using GeometRi;

using System;

​

​

public class Program

{

    public static void Main()

    {

​

        Console.WriteLine("Ptolemy's construction of a pentagon");

        Console.WriteLine("Draw a regular pentagon inside the circle");

        Console.WriteLine();

​

        // We will follow the notation from 

        // https://www.cut-the-knot.org/pythagoras/pentagon.shtml

        // Read this article for detailed description of the algorithm

​

        // Create circle with center in point O and choose arbitrary point C on circle

        // We will start our pentagon from this point

        Point3d O = new Point3d(0, 0, 0);

        Circle3d circle = new Circle3d(O, 1, new Vector3d(0, 0, 1));

        Point3d C = circle.ParametricForm(0);

​

        // Draw a diameter through point C (point D is the opposite point of the diameter)

        Line3d l = new Line3d(C, O);

        Segment3d CD = (Segment3d)l.IntersectionWith(circle);

        Point3d D = CD.P1;

        // Need to check if we get the correct point

        if (D == C)

        {

            D = CD.P2;

        }

​

        // Draw a orthogonal diameter AB

​

        // First, draw two circles with centers in points C and D

        // The intersection of these circles gives a line, orthogonal to the segment CD

        Circle3d circle1 = new Circle3d(C, CD.Length, circle.Normal);

        Circle3d circle2 = new Circle3d(D, CD.Length, circle.Normal);

        Line3d line = ((Segment3d)circle1.IntersectionWith(circle2)).ToLine;

​

        // Now find the intersection of line with our circle

        // This gives points A and B

        Segment3d AB = (Segment3d)line.IntersectionWith(circle);

        Point3d A = AB.P1;

        Point3d B = AB.P2;

​

        // Find the center point of the segment OB (this will be point M)

        Point3d M = (O + B) / 2;

​

        // Draw a circle with center in point M and radius MC

        double radius = M.DistanceTo(C);

        circle1 = new Circle3d(M, radius, circle.Normal);

​

        // Find the intersection with diameter AB (this will be point N)

        Segment3d ON = (Segment3d)circle1.IntersectionWith(AB);

        Point3d N = ON.P1;

        if (N == O)

        {

            N = ON.P2;

        }

​

        // CN is the length of the pentagon's side

        double side_len = C.DistanceTo(N);

​

        // The first point of the pentagon is point C

        Point3d P1 = C;

​

        // Draw a circle from point C with radius 'side_len' and this gives us two more points

        circle1 = new Circle3d(C, side_len, circle.Normal);

        Segment3d tmp_segment = (Segment3d)circle1.IntersectionWith(circle);

        Point3d P2 = tmp_segment.P1;

        Point3d P5 = tmp_segment.P2;

​

        // Repeat this two more times to get remaining points

        circle1 = new Circle3d(P2, side_len, circle.Normal);

        tmp_segment = (Segment3d)circle1.IntersectionWith(circle);

        Point3d P3 = tmp_segment.P1;

        if (P3 == P1)

        {

            P3 = tmp_segment.P2;

        }

​

​

        circle1 = new Circle3d(P3, side_len, circle.Normal);

        tmp_segment = (Segment3d)circle1.IntersectionWith(circle);

        Point3d P4 = tmp_segment.P1;

        if (P4 == P2)

        {

            P4 = tmp_segment.P2;

        }

​

        // Points P1, P2, P3, P4 and P5 define our pentagon

        // Let's check if interion angles are equal 108 degrees (with accuracy 1e-8)

        double angle1 = new Vector3d(P1, P2).AngleToDeg(new Vector3d(P1, P5));

        double angle2 = new Vector3d(P2, P1).AngleToDeg(new Vector3d(P2, P3));

        double angle3 = new Vector3d(P3, P2).AngleToDeg(new Vector3d(P3, P4));

        double angle4 = new Vector3d(P4, P3).AngleToDeg(new Vector3d(P4, P5));

        double angle5 = new Vector3d(P5, P4).AngleToDeg(new Vector3d(P5, P1));

​

        double tolerance = 1e-8;

        if ( Math.Abs(angle1 - 108.0) < tolerance &&

             Math.Abs(angle2 - 108.0) < tolerance &&

             Math.Abs(angle3 - 108.0) < tolerance &&

             Math.Abs(angle4 - 108.0) < tolerance &&

             Math.Abs(angle5 - 108.0) < tolerance )

        {

            Console.WriteLine("Your construction is correct!");

            Console.WriteLine(P1.ToString());

            Console.WriteLine(P2.ToString());

            Console.WriteLine(P3.ToString());

            Console.WriteLine(P4.ToString());

            Console.WriteLine(P5.ToString());

        }

​

​

​

        Console.ReadLine();

​

​

    }

}