Fractals/Computer graphic techniques/2D/algorithms

List:

• line
  • bresenhams-drawing-algorithms
• edge detection
• Filling_contour
• 2D Plane transformations

How to tell whether a point is to the right or left side of a line ?

/* 
  How to tell whether a point is to the right or left side of a line ?

 http://stackoverflow.com/questions/1560492/how-to-tell-whether-a-point-is-to-the-right-or-left-side-of-a-line

  a, b = points
  line = ab
 pont to check = z

  position = sign((Bx - Ax) * (Y - Ay) - (By - Ay) * (X - Ax))
  It is 0 on the line, and +1 on one side, -1 on the other side.

*/

double CheckSide(double Zx, double Zy, double Ax, double Ay, double Bx, double By)
{
  return ((Bx - Ax) * (Zy - Ay) - (By - Ay) * (Zx - Ax));

}

Testing if point is inside triangle

• description
• javascript code

/* 
c console program 
gcc t.c -Wall
./a.out

*/

# include <stdio.h>

// 3 points define triangle 
double Zax = -0.250000000000000;
double Zay = 0.433012701892219;
// left when y
double Zlx = -0.112538773749444;  
double Zly = 0.436719687479814 ;

double Zrx = -0.335875821657728;
double Zry = 0.316782798339332;

// points to test 
// = inside triangle 
double Zx = -0.209881783739630;
double Zy =   +0.4; 

// outside triangle 
double Zxo = -0.193503885412548  ;
double Zyo = 0.521747636163664;

double Zxo2 = -0.338750000000000;
double Zyo2 = +0.440690927838329; 





// ============ http://stackoverflow.com/questions/2049582/how-to-determine-a-point-in-a-2d-triangle
// In general, the simplest (and quite optimal) algorithm is checking on which side of the half-plane created by the edges the point is.
double sign (double  x1, double y1,  double x2, double y2, double x3, double y3)
{
    return (x1 - x3) * (y2 - y3) - (x2 - x3) * (y1 - y3);
}

int  PointInTriangle (double x, double y, double x1, double y1, double x2, double y2, double x3, double y3)
{
    double  b1, b2, b3;

    b1 = sign(x, y, x1, y1, x2, y2) < 0.0;
    b2 = sign(x, y, x2, y2, x3, y3) < 0.0;
    b3 = sign(x, y, x3, y3, x1, y1) < 0.0;

    return ((b1 == b2) && (b2 == b3));
}

int Describe_Position(double Zx, double Zy){
if (PointInTriangle( Zx, Zy, Zax, Zay, Zlx, Zly, Zrx, Zry))
  printf(" Z is inside \n");
  else printf(" Z is outside \n");

return 0;
}



// ======================================



int main(void){

Describe_Position(Zx, Zy);
Describe_Position(Zxo, Zyo);
Describe_Position(Zxo2, Zyo2);

return 0;
}

Orientation and area of the triangle

Orientation and area of the triangle : how to do it ?

// gcc t.c -Wall
// ./a.out
# include <stdio.h>



// http://ncalculators.com/geometry/triangle-area-by-3-points.htm
double GiveTriangleArea(double xa, double ya, double xb, double yb, double xc, double yc)
{
return ((xb*ya-xa*yb)+(xc*yb-xb*yc)+(xa*yc-xc*ya))/2.0;
}


/*

wiki Curve_orientation
[http://mathoverflow.net/questions/44096/detecting-whether-directed-cycle-is-clockwise-or-counterclockwise]


The orientation of a triangle (clockwise/counterclockwise) is the sign of the determinant



matrix = { {1 , x1, y1}, {1 ,x2, y2} , {1,  x3, y3}}
 


where 
(x_1,y_1), (x_2,y_2), (x_3,y_3)$ 
are the Cartesian coordinates of the three vertices of the triangle.

