Coordinates of Shoulder Point

shoulder.html

Exploration

Show that the coordinates of the shoulder point of a conic arc with control points , and and projective discriminant ρ are given by

where is the midpoint of the conic arc's chord and has coordinates

and .

Approach

Create the conic arc and construct the point at parameter t=1/2.

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 conic arc.

Clear[x0,y0,xA,yA,x1,y1,p];
ca1=ConicArc2D[{x0,y0},{xA,yA},{x1,y1},p];

Find the point at parameter t=1/2.

pt1=Point2D[ca1[1/2]] //Simplify

This is the same as the specified point, when simplified.

pt2=Point2D[
  (x0+x1)/2 + p (xA - (x0+x1)/2),
  (y0+y1)/2 + p (yA - (y0+y1)/2)] //Simplify

SameQ[pt1,pt2]