Formula for drawing arcs ending with straight lines, Y as a function of X, starting slope, ending slope, starting point and arc radius?

Question

I am looking for a mathematical formula that, in a graph depicting Y as a function of X, will have a certain slope in front of the specified starting point (X value or even better, X and Y coordinates), then after that it will draw an arc with the specified radius, which will end when it will reach the second given slope, and from the point further there will be another straight line of this second slope.

I know that since it is Y as a function of X, the tilt parameters must be greater than exactly -90 and less than exactly 90 degrees; I am not worried about any violations at (or beyond) these extremes.

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In fact, I would be even happier with a formula that takes the start and end points (2d coordinates), as well as the start and end slopes; and will have two arcs between them (with a straight line between them, when necessary), easily connecting two straight lines (obviously, X of the endpoint should be larger than X for the start point, I don’t care what happens when it’s not So). But I suggest that such a formula may be much harder to come up with than what I first asked.

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ps: along the "arc" - the middle segment of the circle; as if both axes of the graph have the same scale, the arcs will have the correct aspect ratio for a circle with the same radius.

Answer 1

Well, I see it like this:

• calculate P0

  as the intersection of the lines A + t*dA and B - t*dB

• calculate P1 (center of circle)

  this intersection of the translated lines A->P0 and B->P0 perpendicular to the radius r . There are two possibilities, so choose the right one (which leads to a smaller corner of the circular part).

  

• calculate P2,P3

  just the intersection of the lines A-P0 and B-P0 and the perpendicular line from P1 to it

• curve

   // some constants first da=P2-A; db=B-P3; a2=atan2(P2.x-P1.x,P2.y-P1.y); a3=atan2(P3.x-P1.x,P3.y-P1.y); if (a2>a3) a3-=M_PI*2.0; dang=a3-a2; // now (x,y)=curve(t) ... where t = <0,3> if (t<=1.0) { x=A.x+t*da.x; y=A.y+t*da.y; } else if (t<=2.0) { t=a2+((t-1.0)*dang); x=P1.x+r*cos(t); y=P1.y+r*sin(t); } else { t=t-2.0; x=P3.x+t*db.x; y=P3.y+t*db.y; }