arc 5 9 is a quarter of it - nine steps of ten degrees is ninety. arc 5 18 is a half.

A flower

Two quarter arcs with a right angle between them make a petal:

to petal :size
arc :size 9
rt 90
arc :size 9
rt 90
end

Walk it through. Curve round ninety degrees, turn sharply, curve back ninety, turn sharply again - and you are where you began, facing how you began. That is the same closing trick as S22's bar, and it buys the same thing:

repeat 6 [petal 5 rt 60]

The code

to arc :size :steps
repeat :steps [fd :size rt 10]
end

to petal :size
arc :size 9
rt 90
arc :size 9
rt 90
end

cs
ht
repeat 6 [petal 5 rt 60]

Starting from

A fresh boot, at the ? prompt.

What you should see

It takes about ten seconds. Six petals, because six sixties are 360.

The other kind of curve

A circle comes from turning. A wave comes from working out where each point should be and going there - and for that Dr Logo has sin.

?show sin 30
0.5
?show sin 90
1

Degrees, not radians, and the answer is a decimal between -1 and 1. Multiply it by how tall you want the wave, and you have a height for every angle:

to wave :deg
if :deg > 720 [stop]
make "x :deg / 3
make "y sin :deg
setpos se :x - 120 :y * 40
wave :deg + 10
end

se builds the [x y] pair that setpos wants out of the two numbers. Run it from the left-hand edge with the pen already down at the first point:

cs ht pu setpos [-120 0] pd
wave 0

Two full cycles, because 720 degrees is twice round.

Change one thing

• arc 15 36. The circle is bigger than the screen. What happens at the edge, and does Dr Logo tell you?
• Change rt 60 to rt 45 in the flower line and repeat 6 to repeat 8. Then try eight petals with rt 60 still in place and see what a mismatch looks like.
• setscrunch 2, then draw the circle again. The width is untouched and the height doubles - setscrunch stretches the picture vertically, and 1 is what you start with.
• Change sin to cos in wave. The curve starts at the top instead of the middle - and you get an unwanted vertical line at the left. Work out where it came from.

When it goes wrong

Summary

A circle is repeat 36 [fd n rt 10] - thirty-six straight lines, with the step setting the size and the numbers multiplying to 360. Stop early and you have an arc; make a shape that closes and you can repeat it into a flower. For a curve that is not made of turning, sin gives a height for an angle in degrees and setpos puts the turtle there.

Next

S24, patterns - what happens when you repeat something that very nearly closes.