:<math>\mathbf{O} = \begin{bmatrix}

1 & x_{A} & y_{A} \\
1 & x_{B} & y_{B} \\
1 & x_{C} & y_{C}\end{bmatrix}.</math>

A formula for its determinant may be obtained, e.g., using the method of [[cofactor expansion]]:
:<math>\begin{align}
\det(O) &= 1\begin{vmatrix}x_{B}&y_{B}\\x_{C}&y_{C}\end{vmatrix}
-x_{A}\begin{vmatrix}1&y_{B}\\1&y_{C}\end{vmatrix}
+y_{A}\begin{vmatrix}1&x_{B}\\1&x_{C}\end{vmatrix} \\
&= x_{B}y_{C}-y_{B}x_{C}-x_{A}y_{C}+x_{A}y_{B}+y_{A}x_{C}-y_{A}x_{B} \\
&= (x_{B}y_{C}+x_{A}y_{B}+y_{A}x_{C})-(y_{A}x_{B}+y_{B}x_{C}+x_{A}y_{C}).
\end{align}
</math>

If the determinant is negative, then the polygon is oriented clockwise.  If the determinant is positive, the polygon is oriented counterclockwise.  The determinant  is non-zero if points A, B, and C are non-[[collinear]].  In the above example, with points ordered A, B, C, etc., the determinant is negative, and therefore the polygon is clockwise.

*/

double IsTriangleCounterclockwise(double xa, double ya, double xb, double yb, double xc, double yc)
{return  ((xb*yc + xa*yb +ya*xc) - (ya*xb +yb*xc + xa*yc)); }



int DescribeTriangle(double xa, double ya, double xb, double yb, double xc, double yc)
{
 double t = IsTriangleCounterclockwise( xa,  ya, xb,  yb,  xc,  yc);
 double a = GiveTriangleArea( xa,  ya, xb,  yb,  xc,  yc);
 if (t>0)  printf("this triangle is oriented counterclockwise,     determinent = %f ; area = %f\n", t,a);
 if (t<0)  printf("this triangle is oriented clockwise,            determinent = %f; area = %f\n", t,a);
 if (t==0) printf("this triangle is degenerate: colinear or identical points, determinent = %f; area = %f\n", t,a);

 return 0;
}





int main()
{
 // clockwise oriented triangles 
 DescribeTriangle(-94,   0,  92,  68, 400, 180); // https://www-sop.inria.fr/prisme/fiches/Arithmetique/index.html.en
 DescribeTriangle(4.0, 1.0, 0.0, 9.0, 8.0, 3.0); // clockwise orientation https://people.sc.fsu.edu/~jburkardt/datasets/triangles/tex5.txt
 
 //  counterclockwise oriented triangles
 DescribeTriangle(-50.00, 0.00, 50.00,  0.00, 0.00,  0.02); // a "cap" triangle. This example has an area of 1.
 DescribeTriangle(0.0,  0.0, 3.0,  0.0, 0.0,  4.0); // a right triangle. This example has an area of (?? 3 ??)  =  6
 DescribeTriangle(4.0, 1.0, 8.0, 3.0, 0.0, 9.0);  //      https://people.sc.fsu.edu/~jburkardt/datasets/triangles/tex1.txt
 DescribeTriangle(-0.5, 0.0,  0.5,  0.0, 0.0,  0.866025403784439); // an equilateral triangle. This triangle has an area of sqrt(3)/4.

 // degenerate triangles 
 DescribeTriangle(1.0, 0.0, 2.0, 2.0, 3.0, 4.0); // This triangle is degenerate: 3 colinear points. https://people.sc.fsu.edu/~jburkardt/datasets/triangles/tex6.txt
 DescribeTriangle(4.0, 1.0, 0.0, 9.0, 4.0, 1.0); //2 identical points 
 DescribeTriangle(2.0, 3.0, 2.0, 3.0, 2.0, 3.0); // 3 identical points

 return 0; 
}

Testing if point is inside polygon

• stackoverflow