Midpoint of an Arc

arcmidpt.html

Exploration

Graphics saved as "arc12.eps".

Show that the midpoint, P, of a bulge factor arc between points and whose bulge factor is B has coordinates

.

Approach

Construct the perpendicular bisector of the arc's chord. Offset the midpoint of the chord an appropriate direction and distance to find the arc's midpoint.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create the arc end points.

Clear[x0,y0,x1,y1];
P0=Point2D[{x0,y0}];
P1=Point2D[{x1,y1}];

Construct the midpoint of the arc's chord.

PM=Point2D[P0,P1];

Rotate about π/2 radians to find Q, which is on the vector from to P.

Q=Rotate2D[P0,Pi/2,Coordinates2D[PM]]

Offset in the direction of Q by distance h=B d/2, where d is the distance between and .

Clear[B,d];
P=Point2D[PM,Q,B*d/2] /. d->Sqrt[(x0-x1)^2+(y0-y1)^2] //Simplify

The coordinates of the point at the parameter t=1/2 produce the same result.

Arc2D[{x0,y0},{x1,y1},B][1/2] //FullSimplify

Discussion

Example:  Construct the midpoint of the bulge factor arc with end points (4,0) and (0,4) and bulge factor B=2.  First, define a function for computing the midpoint.

ArcMidPoint2D[P0:Point2D[{x0_,y0_}],
              P1:Point2D[{x1_,y1_}],
              B_?IsScalar2D]:=
   Point2D[((x0+x1)-B(y0-y1))/2,((y0+y1)+B(x0-x1))/2];

Construct the midpoint and plot the geometry.

P0=Point2D[p0={4,0}];
P1=Point2D[p1={0,4}];
P=ArcMidPoint2D[P0,P1,2];
Sketch2D[{P0,P1,P,Arc2D[p0,p1,2]}]

Graphics saved as "arcmid01.eps".