Newsgroups: comp.os.msdos.djgpp
Subject: Re: ellipses at an angle
Message-ID: <32F60587.49C4@redestb.es>
From: Jorge Fernandez Suarez <cascoscuro AT redestb DOT es>
Date: Mon, 03 Feb 1997 16:34:31 +0100
Reply-To: cascoscuro AT redestb DOT es
References: <199702030225 DOT UAA22378 AT mail DOT texoma DOT net>
Organization: Universidad de Oviedo
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> > :Well, you could use the original definition of an ellipse (that is, from
> No!  This is not even the definition of an ellipse parallel to a coordinate axis!
> The general equation for an ellipse is:
> 
> Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, B^2 - 4*A*C < 0.
> 
> If B == 0, then this reduces to an ellipse parallel to a coordinate axis.

well, but you're not giving the formula to rotate one ellipse an
angle... this is the
conic general formula (i think)

The way I've do it it's like this:

Ec. of ellipse:   		x^2/a^2 +y^/b^2 = 1

Ec. of rotate a 2D point:  	x'= x*cos(angle)-y*sin(angle)
				y'= x*sin(angle)+y*cos(angle)

... if you plug this 2 formulas into the ellipse ecuation, then you must
iterate across the y-axis to
retrive the two x coordinate for every "y" and draw one horizontal line
between these.

Here's my routine... not optimiced at all, btw:

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void elipse(byte *offset,int cx,int cy,int eje_a,int eje_b,byte theta,byte color) {
  float a,b,c,f,k1,k2,k3; /* Cts for ecuation resolution */
  float aa,bb,cc,dd;      /* idem */
  float ent,rad;          /* real and radikal part */
  int y;                              /* solve ec. for every "y" ... */
  int xi,xf;              /* ... 2 solutions "xi,xf" */
  int yi,yf;              /* minimun and maximum of "y" in elipse */
  byte *ofs;              /* Offset of every Hline */

  /* Ec. rotated ellipse  -->  "A*x^2+ 2*B*x*y + C*y^2 + F= 0 */

  /* Rotation (theta) in radians... */
  aa = (float)coseno[theta]/128/eje_a; 
  bb = (float)  seno[theta]/128/eje_a;
  cc = (float)  seno[theta]/128/eje_b; 
  dd = (float)coseno[theta]/128/eje_b;

  /* rotated coefficients */
  a = aa*aa+cc*cc; b = cc*dd-aa*bb;
  c = bb*bb+dd*dd; f = -1;

  /* solutions... x=k1úy ñ û((k1ý+k2)úyý+k3) */
  k1 = -b/a; k2 = -c/a; k3 = -f/a;

  /* min and max will occur when radikal=0 */
  yf = sqrt(-k3/(k1*k1+k2))-1; yi = -yf;

  /* Iteration since min "y" to max for hlines xi-xf */
  for(y=yi;y<=yf;y++) {
    ent = k1*y;                         /* real part */
    rad = sqrt( (k1*k1+k2)*y*y+k3 );    /* radikal part */

    xf = ( ent+ rad );                  /* Sol. 1 of ec. */
    xi = ( ent- rad );                  /* Sol. 2 of ec. */

    ofs = offset+ANCHO*(cy+y)+cx;       /* Hline start */
    
    hline(ofs, xi, xf, 0, color);
  }
}
                                                       a

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