Martin McBride, 2020-08-28

Tags cardioid

Categories generativepy generative art

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In this article we will look at how to create an image like this using Python and generativepy:

This type of curve is often created using pins and string, but it can be done by drawing lines on a computer image.

We start out with a set of points distributed equally around the circumference of a circle:

In this case we have 10 points, but in the image at the start of this post there are 200 points.

We will assume that the points are on a circle of radius 1 unit, with its centre at the point (0, 0). The (x, y) coordinates of point `i`

are given by:

a = i*2*math.pi/N x = math.cos(a) y = math.sin(a)

`N`

is the total number of points (10 in this case). `a`

is the angle of the point around the circle, measured in radians. The points are numbered 0 to 9.

To create the image, we simply draw a line from each point `i`

to the point `2*i`

. So:

- Point 0 is connected to point 0 (the line has zero length so it isn't drawn)
- Point 1 is connected to point 2
- Point 2 is connected to point 4
- Point 3 is connected to point 6

These 3 lines are shown here:

We then continue, filling in the rest of the points. Note that when we calculate the second point, we use modulo N, so:

- Point 5 is connected to point 0 (10, modulo 10)
- Point 6 is connected to point 2 (12, modulo 10)
- Point 7 is connected to point 4 (14, modulo 10)
- and so on

Here is the complete drawing. You can see the basic shape but it doesn't look much like a cardioid curve yet:

The full shape will be apparent when we add a lot more lines.

To draw the cardioid image, we first need to calculate the points. We will use 400 points, equally spaced around a unit circle. The equation was given above, we just need to convert this to Python code.

We will store each point as an (x, y) tuple, and place all the tuples in an array `points`

. Here is the code:

N = 400 points = [(math.cos(i*2*math.pi/N), math.sin(i*2*math.pi/N)) for i in range(N)]

If you are not familiar with list comprehensions you can read about them here. Here is the equivalent code using a loop - it does the same thing, more verbosely, use whichever you prefer:

N = 200 points = [] for i in range(N): x = (math.cos(i*2*math.pi/N) y = math.sin(i*2*math.pi/N)) point = (x, y) points.append(point)

Next, we need to produce a set of lines joining point[i] to point[i*2] (modulo `N`

):

for i in range(N): j = (i*2) % N Line(ctx).of_start_end(points[i], points[j]).stroke(color, .005)

(This requires Python 3.5 or later, because it uses the extended unpacking syntax in PEP 448).

The code need to be placed in our `draw`

function. Here is the full code:

from generativepy.drawing import make_image, setup from generativepy.color import Color from generativepy.geometry import Line import math N = 400 def draw(ctx, width, height, frame_no, frame_count): setup(ctx, width, height, startx=-1.1, starty=-1.1, width=2.2, background=Color(0.25)) points = [(math.cos(i*2*math.pi/N), math.sin(i*2*math.pi/N)) for i in range(N)] color = Color(0.6, 0.6, 1, 0.25) for i in range(N): j = (i*2) % N Line(ctx).of_start_end(points[i], points[j]).stroke(color, .005) make_image("cardioid.png", draw, 800, 800)

There is one final touch you might have noticed. The stroke colour is set to `(0.6, 0.6, 1, 0.25)`

. This is a light blue, but the fourth values makes the colour transparent. It has 25% opacity (ie it is 75% transparent). This means that where the lines overlap around the border of the cardioid shape, the colour looks brighter.

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