Parabolic Arch

pbarch.html

Exploration

Graphics saved as "parab09.eps".

Find the equation of the parabolic arch of base b and height h as shown in the figure. Assume that b and h are positive.

Approach

Create a parabola rotated -π/2 radians with variables (h,k) and f for the vertex point and focal length. Find the quadratic equation of the parabola. The three given points (0,0), (b/2,h) and (b,0) must satisfy the equation. Solve three equations in the three unknowns h, k and f.

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

Construct the parabola.

Clear[h,k,f];
par1=Parabola2D[{h,k},f,-Pi/2];

Create the equation of the parabola.

Clear[x,y];
eq1=Equation2D[Quadratic2D[par1],{x,y}]

The three points must satisfy the equation of the parabola.

Clear[B,H];
ans=Solve[Map[(eq1 /. #)&,
              {{x->0,y->0},{x->B/2,y->H},{x->B,y->0}}],
          {f,h,k}]

Here's the equation of the parabolic arch.

eq1 /. First[ans]

Discussion

This is an example of the arch with B=4 and H=3.

Sketch2D[{par1 /. First[ans] /. {B->4,H->3}},
   CurveLength2D->9]

Graphics saved as "pbarch01.eps".