Drawing the "Seed of Life" without redrawing anything - recursion

Question

I have a mildly interesting problem which I can't quite figure out (although in fairness, I am pretty drunk)
The "Seed of Life" is a pattern created from drawing circles of equal radius, centred on the intersection of the previous circle.
Language doesn't really matter, the theory is more important here. Anything which can draw a circle will do it. For example, HTML5 + JS canvas can do it. It's a lovely example of how recursion can help solve problems.
The problem is that a naive approach will end up redrawing many, many circles. With 7 layers, you'll end up with over 300,000 circle draws.
A simple approach is to maintain a list of previous circle centre points, and only draw circles which are not in that list.
My question is whether there's a "better" way to approach this? Something which doesn't require checking that list.
A fun problem to ponder.

Answer 1

I think I have this solved thanks to a friend. I'll post here what I'm doing now in case someone ever is curious.
In short, starting from the center and working out, calculate the vertices of a hexagon, and subdivide each edge of the hexagon into i number of places, where i is the layer number.
I drew it in C# using SkiaSharp, but the code is nothing special to the language, there's no reason this couldn't be written in any language. Here's the significant bits:
const float seedAngle = (float)(Math.PI / 3.0);
static void SeedOfLifeDemo(int x, int y) {
//setting up Skia stuff, this will be different depending what language you're using.
var info = new SKImageInfo(x, y);
using var bitmap = FlatImage(info, SKColors.White);
SKCanvas canvas = new SKCanvas(bitmap);
float radius = Math.Min(x, y) / 15;
SKPoint center = new SKPoint(x / 2f, y / 2f);
SKPaint strokePaint = new SKPaint {
Color = SKColors.Black,
Style = SKPaintStyle.Stroke,
StrokeWidth = 1,
IsAntialias = true,
};
int layers = 4;
//Draw the very central circle. This is just a little easier than adding that edge case to SubdividedHexagonAboutPoint
canvas.DrawCircle(center, radius, strokePaint);
for (int i = 1; i <= layers; i++) {
foreach (SKPoint p in SubdividedHexagonAboutPoint(center, radius * i, i)) {
canvas.DrawCircle(p, radius, strokePaint);
}
}
SaveImage(bitmap, "SeedOfLifeFastDemo.Jpg");//More Skia specific stuff
}
//The magic!
static List<SKPoint> SubdividedHexagonAboutPoint(SKPoint centre, float radius, int subdivisions) {
List<SKPoint> points = new List<SKPoint>(6 * subdivisions);
SKPoint? prevPoint = null;
for (int i = 0; i < 7; i++) {//Step around the circle. The 7th step is to close the last edge
float x = (float)(Math.Sin(seedAngle * i) * radius + centre.X);
float y = (float)(Math.Cos(seedAngle * i) * radius + centre.Y);
SKPoint point = new SKPoint(x, y);
if (prevPoint != null) {
points.Add(point);//include the "primary" 6 points
if (subdivisions > 0) {
float xDist = (point.X - prevPoint.Value.X) / subdivisions;
float yDist = (point.Y - prevPoint.Value.Y) / subdivisions;
for (int sub = 1; sub < subdivisions; sub++) {
SKPoint subPoint = new SKPoint(point.X - xDist * sub, point.Y - yDist * sub);
points.Add(subPoint);//include the edge subdivisions
}
}
}
prevPoint = point;
}
return points;
}
This is quite an interesting exercise really, and another example of where recursion can really bite you when used badly.