How do I find the distance from any point on an arc to the chord that creates the arc.

Question

How do I find the distance from any point on an arc to a line that connects the arcs end points (chord)?  I have tried y = h - (4h/w²) x  x² .  Is this correct?  h = height of arc apex, w = width of arc end points and x = the line intersect point, and y = the height from the x point to the arc.

Answer 1

Gary,

 

Let's do the most general case where the cord is not parallel to either the x or y axes.  Then,

Let  U_x & U_y = the unit vectors along the x and y axes.

      (X_k,Y_k) = the beginning and end coordinates of the straight line chord of the arc (k = 1 & 2)

      (X_a, Y_a) = any point on the curving arc (not assumed to be part of a circle necessarily).

              s  =  the distance of (X_a,Y_a) from (X₁,Y₁)

              h  = the distance from (X_a,Y_a) to the chord line along the perpendicular to the chord.

              φ  = the angle at (X₁,Y₁) between the vector to the point on the arc and the vector of the chord line 

 

Now you have a right triangle formed with these relationships:

  

              s = √[(X_a - X₁)² + (Y_a - Y₁)²]  along its hypotenuse

              h = (s) Sinφ

              and Cosφ is determined using the dot product of two vectors:

 

             Vector A = (X_a-X₁)U_x + (Y_a-Y₁)U_y  is the vector from chord Pt#1 to the general point on the arc

             Vector C = (X₂-X₁)U_x + (Y₂-Y₁)U_y is the vector from chord Pt#1 to chord Pt#2

             A dot C = |A| |C| Cosφ

             {(X_a-X₁) (X₂-X₁)  + (Y_a-Y₁) (Y₂-Y₁)} = {√[(X_a-X₁)² + (Y_a-Y₁)²]}{√[(X₂-X₁)² + (Y₂-Y₁)²]}Cosφ,

 

All these values are known, so solve for the angle φ and find h = (s) Sinφ  since s is also known.