How do I know if a point (x, y) is within the filled percentage of a pie chart?

Question

I arrived at this code showing whether a point (x, y) is inside or outside a circle on a Cartesian plane.

def dentro_fora(p, x = 0, y = 0):

    r = cx = cy = 50

    if (x - cx)**2 + (y - cy)**2 < r**2:
        return 'dentro'
    else:
        return 'fora'

But I needed to know if this point is in the filled part of the circle, for example, if I have 33% of the filled circle (assuming it’s a pie chart) how do I find out if the dot (12, 55) is within the filled area?

Whereas the centre of the circle is the point (50, 50) and the radius is 50.

Answer 1

You’re using a basic distance formula, which in practice simply tells you whether the point is in the circle or not.

As it is already enough to determine if it is in the graph, all that remains is to know the angle of the point in relation to the center. It is from this angle that will extract the percentage where the point is.

It has a mathematical function that gives you that ready, is the arc-tangent with 2 parameters:

 angulo = math.atan2( x, y )

whereas x, y are the coordinates of the point in relation to the center of the circle, in the case of the graph of your question, would be ( x - 50, y - 50 ).

Important to consider that the atan2 considers the rightmost point as the beginning, and normally circular charts use the origin at the top, so you can make a move with symmetry:

 angulo = math.atan2( -y, -x )

And with the center setting:

 angulo = math.atan2( -(y - 50), -(x - 50) )

(adjustment as needed, the secret is to reverse the coordinates and put the negative on either or both sides, depending on the source used)

To know in percentage, in relation to the pie/pizza chart, just divide the angle by 2 PI (which represents 360º in Radians), and multiply by 100:

porcentagem = angulo / ( math.pi * 2 ) * 100

(I have not simplified the formula for easy reading)

Of course, if you have several "slices" in the pizza, you will have to add up the percentage of each of them until you pass the "percentage" obtained in the formula above.

Example: if the point is at 62%, and Voce has pieces of 20%, 30% and 40%:

  Pedaço 1 - 20%  (20 >= 62, falso, vamos pro próximo )
  Pedaço 2 - 30%  (30 + 20 >= 62, falso, vamos pro próximo )
  Pedaço 3 - 40%  (40 + 30 + 20 >= 62, verdadeiro, achamos nosso pedaço )

It is probably the case to implement with a loop very simple.